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Four-dimensional variational assimilation in the unstable subspace
(4DVar-AUS) and the optimal subspace dimension: A key a priori information used in 4DVar is the knowledge of the system's
evolution equations. In this paper we propose a method for taking full
advantage of the knowledge of the system's dynamical instabilities in order to
improve the quality of the analysis. We present an algorithm, four-dimensional
variational assimilation in the unstable subspace (4DVar-AUS), that consists in
confining in this subspace the increment of the control variable. The existence
of an optimal subspace dimension for this confinement is hypothesized.
Theoretical arguments in favor of the present approach are supported by
numerical experiments in a simple perfect non-linear model scenario. It is
found that the RMS analysis error is a function of the dimension N of the
subspace where the analysis is confined and is minimum for N approximately
equal to the dimension of the unstable and neutral manifold. For all
assimilation windows, from 1 to 5 days, 4DVar-AUS performs better than standard
4DVar. In the presence of observational noise, the 4DVar solution, while being
closer to the observations, if farther away from the truth. The implementation
of 4DVar-AUS does not require the adjoint integration. | nlin_CD |
Chaotic dynamics of graphene and graphene nanoribbons: We study the chaotic dynamics of graphene structures, considering both a
periodic, defect free, graphene sheet and graphene nanoribbons (GNRs) of
various widths. By numerically calculating the maximum Lyapunov exponent, we
quantify the chaoticity for a spectrum of energies in both systems. We find
that for all cases, the chaotic strength increases with the energy density, and
that the onset of chaos in graphene is slow, becoming evident after more than
$10^4$ natural oscillations of the system. For the GNRs, we also investigate
the impact of the width and chirality (armchair or zigzag edges) on their
chaotic behavior. Our results suggest that due to the free edges the chaoticity
of GNRs is stronger than the periodic graphene sheet, and decreases by
increasing width, tending asymptotically to the bulk value. In addition, the
chaotic strength of armchair GNRs is higher than a zigzag ribbon of the same
width. Further, we show that the composition of ${}^{12}C$ and ${}^{13}C$
carbon isotopes in graphene has a minor impact on its chaotic strength. | nlin_CD |
Multi-stabilities and symmetry-broken one-colour and two-colour states
in closely coupled single-mode lasers: We theoretically investigate the dynamics of two mutually coupled identical
single-mode semi-conductor lasers. For small separation and large coupling
between the lasers, symmetry-broken one-colour states are shown to be stable.
In this case the light output of the lasers have significantly different
intensities while at the same time the lasers are locked to a single common
frequency. For intermediate coupling we observe stable symmetry-broken
two-colour states, where both lasers lase simultaneously at two optical
frequencies which are separated by up to 150~GHz. Using a five dimensional
model we identify the bifurcation structure which is responsible for the
appearance of symmetric and symmetry-broken one-colour and two-colour states.
Several of these states give rise to multi-stabilities and therefore allow for
the design of all-optical memory elements on the basis of two coupled
single-mode lasers. The switching performance of selected designs of optical
memory elements is studied numerically. | nlin_CD |
Scaling of global momentum transport in Taylor-Couette and pipe flow: We interpret measurements of the Reynolds number dependence of the torque in
Taylor-Couette flow by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] and of
the pressure drop in pipe flow by Smits and Zagarola, [Phys. Fluids 10, 1045
(1998)] within the scaling theory of Grossmann and Lohse [J. Fluid Mech. 407,
27 (2000)], developed in the context of thermal convection. The main idea is to
split the energy dissipation into contributions from a boundary layer and the
turbulent bulk. This ansatz can account for the observed scaling in both cases
if it is assumed that the internal wind velocity $U_w$ introduced through the
rotational or pressure forcing is related to the the external (imposed)
velocity U, by $U_w/U \propto Re^\xi$ with xi = -0.051 and xi = -0.041 for the
Taylor-Couette (U inner cylinder velocity) and pipe flow (U mean flow velocity)
case, respectively. In contrast to the Rayleigh-Benard case the scaling
exponents cannot (yet) be derived from the dynamical equations. | nlin_CD |
A new generator of chaotic bit sequences with mixed-mode inputs: This paper presents a new generator of chaotic bit sequences with mixed-mode
(continuous and discrete) inputs. The generator has an improved level of
chaotic properties in comparison with the existing single source (input)
digital chaotic bit generators. The 0-1 test is used to show the improved
chaotic behavior of our generator having a chaotic continuous input (Chua,
R\"{o}ssler or Lorenz system) intermingled with a discrete input (logistic,
Tinkerbell or Henon map) with various parameters. The obtained sequences of
chaotic bits show some features of random processes with increased entropy
levels, even in the cases of small numbers of bit representations. The
properties of the new generator and its binary sequences compare well with
those obtained from a truly random binary reference quantum generator, as
evidenced by the results of the $ent$ tests. | nlin_CD |
Efficiency of Monte Carlo Sampling in Chaotic Systems: In this paper we investigate how the complexity of chaotic phase spaces
affect the efficiency of importance sampling Monte Carlo simulations. We focus
on a flat-histogram simulation of the distribution of finite-time Lyapunov
exponent in a simple chaotic system and obtain analytically that the
computational effort of the simulation: (i) scales polynomially with the
finite-time, a tremendous improvement over the exponential scaling obtained in
usual uniform sampling simulations; and (ii) the polynomial scaling is
sub-optimal, a phenomenon known as critical slowing down. We show that critical
slowing down appears because of the limited possibilities to issue a local
proposal on the Monte Carlo procedure in chaotic systems. These results remain
valid in other methods and show how generic properties of chaotic systems limit
the efficiency of Monte Carlo simulations. | nlin_CD |
Comparing the efficiency of numerical techniques for the integration of
variational equations: We present a comparison of different numerical techniques for the integration
of variational equations. The methods presented can be applied to any
autonomous Hamiltonian system whose kinetic energy is quadratic in the
generalized momenta, and whose potential is a function of the generalized
positions. We apply the various techniques to the well-known H\'enon-Heiles
system, and use the Smaller Alignment Index (SALI) method of chaos detection to
evaluate the percentage of its chaotic orbits. The accuracy and the speed of
the integration schemes in evaluating this percentage are used to investigate
the numerical efficiency of the various techniques. | nlin_CD |
An Optimal Control Formulation for Inviscid Incompressible Ideal Fluid
Flow: In this paper we consider the Hamiltonian formulation of the equations of
incompressible ideal fluid flow from the point of view of optimal control
theory. The equations are compared to the finite symmetric rigid body equations
analyzed earlier by the authors. We discuss various aspects of the Hamiltonian
structure of the Euler equations and show in particular that the optimal
control approach leads to a standard formulation of the Euler equations -- the
so-called impulse equations in their Lagrangian form. We discuss various other
aspects of the Euler equations from a pedagogical point of view. We show that
the Hamiltonian in the maximum principle is given by the pairing of the
Eulerian impulse density with the velocity. We provide a comparative discussion
of the flow equations in their Eulerian and Lagrangian form and describe how
these forms occur naturally in the context of optimal control. We demonstrate
that the extremal equations corresponding to the optimal control problem for
the flow have a natural canonical symplectic structure. | nlin_CD |
Disentangling regular and chaotic motion in the standard map using
complex network analysis of recurrences in phase space: Recurrence in the phase space of complex systems is a well-studied
phenomenon, which has provided deep insights into the nonlinear dynamics of
such systems. For dissipative systems, characteristics based on recurrence
plots have recently attracted much interest for discriminating qualitatively
different types of dynamics in terms of measures of complexity, dynamical
invariants, or even structural characteristics of the underlying attractor's
geometry in phase space. Here, we demonstrate that the latter approach also
provides a corresponding distinction between different co-existing dynamical
regimes of the standard map, a paradigmatic example of a low-dimensional
conservative system. Specifically, we show that the recently developed approach
of recurrence network analysis provides potentially useful geometric
characteristics distinguishing between regular and chaotic orbits. We find that
chaotic orbits in an intermittent laminar phase (commonly referred to as sticky
orbits) have a distinct geometric structure possibly differing in a subtle way
from those of regular orbits, which is highlighted by different recurrence
network properties obtained from relatively short time series. Thus, this
approach can help discriminating regular orbits from laminar phases of chaotic
ones, which presents a persistent challenge to many existing chaos detection
techniques. | nlin_CD |
The structure and evolution of confined tori near a Hamiltonian Hopf
Bifurcation: We study the orbital behavior at the neighborhood of complex unstable
periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a
transition of a family of periodic orbits from stability to complex instability
(also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable
periodic orbits move out of the unit circle. Then the periodic orbits become
complex unstable. In this paper we first integrate initial conditions close to
the ones of a complex unstable periodic orbit, which is close to the transition
point. Then, we plot the consequents of the corresponding orbit in a 4D surface
of section. To visualize this surface of section we use the method of color and
rotation [Patsis and Zachilas 1994]. We find that the consequents are contained
in 2D "confined tori". Then, we investigate the structure of the phase space in
the neighborhood of complex unstable periodic orbits, which are further away
from the transition point. In these cases we observe clouds of points in the 4D
surfaces of section. The transition between the two types of orbital behavior
is abrupt. | nlin_CD |
Statistics of the Island-Around-Island Hierarchy in Hamiltonian Phase
Space: The phase space of a typical Hamiltonian system contains both chaotic and
regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon
is the algebraic decay of correlations and recurrence time distributions. For
area-preserving maps, this has been attributed to the stickiness of boundary
circles, which separate chaotic and regular components. Though such dynamics
has been extensively studied, a full understanding depends on many fine details
that typically are beyond experimental and numerical resolution. This calls for
a statistical approach, the subject of the present work. We calculate the
statistics of the boundary circle winding numbers, contrasting the distribution
of the elements of their continued fractions to that for uniformly selected
irrationals. Since phase space transport is of great interest for dynamics, we
compute the distributions of fluxes through island chains. Analytical fits show
that the "level" and "class" distributions are distinct, and evidence for their
universality is given. | nlin_CD |
Rotational random walk of the harmonic three body system: When Robert Brown first observed colloidal pollen grains in water he
inaccurately concluded that their motion arose "neither from currents in the
fluid, nor from its gradual evaporation, but belonged to the particle itself".
In this work we study the dynamics of a classical molecule consisting of three
masses and three harmonic springs in free space that does display a rotational
random walk "belonging to the particle itself". The geometric nonlinearities
arising from the non-zero rest lengths of the springs connecting the masses
break the integrability of the harmonic system and lead to chaotic dynamics in
many regimes of phase space. The non-trivial connection of the system's shape
space allows it, much like falling cats, to rotate with zero angular momentum
and manifest its chaotic dynamics as an orientational random walk. In the
transition to chaos the system displays random orientation reversals and
provides a simple realization of L\'{e}vy walks. | nlin_CD |
Chaotic motion of three-body problem : an origin of macroscopic
randomness of the universe: The famous three-body problem is investigated by means of a numerical
approach with negligible numerical noises in a long enough time interval,
namely the Clean Numerical Simulation (CNS). From physical viewpoints, position
of any bodies contains inherent micro-level uncertainty. The evaluations of
such kind of inherent micro-level uncertainty are accurately simulated by means
of the CNS. Our reliable, very accurate CNS results indicate that the inherent
micro-level uncertainty of position of a star/planet might transfer into
macroscopic randomness. Thus, the inherent micro-level uncertainty of a body
might be an origin of macroscopic randomness of the universe. In addition, from
physical viewpoints, orbits of some three-body systems at large time are
inherently random, and thus it has no physical meanings to talk about the
accurate long-term prediction of the chaotic orbits. Note that such kind of
uncertainty and randomness has nothing to do with the ability of human being.
All of these might enrich our knowledge and deepen our understandings about not
only the three-body problem but also chaos. | nlin_CD |
The Scalings of Scalar Structure Functions in a Velocity Field with
Coherent Vortical Structures: In planar turbulence modelled as an isotropic and homogeneous collection of
2-D non-interacting compact vortices, the structure functions S_p(r) of a
statistically stationary passive scalar field have the following scaling
behaviour in the limit where the P\'eclet number Pe -> \infty S_p(r) ~
constant+\ln({\frac{r}{LPe^{-1/3}}}) for LPe^{-1/3} << L, S_p(r) ~
({\frac{r}{LPe^{-1/3}}})^{6(1-D)} for LPe^{-1/2} << LPe^{-1/3}, where L is a
large scale and D is the fractal co-dimension of the spiral scalar structures
generated by the vortices (1/2 <= D < 2/3). Note that LPe^{-1/2} is the scalar
Taylor microscale which stems naturally from our analytical treatment of the
advection-diffusion equation. The essential ingredients of our theory are the
locality of inter-scale transfer and Lundgren's time average assumption. A
phenomenological theory explicitly based only on these two ingredients
reproduces our results and a generalisation of this phenomenology to spatially
smooth chaotic flows yields (k\ln k)^{-1} generalised power spectra for the
advected scalar fields. | nlin_CD |
Chaotic behavior of three interacting vortices in a confined
Bose-Einstein condensate: Motivated by recent experimental works, we investigate a system of vortex
dynamics in an atomic Bose-Einstein condensate (BEC), consisting of three
vortices, two of which have the same charge. These vortices are modeled as a
system of point particles which possesses a Hamiltonian structure. This tripole
system constitutes a prototypical model of vortices in BECs exhibiting chaos.
By using the angular momentum integral of motion we reduce the study of the
system to the investigation of a two degree of freedom Hamiltonian model and
acquire quantitative results about its chaotic behavior. Our investigation tool
is the construction of scan maps by using the Smaller ALignment Index (SALI) as
a chaos indicator. Applying this approach to a large number of initial
conditions we manage to accurately and efficiently measure the extent of chaos
in the model and its dependence on physically important parameters like the
energy and the angular momentum of the system. | nlin_CD |
Constraints on the spectral distribution of energy and enstrophy
dissipation in forced two-dimensional turbulence: We study two-dimensional turbulence in a doubly periodic domain driven by a
monoscale-like forcing and damped by various dissipation mechanisms of the form
$\nu_{\mu}(-\Delta)^{\mu}$. By ``monoscale-like'' we mean that the forcing is
applied over a finite range of wavenumbers $k_\min \leq k \leq k_\max$, and
that the ratio of enstrophy injection $\eta \geq 0$ to energy injection
$\epsilon \geq 0$ is bounded by $k_\min^2 \epsilon \leq \eta \leq k_\max^2
\epsilon$. It is shown that for $\mu\geq 0$ the asymptotic behaviour satisfies
(eqnarray) \norm u_1^2&\leq&k_\max^2\norm u^2,(eqnarray) where $\norm u^2$ and
$\norm u_1^2$ are the energy and enstrophy, respectively. It is also shown that
for Navier-Stokes turbulence ($\mu = 1$), the time-mean enstrophy dissipation
rate is bounded from above by $2\nu_1 k_\max^2$. These results place strong
constraints on the spectral distribution of energy and enstrophy and of their
dissipation, and thereby on the existence of energy and enstrophy cascades, in
such systems. In particular, the classical dual cascade picture is shown to be
invalid for forced two-dimensional Navier--Stokes turbulence ($\mu=1$) when it
is forced in this manner. Inclusion of Ekman drag ($\mu=0$) along with
molecular viscosity permits a dual cascade, but is incompatible with the
log-modified -3 power law for the energy spectrum in the enstrophy-cascading
inertial range. In order to achieve the latter, it is necessary to invoke an
inverse viscosity ($\mu<0$). | nlin_CD |
The perturbed restricted three-body problem with angular velocity:
Analysis of basins of convergence linked to the libration points: The analysis of the affect of angular velocity on the geometry of the basins
of convergence (BoC) linked to the equilibrium points in the restricted
three-body problem is illustrated when the primaries are source of radiation.
The bivariate scheme of the Newton-Raphson (N-R) iterative method has been used
to discuss the topology of the basins of convergence. The parametric evolution
of the fractality of the convergence plane is also presented where the degree
of fractality is illustrated by evaluating the basin entropy of the convergence
plane. | nlin_CD |
RS Flip-Flop Circuit Dynamics Revisited: Logical RS flip-flop circuits are investigated once again in the context of
discrete planar dynamical systems, but this time starting with simple bilinear
(minimal) component models based on fundamental principles. The dynamics of the
minimal model is described in detail, and shown to exhibit some of the expected
properties, but not the chaotic regimes typically found in simulations of
physical realizations of chaotic RS flip-flop circuits. Any physical
realization of a chaotic logical circuit must necessarily involve small
perturbations - usually with quite large or even nonexisting derivatives - and
possibly some symmetry-breaking. Therefore, perturbed forms of the minimal
model are also analyzed in considerable detail. It is proved that perturbed
minimal models can exhibit chaotic regimes, sometimes associated with chaotic
strange attractors, as well as some of the bifurcation features present in
several more elaborate and less fundamentally grounded dynamical models that
have been investigated in the recent literature. Validation of the approach
developed is provided by some comparisons with (mainly simulated) dynamical
results obtained from more traditional investigations. | nlin_CD |
Zero delay synchronization of chaos in coupled map lattices: We show that two coupled map lattices that are mutually coupled to one
another with a delay can display zero delay synchronization if they are driven
by a third coupled map lattice. We analytically estimate the parametric regimes
that lead to synchronization and show that the presence of mutual delays
enhances synchronization to some extent. The zero delay or isochronal
synchronization is reasonably robust against mismatches in the internal
parameters of the coupled map lattices and we analytically estimate the
synchronization error bounds. | nlin_CD |
Semiclassical calculation of time delay statistics in chaotic quantum
scattering: We present a semiclassical calculation, based on classical action
correlations implemented by means of a matrix integral, of all moments of the
Wigner--Smith time delay matrix, $Q$, in the context of quantum scattering
through systems with chaotic dynamics. Our results are valid for broken time
reversal symmetry and depend only on the classical dwell time and the number of
open channels, $M$, which is arbitrary. Agreement with corresponding random
matrix theory reduces to an identity involving some combinatorial concepts,
which can be proved in special cases. | nlin_CD |
Variations on the Fermi-Pasta-Ulam chain, a survey: We will present a survey of low energy periodic Fermi-Pasta-Ulam chains with
leading idea the "breaking of symmetry". The classical periodic FPU-chain
(equal masses for all particles) was analysed by Rink in 2001 with main
conclusions that the normal form of the beta-chain is always integrable and
that in many cases this also holds for the alfa-chain. The FPU-chain with
alternating masses already shows a certain breaking of symmetry. Three exact
families of periodic solutions can be identified and a few exact invariant
manifolds which are related to the results of Chechin et al.~(1998-2005) on
bushes of periodic solutions. An alternating chain of 2n particles is present
as submanifold in chains with k 2n particles, k=2, 3, ... . Interaction between
the optical and acoustical group in the case of large mass m is demonstrated.
The part played by resonance suggests the role of the mass ratios. The
1:1:1:...:1 resonance does not arise for any number of particles and mass
ratios. An interesting case is the 1:2:3 resonance that produces after a
Hamilton-Hopf bifurcation and breaking symmetry chaotic behaviour in the sense
of Shilnikov-Devaney. Another interesting case is the 1:2:4 resonance. As
expected the analysis of various cases has a significant impact on recurrence
phenomena; this will be illustrated by numerical results. | nlin_CD |
Predictability of a system with transitional chaos: The paper is focused on the discussion of the phenomenon of transitional
chaos in dynamic autonomous and non-autonomous systems. This phenomenon
involves the disappearance of chaotic oscillations in specific time periods and
the system becoming predictable. Variable dynamics of the system may be used to
control the process. | nlin_CD |
A Structure behind Primitive Chaos: Recently, a new concept, primitive chaos, was proposed, as a concept closely
related to the fundamental problems of physics itself such as determinism,
causality, free will, predictability, and irreversibility [J. Phys. Soc. Jpn.
{\bf 79}, 15002 (2010)]. This letter reveals a structure hidden behind the
primitive chaos; under some conditions, a new primitive chaos is constructed
from the original primitive chaos, this procedure can be repeated, and the
hierarchic structure of the primitive chaos is obtained. This implies such a
picture that new events and causality is constructed from the old ones, with
the aid of the concept of a coarse graining. As an application of this
structure, interesting facts are revealed for the essential condition of the
primitive chaos and for the chaotic behaviors. | nlin_CD |
A comment on the arguments about the reliability and convergence of
chaotic simulations: Yao and Hughes commented (Tellus-A, 60: 803 - 805, 2008) that "all chaotic
responses are simply numerical noise and have nothing to do with the solutions
of differential equations". However, using 1200 CPUs of the National
Supercomputer TH-A1 and a parallel integral algorithm of the so-called "Clean
Numerical Simulation" (CNS) based on the 3500th-order Taylor expansion and data
in 4180-digit multiple precision, one can gain reliable, convergent chaotic
solution of Lorenz equation in a rather long interval [0,10000]. This supports
Lorenz's optimistic viewpoint (Tellus-A, 60: 806 - 807, 2008): "numerical
approximations can converge to a chaotic true solution throughout any finite
range of time". | nlin_CD |
Nonlinear stiffness, Lyapunov exponents, and attractor dimension: I propose that stiffness may be defined and quantified for nonlinear systems
using Lyapunov exponents, and demonstrate the relationship that exists between
stiffness and the fractal dimension of a strange attractor: that stiff chaos is
thin chaos. | nlin_CD |
Sample and Hold Errors in the Implementation of Chaotic Maps: Though considerable effort has recently been devoted to hardware realization
of chaotic maps, the analysis generally neglects the influence of
implementation inaccuracies. Here we investigate the consequences of S/H errors
on Bernoulli shift, tent map and tailed tent map systems: an error model is
proposed and implementations are characterized under its assumptions. | nlin_CD |
Multiple Perron-Frobenius operators: A cycle expansion technique for discrete sums of several PF operators,
similar to the one used in standard classical dynamical zeta-function formalism
is constructed. It is shown that the corresponding expansion coefficients show
an interesting universal behavior, which illustrates the details of the
interference between the particlar mappings entering the sum. | nlin_CD |
Appearance of chaos and hyperchaos in evolving pendulum network: The study of deterministic chaos continues to be one of the important
problems in the field of nonlinear dynamics. Interest in the study of chaos
exists both in low-dimensional dynamical systems and in large ensembles of
coupled oscillators. In this paper, we study the emergence of spatio-temporal
chaos in chains of locally coupled identical pendulums with constant torque.
The study of the scenarios of the emergence (disappearance) and properties of
chaos is done as a result of changes in: (i) the individual properties of
elements due to the influence of dissipation in this problem, and (ii) the
properties of the entire ensemble under consideration, determined by the number
of interacting elements and the strength of the connection between them. It is
shown that an increase of dissipation in an ensemble with a fixed coupling
force and elements number can lead to the appearance of chaos as a result of a
cascade of period doubling bifurcations of periodic rotational motions or as a
result of invariant tori destruction bifurcation. Chaos and hyperchaos can
occur in an ensemble by adding or excluding one or more elements. Moreover,
chaos arises hard, since in this case the control parameter is discrete. The
influence of the coupling strength on the occurrence of chaos is specific. The
appearance of chaos occurs with small and intermediate coupling and is caused
by the overlap of the various out-of-phase rotational modes regions existence.
The boundaries of these areas are determined analytically and confirmed in a
numerical experiment. Chaotic regimes in the chain do not exist if the coupling
strength is strong enough. | nlin_CD |
Percolation model for nodal domains of chaotic wave functions: Nodal domains are regions where a function has definite sign. In recent paper
[nlin.CD/0109029] it is conjectured that the distribution of nodal domains for
quantum eigenfunctions of chaotic systems is universal. We propose a
percolation-like model for description of these nodal domains which permits to
calculate all interesting quantities analytically, agrees well with numerical
simulations, and due to the relation to percolation theory opens the way of
deeper understanding of the structure of chaotic wave functions. | nlin_CD |
A route to thermalization in the $α$-Fermi-Pasta-Ulam system: We study the original $\alpha$-Fermi-Pasta-Ulam (FPU) system with $N=16,32$
and $64$ masses connected by a nonlinear quadratic spring. Our approach is
based on resonant wave-wave interaction theory, i.e. we assume that, in the
weakly nonlinear regime (the one in which Fermi was originally interested), the
large time dynamics is ruled by exact resonances. After a detailed analysis of
the $\alpha$-FPU equation of motion, we find that the first non trivial
resonances correspond to six-wave interactions. Those are precisely the
interactions responsible for the thermalization of the energy in the spectrum.
We predict that for small amplitude random waves the time scale of such
interactions is extremely large and it is of the order of $1/\epsilon^8$, where
$\epsilon$ is the small parameter in the system. The wave-wave interaction
theory is not based on any threshold: equipartition is predicted for arbitrary
small nonlinearity. Our results are supported by extensive numerical
simulations. A key role in our finding is played by the {\it Umklapp} (flip
over) resonant interactions, typical of discrete systems. The thermodynamic
limit is also briefly discussed. | nlin_CD |
Heterogeneity and chaos in the Peyrard-Bishop-Dauxois DNA model: We discuss the effect of heterogeneity on the chaotic properties of the
Peyrard-Bishop-Dauxois nonlinear model of DNA. Results are presented for the
maximum Lyapunov exponent and the deviation vector distribution. Different
compositions of adenine-thymine (AT) and guanine-cytosine (GC) base pairs are
examined for various energies up to the melting point of the corresponding
sequence. We also consider the effect of the alternation index, which measures
the heterogeneity of the DNA chain through the number of alternations between
different types (AT or GC) of base pairs, on the chaotic behavior of the
system. Biological gene promoter sequences have been also investigated, showing
no distinct behavior of the maximum Lyapunov exponent. | nlin_CD |
Transition to anomalous dynamics in a simple random map: The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest
deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise
linear time-discrete map on the unit interval with a uniform slope larger than
one, hence expanding, with a positive Lyapunov exponent and a uniform invariant
density. If the slope is less than one the map becomes contracting, the
Lyapunov exponent is negative, and the density trivially collapses onto a fixed
point. Sampling from these two different types of maps at each time step by
randomly selecting the expanding one with probability $p$, and the contracting
one with probability $1-p$, gives a prototype of a random dynamical system.
Here we calculate the invariant density of this simple random map, as well as
its position autocorrelation function, analytically and numerically under
variation of $p$. We find that the map exhibits a non-trivial transition from
fully chaotic to completely regular dynamics by generating a long-time
anomalous dynamics at a critical sampling probability $p_c$, defined by a zero
Lyapunov exponent. This anomalous dynamics is characterised by an infinite
invariant density, weak ergodicity breaking and power law correlation decay. | nlin_CD |
Predicting the outcome of roulette: There have been several popular reports of various groups exploiting the
deterministic nature of the game of roulette for profit. Moreover, through its
history the inherent determinism in the game of roulette has attracted the
attention of many luminaries of chaos theory. In this paper we provide a short
review of that history and then set out to determine to what extent that
determinism can really be exploited for profit. To do this, we provide a very
simple model for the motion of a roulette wheel and ball and demonstrate that
knowledge of initial position, velocity and acceleration is sufficient to
predict the outcome with adequate certainty to achieve a positive expected
return. We describe two physically realisable systems to obtain this knowledge
both incognito and {\em in situ}. The first system relies only on a mechanical
count of rotation of the ball and the wheel to measure the relevant parameters.
By applying this techniques to a standard casino-grade European roulette wheel
we demonstrate an expected return of at least 18%, well above the -2.7%
expected of a random bet. With a more sophisticated, albeit more intrusive,
system (mounting a digital camera above the wheel) we demonstrate a range of
systematic and statistically significant biases which can be exploited to
provide an improved guess of the outcome. Finally, our analysis demonstrates
that even a very slight slant in the roulette table leads to a very pronounced
bias which could be further exploited to substantially enhance returns. | nlin_CD |
A local Echo State Property through the largest Lyapunov exponent: Echo State Networks are efficient time-series predictors, which highly depend
on the value of the spectral radius of the reservoir connectivity matrix. Based
on recent results on the mean field theory of driven random recurrent neural
networks, enabling the computation of the largest Lyapunov exponent of an ESN,
we develop a cheap algorithm to establish a local and operational version of
the Echo State Property. | nlin_CD |
Telegraph-type versus diffusion-type models of turbulent relative
dispersion: Properties of two equations describing the evolution of the probability
density function (PDF) of the relative dispersion in turbulent flow are
compared by investigating their solutions: the Richardson diffusion equation
with the drift term and the self-similar telegraph equation derived by
Ogasawara and Toh [J. Phys. Soc. Jpn. 75, 083401 (2006)]. The solution of the
self-similar telegraph equation vanishes at a finite point, which represents
persistent separation of a particle pair, while that of the Richardson equation
extends infinitely just after the initial time. Each equation has a similarity
solution, which is found to be an asymptotic solution of the initial value
problem. The time lag has a dominant effect on the relaxation process into the
similarity solution. The approaching time to the similarity solution can be
reduced by advancing the time of the similarity solution appropriately.
Batchelor scaling, a scaling law relevant to initial separation, is observed
only for the telegraph case. For both models, we estimate the Richardson
constant, based on their similarity solutions. | nlin_CD |
Dynamics of a Gear System with Faults in Meshing Stiffness: Gear box dynamics is characterised by a periodically changing stiffness. In
real gear systems, a backlash also exists that can lead to a loss in contact
between the teeth. Due to this loss of contact the gear has piecewise linear
stiffness characteristics, and the gears can vibrate regularly and chaotically.
In this paper we examine the effect of tooth shape imperfections and defects.
Using standard methods for nonlinear systems we examine the dynamics of gear
systems with various faults in meshing stiffness. | nlin_CD |
Parametric excitation and chaos through dust-charge fluctuation in a
dusty plasma: We consider a van der Pol-Mathieu (vdPM) equation with parametric forcing,
which arises in a simplified model of dusty plasma with dust-charge
fluctuation. We make a detailed numerical investigation and show that the
system can be driven to chaos either through a period doubling cascade or
though a subcritical pitchfork bifurcation over an wide range of parameter
space. We also discuss the frequency entrainment or frequency-locked phase of
the dust-charge fluctuation dynamics and show that the system exhibits 2:1
parametric resonance away from the chaotic regime. | nlin_CD |
Structure, size, and statistical properties of chaotic components in a
mixed-type Hamiltonian system: We perform a detailed study of the chaotic component in mixed-type
Hamiltonian systems on the example of a family of billiards [introduced by
Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)]. The phase space is divided
into a grid of cells and a chaotic orbit is iterated a large number of times.
The structure of the chaotic component is discerned from the cells visited by
the chaotic orbit. The fractal dimension of the border of the chaotic component
for various values of the billiard shape parameter is determined with the
box-counting method. The cell-filling dynamics is compared to a model of
uncorrelated motion, the so-called random model [Robnik et al. J. Phys. A:
Math. Gen. 30, L803 (1997)], and deviations attributed to sticky objects in the
phase space are found. The statistics of the number of orbit visits to the
cells is analyzed and found to be in agreement with the random model in the
long run. The stickiness of the various structures in the phase space is
quantified in terms of the cell recurrence times. The recurrence time
distributions in a few selected cells as well as the mean and standard
deviation of recurrence times for all cells are analyzed. The standard
deviation of cell recurrence time is found to be a good quantifier of
stickiness on a global scale. Three methods for determining the measure of the
chaotic component are compared and the measure is calculated for various values
of the billiard shape parameter. Lastly, the decay of correlations and the
diffusion of momenta is analyzed. | nlin_CD |
Stability of quantum motion and correlation decay: We derive a simple and general relation between the fidelity of quantum
motion, characterizing the stability of quantum dynamics with respect to
arbitrary static perturbation of the unitary evolution propagator, and the
integrated time auto-correlation function of the generator of perturbation.
Surprisingly, this relation predicts the slower decay of fidelity the faster
decay of correlations is. In particular, for non-ergodic and non-mixing
dynamics, where asymptotic decay of correlations is absent, a qualitatively
different and faster decay of fidelity is predicted on a time scale 1/delta as
opposed to mixing dynamics where the fidelity is found to decay exponentially
on a time-scale 1/delta^2, where delta is a strength of perturbation. A
detailed discussion of a semi-classical regime of small effective values of
Planck constant is given where classical correlation functions can be used to
predict quantum fidelity decay. Note that the correct and intuitively expected
classical stability behavior is recovered in the classical limit hbar->0, as
the two limits delta->0 and hbar->0 do not commute. In addition we also discuss
non-trivial dependence on the number of degrees of freedom. All the theoretical
results are clearly demonstrated numerically on a celebrated example of a
quantized kicked top. | nlin_CD |
Spatiotemporal phase synchronization in a large array of convective
oscillators: In a quasi-1D thermal convective system consisting of a large array of
nonlinearly coupled oscillators, clustering is the way to achieve a regime of
mostly antiphase synchronized oscillators. This regime is characterized by a
spatiotemporal doubling of traveling modes. As the dynamics is explored beyond
a spatiotemporal chaos regime with weak coupling, new interacting modes emerge
through a supercritical bifurcation. In this new regime, the system exhibits
coherent subsystems of antiphase synchronized oscillators, which are stationary
clusters following a spatiotemporal beating phenomena. This regime is the
result of a stronger coupling. We show from a phase mismatch model applied to
each oscillator, that these phase coherent domains undergo a global phase
instability meanwhile the interactions between oscillators become nonlocal. For
each value of the control parameter we find out the time-varying topology (link
matrix) from the contact interactions between oscillators. The new
characteristic spatiotemporal scales are extracted from the antiphase
correlations at the time intervals defined by the link matrix. The
interpretation of these experimental results contributes to widen the
understanding of other complex systems exhibiting similar phase chaotic
dynamics in 2D and 3D. | nlin_CD |
Dynamical Tunneling in Many-Dimensional Chaotic Systems: We investigate dynamical tunneling in many dimensional systems using a
quasi-periodically modulated kicked rotor, and find that the tunneling rate
from the torus to the chaotic region is drastically enhanced when the chaotic
states become delocalized as a result of the Anderson transition. This result
strongly suggests that amphibious states, which were discovered for a
one-dimensional kicked rotor with transporting islands [L. Hufnagel et al.,
Phys. Rev. Lett. 89, 154101 (2002)], quite commonly appear in many dimensional
systems. | nlin_CD |
Beyond Lyapunov: Ergodic parameters like the Lyapunov and the conditional exponents are global
functions of the invariant measure, but the invariant measure itself contains
more information. A more complete characterization of the dynamics by new
families of ergodic parameters is discussed, as well as their relation to the
dynamical R\'{e}nyi entropies and measures of self-organization. A
generalization of the Pesin formula is derived which holds under some weak
correlation conditions. | nlin_CD |
Numerical Investigation of Scaling Regimes in a Model of Anisotropically
Advected Vector Field: Influence of strong uniaxial small-scale anisotropy on the stability of
inertial-range scaling regimes in a model of a passive transverse vector field
advected by an incompressible turbulent flow is investigated by means of the
field theoretic renormalization group. Turbulent fluctuations of the velocity
field are taken in the form of a Gaussian statistics with zero mean and defined
noise with finite correlations in time. It is shown that stability of the
inertial-range scaling regimes in the three-dimensional case is not destroyed
by anisotropy but the corresponding stability of the two-dimensional system can
be corrupted by the presence of anisotropy. A borderline dimension $d_c$ below
which the stability of the scaling regime is not present is calculated as a
function of anisotropy parameters. | nlin_CD |
Pseudo-Random Bit Generation based on 2D chaotic maps of logistic type
and its Applications in Chaotic Cryptography: Pseudo-Random Bit Generation (PRBG) is required in many aspects of
cryptography as well as in other applications of modern security engineering.
In this work, PRBG based on 2D symmetrical chaotic mappings of logistic type is
considered. The sequences generated with a chaotic PRBG of this type, are
statistically tested and the computational effectiveness of the generators is
estimated. Considering this PRBG valid for cryptography, the size of the
available key space is also calculated. Different cryptographic applications
can be suitable to this PRBG, being a stream cipher probably the most immediate
of them. | nlin_CD |
Effect of asymmetry parameter on the dynamical states of nonlocally
coupled nonlinear oscillators: We show that coexisting domains of coherent and incoherent oscillations can
be induced in an ensemble of any identical nonlinear dynamical systems using
the nonlocal rotational matrix coupling with an asymmetry parameter. Further,
chimera is shown to emerge in a wide range of the asymmetry parameter in
contrast to near $\frac{\pi}{2}$ values of it employed in the earlier works. We
have also corroborated our results using the strength of incoherence in the
frequency domain ($S_{\omega}$) and in the amplitude domain ($S$) thereby
distinguishing the frequency and amplitude chimeras. The robust nature of the
asymmetry parameter in inducing chimeras in any generic dynamical system is
established using ensembles of identical R\"ossler oscillators, Lorenz systems,
and Hindmarsh-Rose (HR) neurons in their chaotic regimes. | nlin_CD |
Dynamical properties of the Molniya satellite constellation: long-term
evolution of the semi-major axis: We describe the phase space structures related to the semi-major axis of
Molniya-like satellites subject to tesseral and lunisolar resonances. In
particular, the questions answered in this contribution are: (i) we study the
indirect interplay of the critical inclination resonance on the
semi-geosynchronous resonance using a hierarchy of more realistic dynamical
systems, thus discussing the dynamics beyond the integrable approximation. By
introducing ad hoc tractable models averaged over fast angles, (ii) we
numerically demarcate the hyperbolic structures organising the long-term
dynamics via Fast Lyapunov Indicators cartography. Based on the publicly
available two-line elements space orbital data, (iii) we identify two
satellites, namely Molniya 1-69 and Molniya 1-87, displaying fingerprints
consistent with the dynamics associated to the hyperbolic set. Finally, (iv)
the computations of their associated dynamical maps highlight that the
spacecraft are trapped within the hyperbolic tangle. This research therefore
reports evidence of actual artificial satellites in the near-Earth environment
whose dynamics are ruled by manifolds and resonant mechanisms. The tools,
formalism and methodologies we present are exportable to other region of space
subject to similar commensurabilities as the geosynchronous region. | nlin_CD |
Universality in active chaos: Many examples of chemical and biological processes take place in large-scale
environmental flows. Such flows generate filamental patterns which are often
fractal due to the presence of chaos in the underlying advection dynamics. In
such processes, hydrodynamical stirring strongly couples into the reactivity of
the advected species and might thus make the traditional treatment of the
problem through partial differential equations difficult. Here we present a
simple approach for the activity in in-homogeneously stirred flows. We show
that the fractal patterns serving as skeletons and catalysts lead to a rate
equation with a universal form that is independent of the flow, of the particle
properties, and of the details of the active process. One aspect of the
universality of our appraoch is that it also applies to reactions among
particles of finite size (so-called inertial particles). | nlin_CD |
Itinerant chimeras in an adaptive network of pulse-coupled oscillators: In a network of pulse-coupled oscillators with adaptive coupling, we a
dynamical regime which we call an `itinerant chimera'. Similarly as in
classical chimera states, the network splits into two domains, the coherent and
the incoherent ones. The drastic difference is that the composition of the
domains is volatile, i.e. the oscillators demonstrate spontaneous switching
between the domains. This process can be seen as traveling of the oscillators
from one domain to another, or as traveling of the chimera core across network.
We explore the basic features of the itinerant chimeras, such as the mean and
the variance of the core size, and the oscillators lifetime within the core. We
also study the scaling behavior of the system and show that the observed regime
is not a finite-size effect but a key feature of the collective dynamics which
persists even in large networks. | nlin_CD |
Dispersion and collapse in stochastic velocity fields on a cylinder: The dynamics of fluid particles on cylindrical manifolds is investigated. The
velocity field is obtained by generalizing the isotropic Kraichnan ensemble,
and is therefore Gaussian and decorrelated in time. The degree of
compressibility is such that when the radius of the cylinder tends to infinity
the fluid particles separate in an explosive way. Nevertheless, when the radius
is finite the transition probability of the two-particle separation converges
to an invariant measure. This behavior is due to the large-scale
compressibility generated by the compactification of one dimension of the
space. | nlin_CD |
Nonlinear chaos in temperature time series: Part I: Case studies: In this work we present 3 case studies of local temperature time series
obtained from stations in Europe and Israel. The nonlinear nature of the series
is presented along with model based forecasting. Data is nonlinearly filtered
using high dimensional projection and analysis is performed on the filtered
data. A lorenz type model of 3 first order ODEs is then fitted. Forecasts are
shown for periods of 100 days ahead, outperforming any existing forecast method
known today. While other models fail at forecasting periods above 11 days, ours
shows remarkable stability 100 days ahead.
Thus finally a local dynamical system if found for local temperature
forecasting not requiring solution of Navier-Stokes equations. Thus saving
computational costs. | nlin_CD |
Quantum Properties of Double Kicked Systems with Classical Translational
Invariance in Momentum: Double kicked rotors (DKRs) appear to be the simplest nonintegrable
Hamiltonian systems featuring classical translational symmetry in phase space
(i.e., in angular momentum) for an \emph{infinite} set of values (the rational
ones) of a parameter $\eta$. The experimental realization of quantum DKRs by
atom-optics methods motivates the study of the double kicked particle (DKP).
The latter reduces, at any fixed value of the conserved quasimomentum
$\beta\hbar$, to a generalized DKR, the \textquotedblleft $\beta
$-DKR\textquotedblright . We determine general quantum properties of $\beta
$-DKRs and DKPs for arbitrary rational $\eta $. The quasienergy problem of
$\beta $-DKRs is shown to be equivalent to the energy eigenvalue problem of a
finite strip of coupled lattice chains. Exact connections are then obtained
between quasienergy spectra of $\beta $-DKRs for all $\beta $ in a generically
infinite set. The general conditions of quantum resonance for $\beta $-DKRs are
shown to be the simultaneous rationality of $\eta $, $\beta$, and a scaled
Planck constant $\hbar _{\mathrm{S}}$. For rational $\hbar _{\mathrm{S}}$ and
generic values of $\beta $, the quasienergy spectrum is found to have a
staggered-ladder structure. Other spectral structures, resembling Hofstadter
butterflies, are also found. Finally, we show the existence of particular DKP
wave-packets whose quantum dynamics is \emph{free}, i.e., the evolution
frequencies of expectation values in these wave-packets are independent of the
nonintegrability. All the results for rational $\hbar _{\mathrm{S}}$ exhibit
unique number-theoretical features involving $\eta $, $\hbar _{\mathrm{S}}$,
and $\beta $. | nlin_CD |
Chaotic advection of inertial particles in two dimensional flows: We study the dynamics of inertial particles in two dimensional incompressible
flows. The Maxey-Riley equation describing the motion of inertial particles is
used to construct a four dimensional dissipative bailout embedding map. This
map models the dynamics of the inertial particles while the base flow is
represented by a 2-d area preserving map. The dynamics of particles heavier
than the fluid, the aerosols, as well as that of bubbles, particles lighter
than the fluid, can be classified into 3 main dynamical regimes - periodic
orbits, chaotic structures and mixed regions. A phase diagram in the parameter
space is constructed with the Lyapunov characteristic exponents of the 4-d map
in which these dynamical regimes are distinctly identified. The embedding map
can target periodic orbits, as well as chaotic structures, in both the aerosol
and bubble regimes, at suitable values of the dissipation parameter. | nlin_CD |
Persistent topological features of dynamical systems: A general method for constructing simplicial complex from observed time
series of dynamical systems based on the delay coordinate reconstruction
procedure is presented. The obtained simplicial complex preserves all pertinent
topological features of the reconstructed phase space and it may be analyzes
from topological, combinatorial and algebraic aspects. In focus of this study
is the computation of homology of the invariant set of some well known
dynamical systems which display chaotic behavior. Persistent homology of
simplicial complex and its relationship with the embedding dimensions are
examined by studying the lifetime of topological features and topological
noise. The consistency of topological properties for different dynamic regimes
and embedding dimensions is examined. The obtained results shed new light on
the topological properties of the reconstructed phase space and open up new
possibilities for application of advanced topological methods. the method
presented here may be used as a generic method for constructing simplicial
complex from a scalar time series which has a number of advantages compared to
the mapping of the time series to a complex network. | nlin_CD |
Coupled networks and networks with bimodal frequency distributions are
equivalent: Populations of oscillators can display a variety of synchronization patterns
depending on the oscillators' intrinsic coupling and the coupling between them.
We consider two coupled, symmetric (sub)populations with unimodal frequency
distributions and show that the resulting synchronization patterns may resemble
those of a single population with bimodally distributed frequencies. Our proof
of the equivalence of their stability, dynamics, and bifurcations, is based on
an Ott-Antonsen ansatz. The generalization to networks consisting of multiple
(sub)populations vis-\`a-vis networks with multimodal frequency distributions,
however, appears impossible. | nlin_CD |
Quantification of depth of anesthesia by nonlinear time series analysis
of brain electrical activity: We investigate several quantifiers of the electroencephalogram (EEG) signal
with respect to their ability to indicate depth of anesthesia. For 17 patients
anesthetized with Sevoflurane, three established measures (two spectral and one
based on the bispectrum), as well as a phase space based nonlinear correlation
index were computed from consecutive EEG epochs. In absence of an independent
way to determine anesthesia depth, the standard was derived from measured blood
plasma concentrations of the anesthetic via a pharmacokinetic/pharmacodynamic
model for the estimated effective brain concentration of Sevoflurane. In most
patients, the highest correlation is observed for the nonlinear correlation
index D*. In contrast to spectral measures, D* is found to decrease
monotonically with increasing (estimated) depth of anesthesia, even when a
"burst-suppression" pattern occurs in the EEG. The findings show the potential
for applications of concepts derived from the theory of nonlinear dynamics,
even if little can be assumed about the process under investigation. | nlin_CD |
Thermodynamics of a time dependent and dissipative oval billiard: a heat
transfer and billiard approach: We study some statistical properties for the behavior of the average squared
velocity -- hence the temperature -- for an ensemble of classical particles
moving in a billiard whose boundary is time dependent. We assume the collisions
of the particles with the boundary of the billiard are inelastic leading the
average squared velocity to reach a steady state dynamics for large enough
time. The description of the stationary state is made by using two different
approaches: (i) heat transfer motivated by the Fourier law and, (ii) billiard
dynamics using either numerical simulations and theoretical description. | nlin_CD |
Microscopic theory of the Andreev gap: We present a microscopic theory of the Andreev gap, i.e. the phenomenon that
the density of states (DoS) of normal chaotic cavities attached to
superconductors displays a hard gap centered around the Fermi energy. Our
approach is based on a solution of the quantum Eilenberger equation in the
regime $t_D\ll t_E$, where $t_D$ and $t_E$ are the classical dwell time and
Ehrenfest-time, respectively. We show how quantum fluctuations eradicate the
DoS at low energies and compute the profile of the gap to leading order in the
parameter $t_D/t_E$ . | nlin_CD |
Thermalization in one- plus two-body ensembles for dense interacting
boson systems: Employing one plus two-body random matrix ensembles for bosons, temperature
and entropy are calculated, using different definitions, as a function of the
two-body interaction strength \lambda for a system with 10 bosons (m=10) in
five single particle levels (N=5). It is found that in a region \lambda \sim
\lambda_t, different definitions give essentially same values for temperature
and entropy, thus defining a thermalization region. Also, (m,N) dependence of
\lambda_t has been derived. It is seen that \lambda_t is much larger than the
\lambda values where level fluctuations change from Poisson to GOE and strength
functions change from Breit-Wigner to Gaussian. | nlin_CD |
Semiclassical spatial correlations in chaotic wave functions: We study the spatial autocorrelation of energy eigenfunctions $\psi_n({\bf
q})$ corresponding to classically chaotic systems in the semiclassical regime.
Our analysis is based on the Weyl-Wigner formalism for the spectral average
$C_{\epsilon}({\bf q^{+}},{\bf q^{-}},E)$ of $\psi_n({\bf q}^{+})\psi_n^*({\bf
q}^{-})$, defined as the average over eigenstates within an energy window
$\epsilon$ centered at $E$. In this framework $C_{\epsilon}$ is the Fourier
transform in momentum space of the spectral Wigner function $W({\bf
x},E;\epsilon)$. Our study reveals the chord structure that $C_{\epsilon}$
inherits from the spectral Wigner function showing the interplay between the
size of the spectral average window, and the spatial separation scale. We
discuss under which conditions is it possible to define a local system
independent regime for $C_{\epsilon}$. In doing so, we derive an expression
that bridges the existing formulae in the literature and find expressions for
$C_{\epsilon}({\bf q^{+}}, {\bf q^{-}},E)$ valid for any separation size $|{\bf
q^{+}}-{\bf q^{-}}|$. | nlin_CD |
Combinatorial problems in the semiclassical approach to quantum chaotic
transport: A semiclassical approach to the calculation of transport moments $M_m={\rm
Tr}[(t^\dag t)^m]$, where $t$ is the transmission matrix, was developed in [M.
Novaes, Europhys. Lett. 98, 20006 (2012)] for chaotic cavities with two leads
and broken time-reversal symmetry. The result is an expression for $M_m$ as a
perturbation series in 1/N, where N is the total number of open channels, which
is in agreement with random matrix theory predictions. The coefficients in this
series were related to two open combinatorial problems. Here we expand on this
work, including the solution to one of the combinatorial problems. As a
by-product, we also present a conjecture relating two kinds of factorizations
of permutations. | nlin_CD |
Fractional Equations of Kicked Systems and Discrete Maps: Starting from kicked equations of motion with derivatives of non-integer
orders, we obtain "fractional" discrete maps. These maps are generalizations of
well-known universal, standard, dissipative, kicked damped rotator maps. The
main property of the suggested fractional maps is a long-term memory. The
memory effects in the fractional discrete maps mean that their present state
evolution depends on all past states with special forms of weights. These forms
are represented by combinations of power-law functions. | nlin_CD |
A spatiotemporal-chaos-based cryptosystem taking advantages of both
synchronous and self-synchronizing schemes: Two-dimensional one-way coupled map lattices are used for cryptograph where
multiple space units produce chaotic outputs in parallel. One of the outputs
plays the role of driving for synchronization of the decryption system while
the others perform the function of information encoding. With this separation
of functions the receiver can establish a self-checking and self-correction
mechanism, and enjoys the advantages of both synchronous and self-synchronizing
schemes. A comparison between the present system with the system of Advanced
Encryption Standard, AES, is presented in the aspect of channel noise
influence. Numerical investigations show that our system is much stronger than
AES against channel noise perturbations, and thus can be better used for secure
communications with large noise channel exists in open channels, e.g.,
mobile-phone secure communications. | nlin_CD |
Controlling vortical motion of particles in two-dimensional driven
superlattices: We demonstrate the control of vortical motion of neutral classical particles
in driven superlattices. Our superlattice consists of a superposition of
individual lattices whose potential depths are modulated periodically in time
but with different phases. This driving scheme breaks the spatial reflection
symmetries and allows an ensemble of particles to rotate with an average
angular velocity. An analysis of the underlying dynamical attractors provides
an efficient method to control the angular velocities of the particles by
changing the driving amplitude. As a result, spatially periodic patterns of
particles showing different vortical motion can be created. Possible
experimental realizations include holographic optical lattice based setups for
colloids or cold atoms. | nlin_CD |
Predictability and suppression of extreme events in complex systems: In many complex systems, large events are believed to follow power-law,
scale-free probability distributions, so that the extreme, catastrophic events
are unpredictable. Here, we study coupled chaotic oscillators that display
extreme events. The mechanism responsible for the rare, largest events makes
them distinct and their distribution deviates from a power-law. Based on this
mechanism identification, we show that it is possible to forecast in real time
an impending extreme event. Once forecasted, we also show that extreme events
can be suppressed by applying tiny perturbations to the system. | nlin_CD |
Discussion on Nonlinear Dynamic Behavior of Suspension Based Bridge
Model: In this paper we explore the numerical study. of the Nonlinear Behavior of
Suspension Bridge Models. The study of suspension bridges is one of the classic
problems of mechanical vibrations, one of the most famous collapses of which
was that of the Tacoma Narrows Bridge. This paper covers an initial explanation
of vibrations in a suspension bridge. To do this, three different systems are
going to be simulated: The first being a system where only the vertical
vibrations of the bridge deck are taken into account, the second covering the
vibrations of the main cable and the roadbed, and lastly, a system that takes
both vertical and torsional vibrations into account. A time-frequency analysis
will also be done on all systems with temporal response, Fast Fourier Transform
(FFT) and Continuous Wavelet Transform (CWT), plus in a specific case the use
of Hilbert-Huang transform (HHT). Poincare maps and Lyapunov exponents are used
to characterize the dynamics of the system. In particular, in the vertical and
torsional system, an explanation of why the Tacoma Bridge oscillations have
undergone an abrupt change from vertical to torsional oscillations. Thus,
extremely rich dynamic behaviors are studied by numerical simulation in the
time and frequency domains. | nlin_CD |
Basin boundary, edge of chaos, and edge state in a two-dimensional model: In shear flows like pipe flow and plane Couette flow there is an extended
range of parameters where linearly stable laminar flow coexists with a
transient turbulent dynamics. When increasing the amplitude of a perturbation
on top of the laminar flow, one notes a a qualitative change in its lifetime,
from smoothly varying and short one on the laminar side to sensitively
dependent on initial conditions and long on the turbulent side. The point of
transition defines a point on the edge of chaos. Since it is defined via the
lifetimes, the edge of chaos can also be used in situations when the turbulence
is not persistent. It then generalises the concept of basin boundaries, which
separate two coexisting attractors, to cases where the dynamics on one side
shows transient chaos and almost all trajectories eventually end up on the
other side. In this paper we analyse a two-dimensional map which captures many
of the features identified in laboratory experiments and direct numerical
simulations of hydrodynamic flows. The analysis of the map shows that different
dynamical situations in the edge of chaos can be combined with different
dynamical situations in the turbulent region. Consequently, the model can be
used to develop and test further characterisations that are also applicable to
realistic flows. | nlin_CD |
A cellular automaton identification of the universality classes of
spatiotemporal intermittency: The phase diagram of the coupled sine circle map lattice shows
spatio-temporal intermittency of two distinct types: spatio-temporal
intermittency of the directed percolation (DP) class, and spatial intermittency
which does not belong to this class. These two types of behaviour are seen to
be special cases of the spreading and non-spreading regimes seen in the system.
In the spreading regime, each site can infect its neighbours permitting an
initial disturbance to spread, whereas in the non-spreading regime no infection
is possible. The two regimes are separated by a line which we call the
infection line. The coupled map lattice can be mapped on to an equivalent
cellular automaton which shows a transition from a probabilistic cellular
automaton (PCA) to a deterministic cellular automaton (DCA) at the infection
line. Thus the existence of the DP and non-DP universality classes in the same
system is signalled by the PCA to DCA transition. We also discuss the dynamic
origin of this transition. | nlin_CD |
Nontwist non-Hamiltonian systems: We show that the nontwist phenomena previously observed in Hamiltonian
systems exist also in time-reversible non-Hamiltonian systems. In particular,
we study the two standard collision/reconnection scenarios and we compute the
parameter space breakup diagram of the shearless torus. Besides the Hamiltonian
routes, the breakup may occur due to the onset of attractors. We study these
phenomena in coupled phase oscillators and in non-area-preserving maps. | nlin_CD |
Synchronization of extended chaotic systems with long-range
interactions: an analogy to Levy-flight spreading of epidemics: Spatially extended chaotic systems with power-law decaying interactions are
considered. Two coupled replicas of such systems synchronize to a common
spatio-temporal chaotic state above a certain coupling strength. The
synchronization transition is studied as a nonequilibrium phase transition and
its critical properties are analyzed at varying the interaction range. The
transition is found to be always continuous, while the critical indexes vary
with continuity with the power law exponent characterizing the interaction.
Strong numerical evidences indicate that the transition belongs to the {\it
anomalous directed percolation} family of universality classes found for
L{\'e}vy-flight spreading of epidemic processes. | nlin_CD |
Bushes of vibrational modes for Fermi-Pasta-Ulam chains: Some exact solutions and multi-mode invariant submanifolds were found for the
Fermi-Pasta-Ulam (FPU) beta-model by Poggi and Ruffo in Phys. D 103 (1997) 251.
In the present paper we demonstrate how results of such a type can be obtained
for an arbitrary N-particle chain with periodic boundary conditions with the
aid of our group-theoretical approach [Phys. D 117 (1998) 43] based on the
concept of bushes of normal modes for mechanical systems with discrete
symmetry. The integro-differential equation describing the FPU-alfa dynamics in
the modal space is derived. The loss of stability of the bushes of modes for
the FPU-alfa model, in particular, for the limiting case N >> 1 for the
dynamical regime with displacement pattern having period twice the lattice
spacing (Pi-mode) is studied. Our results for the FPU-alfa chain are compared
with those by Poggi and Ruffo for the FPU-beta chain. | nlin_CD |
Synchronization of fractional order chaotic systems: The chaotic dynamics of fractional order systems begin to attract much
attentions in recent years. In this brief report, we study the master-slave
synchronization of fractional order chaotic systems. It is shown that
fractional order chaotic systems can also be synchronized. | nlin_CD |
On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems: The Kolmogorov-Sinai (K-S) entropy is a central measure of complexity and
chaos. Its calculation for many-body systems is an interesting and important
challenge. In this paper, the evaluation is formulated by considering
$N$-dimensional symplectic maps and deriving a transfer matrix formalism for
the stability problem. This approach makes explicit a duality relation that is
exactly analogous to one found in a generalized Anderson tight-binding model,
and leads to a formally exact expression for the finite-time K-S entropy.
Within this formalism there is a hierarchy of approximations, the final one
being a diagonal approximation that only makes use of instantaneous Hessians of
the potential to find the K-S entropy. By way of a non-trivial illustration,
the K-S entropy of $N$ identically coupled kicked rotors (standard maps) is
investigated. The validity of the various approximations with kicking strength,
particle number, and time are elucidated. An analytic formula for the K-S
entropy within the diagonal approximation is derived and its range of validity
is also explored. | nlin_CD |
A new self-synchronizing stream cipher: A new self-synchronizing stream cipher (SSSC) is proposed based on one-way
and nearest neighbor coupled integer maps. Some ideas of spatiotemporal chaos
synchronization and chaotic cryptography are applied in this new SSSC system.
Several principles of constructing optimal SSSC are discussed, and the methods
realizing these principles are specified. This SSSC is compared with several
SSSC systems in security by applying chosen-ciphertext attacks. It is shown
that our new system can provide SSSC with high security and fairly fast
performance. | nlin_CD |
Current Statistics for Quantum Transport through Two-Dimensional Open
Chaotic Billiards: The probability current statistics of two-dimensional open chaotic ballistic
billiards is studied both analytically and numerically. Assuming that the real
and imaginary parts of the scattering wave function are both random Gaussian
fields, we find a universal distribution function for the probability current.
In by-passing we recover previous analytic forms for wave function statistics.
The expressions bridge the entire region from GOE to GUE type statistics. Our
analytic expressions are verified numerically by explicit quantum-mechanical
calculations of transport through a Bunimovich billiard. | nlin_CD |
Perturbative Dynamics of Stationary States in Nonlinear Parity-Time
Symmetric Coupler: We investigate the nonlinear parity-time (PT) symmetric coupler from a
dynamical perspective. As opposed to linear PT-coupler where the PT threshold
dictates the evolutionary characteristics of optical power in the two
waveguides, in a nonlinear coupler, the PT threshold governs the existence of
stationary points. We have found that the stability of the ground state
undergoes a phase transition when the gain/loss coefficient is increased from
zero to beyond the PT threshold. Moreover, we found that instabilities in
initial conditions can lead to aperiodic oscillations as well as exponential
growth and decay of optical power. At the PT threshold, we observed the
existence of a stable attractor under the influence of fluctuating gain/loss
coefficient. Phase plane analysis has shown us the presence of a toroidal
chaotic attractor. The chaotic dynamics can be controlled by a judicious choice
of the waveguide parameters. | nlin_CD |
Violations of local equilibrium and stochastic thermostats: We quantitatively investigate the violations of local equilibrium in the
$\phi^4$ theory under thermal gradients, using stochastic thermostats. We find
that the deviations from local equilibrium can be quite well described by a
behavior $\sim(\nabla T)^2$. The dependence of the quantities on the thermostat
type is analyzed and its physical implications are discussed. | nlin_CD |
Perturbed phase-space dynamics of hard-disk fluids: The Lyapunov spectrum describes the exponential growth, or decay, of
infinitesimal phase-space perturbations. The perturbation associated with the
maximum Lyapunov exponent is strongly localized in space, and only a small
fraction of all particles contributes to the perturbation growth at any instant
of time. This fraction converges to zero in the thermodynamic
large-particle-number limit. For hard-disk and hard-sphere systems the
perturbations belonging to the small positive and large negative exponents are
coherently spread out and form orthogonal periodic structures in space, the
``Lyapunov modes''. There are two types of mode polarizations, transverse and
longitudinal. The transverse modes do not propagate, but the longitudinal modes
do with a speed about one third of the sound speed. We characterize the
symmetry and the degeneracy of the modes. In the thermodynamic limit the
Lyapunov spectrum has a diverging slope near the intersection with the
abscissa. No positive lower bound exists for the positive exponents. The mode
amplitude scales with the inverse square root of the particle number as
expected from the normalization of the perturbation vectors. | nlin_CD |
Elastic turbulence in curvilinear flows of polymer solutions: Following our first report (A. Groisman and V. Steinberg, $\sl Nature$ $\bf
405$, 53 (2000)) we present an extended account of experimental observations of
elasticity induced turbulence in three different systems: a swirling flow
between two plates, a Couette-Taylor (CT) flow between two cylinders, and a
flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of
width of the region available for flow to radius of curvature of the
streamlines. The experiments were carried out with dilute solutions of high
molecular weight polyacrylamide in concentrated sugar syrups. High polymer
relaxation time and solution viscosity ensured prevalence of non-linear elastic
effects over inertial non-linearity, and development of purely elastic
instabilities at low Reynolds number (Re) in all three flows. Above the elastic
instability threshold, flows in all three systems exhibit features of developed
turbulence. Those include: (i)randomly fluctuating fluid motion excited in a
broad range of spatial and temporal scales; (ii) significant increase in the
rates of momentum and mass transfer (compared to those expected for a steady
flow with a smooth velocity profile). Phenomenology, driving mechanisms, and
parameter dependence of the elastic turbulence are compared with those of the
conventional high Re hydrodynamic turbulence in Newtonian fluids. | nlin_CD |
Phase controlling current reversals in a chaotic ratchet transport: We consider a deterministic chaotic ratchet model for which the driving force
is designed to allow the rectification of current as well as the control of
chaos of the system. Besides the amplitude of the symmetric driving force which
is often used in this framework as control parameter, a phase has been newly
included here. Exploring this phase, responsible of the asymmetry of the driven
force, a number of interesting departures have been revealed. Remarkably, it
becomes possible to drive the system into one of the following regime: the
state of zero transport, the state of directed transport and most importantly
the state of reverse transport (current reversal). To have a full control of
the system, a current reversal diagram has been computed thereby clearly
showing the entire transport spectrum which is expected to be of interest for
possible experiments in this model. | nlin_CD |
Magnetic field gradients in solar wind plasma and geophysics periods: Using recent data obtained by Advanced Composition Explorer (ACE) the pumping
scale of the magnetic field gradients of the solar wind plasma has been
calculated. This pumping scale is found to be equal to 24h $\pm$ 2h. The ACE
spacecraft orbits at the L1 libration point which is a point of Earth-Sun
gravitational equilibrium about 1.5 million km from Earth. Since the Earth's
magnetosphere extends into the vacuum of space from approximately 80 to 60,000
kilometers on the side toward the Sun the pumping scale cannot be a consequence
of the 24h-period of the Earth's rotation. Vise versa, a speculation is
suggested that for the very long time of the coexistence of Earth and of the
solar wind the weak interaction between the solar wind and Earth could lead to
stochastic synchronization between the Earth's rotation and the pumping scale
of the solar wind magnetic field gradients. This synchronization could
transform an original period of the Earth's rotation to the period close to the
pumping scale of the solar wind magnetic field gradients. | nlin_CD |
Lorenz System Parameter Determination and Application to Break the
Security of Two-channel Chaotic Cryptosystems: This paper describes how to determine the parameter values of the chaotic
Lorenz system used in a two-channel cryptosystem. The geometrical properties of
the Lorenz system are used firstly to reduce the parameter search space, then
the parameters are exactly determined, directly from the ciphertext, through
the minimization of the average jamming noise power created by the encryption
process. | nlin_CD |
Maximizing coherence of oscillations by external locking: We study how the coherence of noisy oscillations can be optimally enhanced by
external locking. Basing on the condition of minimizing the phase diffusion
constant, we find the optimal forcing explicitly in the limits of small and
large noise, in dependence of phase sensitivity of the oscillator. We show that
the form of the optimal force bifurcates with the noise intensity. In the limit
of small noise, the results are compared with purely deterministic conditions
of optimal locking. | nlin_CD |
Jets or vortices - what flows are generated by an inverse turbulent
cascade?: An inverse cascade - energy transfer to progressively larger scales - is a
salient feature of two-dimensional turbulence. If the cascade reaches the
system scale, it creates a coherent flow expected to have the largest available
scale and conform with the symmetries of the domain. In a doubly periodic
rectangle, the mean flow with zero total momentum was therefore believed to be
unidirectional, with two jets along the short side; while for an aspect ratio
close to unity, a vortex dipole was expected. Using direct numerical
simulations, we show that in fact neither the box symmetry is respected nor the
largest scale is realized: the flow is never purely unidirectional since the
inverse cascade produces coherent vortices, whose number and relative motion
are determined by the aspect ratio. This spontaneous symmetry breaking is
closely related to the hierarchy of averaging times. Long-time averaging
restores translational invariance due to vortex wandering along one direction,
and gives jets whose profile, however, can be deduced neither from the
largest-available-scale argument, nor from the often employed maximum-entropy
principle or quasi-linear approximation. | nlin_CD |
Heterogeneous delays making parents synchronized: A coupled maps on
Cayley tree model: We study the phase synchronized clusters in the diffusively coupled maps on
the Cayley tree networks for heterogeneous delay values. Cayley tree networks
comprise of two parts: the inner nodes and the boundary nodes. We find that
heterogeneous delays lead to various cluster states, such as; (a) cluster state
consisting of inner nodes and boundary nodes, and (b) cluster state consisting
of only boundary nodes. The former state may comprise of nodes from all the
generations forming self-organized cluster or nodes from few generations
yielding driven clusters depending upon on the parity of heterogeneous delay
values. Furthermore, heterogeneity in delays leads to the lag synchronization
between the siblings lying on the boundary by destroying the exact
synchronization among them. The time lag being equal to the difference in the
delay values. The Lyapunov function analysis sheds light on the destruction of
the exact synchrony among the last generation nodes. To the end we discuss the
relevance of our results with respect to their applications in the family
business as well as in understanding the occurrence of genetic diseases. | nlin_CD |
Chaos detection tools: application to a self-consistent triaxial model: Together with the variational indicators of chaos, the spectral analysis
methods have also achieved great popularity in the field of chaos detection.
The former are based on the concept of local exponential divergence. The latter
are based on the numerical analysis of some particular quantities of a single
orbit, e.g. its frequency. In spite of having totally different conceptual
bases, they are used for the very same goals such as, for instance, separating
the chaotic and the regular component. In fact, we show herein that the
variational indicators serve to distinguish both components of a Hamiltonian
system in a more reliable fashion than a spectral analysis method does. We
study two start spaces for different energy levels of a self-consistent
triaxial stellar dynamical model by means of some selected variational
indicators and a spectral analysis method. In order to select the appropriate
tools for this paper, we extend previous studies where we make a comparison of
several variational indicators on different scenarios. Herein, we compare the
Average Power Law Exponent (APLE) and an alternative quantity given by the Mean
Exponential Growth factor of Neary Orbits (MEGNO): the MEGNO's Slope Estimation
of the largest Lyapunov Characteristic Exponent (SElLCE). The spectral analysis
method selected for the investigation is the Frequency Modified Fourier
Transform (FMFT). Besides a comparative study of the APLE, the Fast Lyapunov
Indicator (FLI), the Orthogonal Fast Lyapunov Indicator (OFLI) and the
MEGNO/SElLCE, we show that the SElLCE could be an appropriate alternative to
the MEGNO when studying large samples of initial conditions. The SElLCE
separates the chaotic and the regular components reliably and identifies the
different levels of chaoticity. We show that the FMFT is not as reliable as the
SElLCE to describe clearly the chaotic domains in the experiments. | nlin_CD |
Structure of characteristic Lyapunov vectors in anharmonic Hamiltonian
lattices: In this work we perform a detailed study of the scaling properties of
Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices:
the Fermi-Pasta-Ulam and $\Phi^4$ models. In this case, characteristic (also
called covariant) LVs exhibit qualitative similarities with those of
dissipative lattices but the scaling exponents are different and seemingly
nonuniversal. In contrast, backward LVs (obtained via Gram-Schmidt
orthonormalizations) present approximately the same scaling exponent in all
cases, suggesting it is an artificial exponent produced by the imposed
orthogonality of these vectors. We are able to compute characteristic LVs in
large systems thanks to a `bit reversible' algorithm, which completely obviates
computer memory limitations. | nlin_CD |
Plykin-like attractor in non-autonomous coupled oscillators: A system of two coupled non-autonomous oscillators is considered. Dynamics of
complex amplitudes is governed by differential equations with periodic
piecewise continuous dependence of the coefficients on time. The Poincar\'{e}
map is derived explicitly. With exclusion of the overall phase, on which the
evolution of other variables does not depend, the Poincar\'{e} map is reduced
to 3D mapping. It possesses an attractor of Plykin type located on an invariant
sphere. Computer verification of the cone criterion confirms the hyperbolic
nature of the attractor in the 3D map. Some results of numerical studies of the
dynamics for the coupled oscillators are presented, including the attractor
portraits, Lyapunov exponents, and the power spectral density. | nlin_CD |
Randomness, chaos, and structure: We show how a simple scheme of symbolic dynamics distinguishes a chaotic from
a random time series and how it can be used to detect structural relationships
in coupled dynamics. This is relevant for the question at which scale in
complex dynamics regularities and patterns emerge. | nlin_CD |
Pattern formation in oscillatory complex networks consisting of
excitable nodes: Oscillatory dynamics of complex networks has recently attracted great
attention. In this paper we study pattern formation in oscillatory complex
networks consisting of excitable nodes. We find that there exist a few center
nodes and small skeletons for most oscillations. Complicated and seemingly
random oscillatory patterns can be viewed as well-organized target waves
propagating from center nodes along the shortest paths, and the shortest loops
passing through both the center nodes and their driver nodes play the role of
oscillation sources. Analyzing simple skeletons we are able to understand and
predict various essential properties of the oscillations and effectively
modulate the oscillations. These methods and results will give insights into
pattern formation in complex networks, and provide suggestive ideas for
studying and controlling oscillations in neural networks. | nlin_CD |
Sampling local properties of attractors via Extreme Value Theory: We provide formulas to compute the coefficients entering the affine scaling
needed to get a non-degenerate function for the asymptotic distribution of the
maxima of some kind of observable computed along the orbit of a randomly
perturbed dynamical system. This will give information on the local geometrical
properties of the stationary measure. We will consider systems perturbed with
additive noise and with observational noise. Moreover we will apply our
techniques to chaotic systems and to contractive systems, showing that both
share the same qualitative behavior when perturbed. | nlin_CD |
Invited review: Fluctuation-induced transport. From the very small to
the very large scales: The study of fluctuation-induced transport is concerned with the directed
motion of particles on a substrate when subjected to a fluctuating external
field. Work over the last two decades provides now precise clues on how the
average transport depends on three fundamental aspects: the shape of the
substrate, the correlations of the fluctuations and the mass, geometry,
interaction and density of the particles. These three aspects, reviewed here,
acquire additional relevance because the same notions apply to a bewildering
variety of problems at very different scales, from the small nano or
micro-scale, where thermal fluctuations effects dominate, up to very large
scales including ubiquitous cooperative phenomena in granular materials. | nlin_CD |
Lyapunov instabilities in lattices of interacting classical spins at
infinite temperature: We numerically investigate Lyapunov instabilities for one-, two- and
three-dimensional lattices of interacting classical spins at infinite
temperature. We obtain the largest Lyapunov exponents for a very large variety
of nearest-neighbor spin-spin interactions and complete Lyapunov spectra in a
few selected cases. We investigate the dependence of the largest Lyapunov
exponents and whole Lyapunov spectra on the lattice size and find that both
quickly become size-independent. Finally, we analyze the dependence of the
largest Lyapunov exponents on the anisotropy of spin-spin interaction with the
particular focus on the difference between bipartite and nonbipartite lattices. | nlin_CD |
Surprising relations between parametric level correlations and fidelity
decay: Unexpected relations between fidelity decay and cross form--factor, i.e.,
parametric level correlations in the time domain are found both by a heuristic
argument and by comparing exact results, using supersymmetry techniques, in the
framework of random matrix theory. A power law decay near Heisenberg time, as a
function of the relevant parameter, is shown to be at the root of revivals
recently discovered for fidelity decay. For cross form--factors the revivals
are illustrated by a numerical study of a multiply kicked Ising spin chain. | nlin_CD |
Statistical conservation laws in turbulent transport: We address the statistical theory of fields that are transported by a
turbulent velocity field, both in forced and in unforced (decaying)
experiments. We propose that with very few provisos on the transporting
velocity field, correlation functions of the transported field in the forced
case are dominated by statistically preserved structures. In decaying
experiments (without forcing the transported fields) we identify infinitely
many statistical constants of the motion, which are obtained by projecting the
decaying correlation functions on the statistically preserved functions. We
exemplify these ideas and provide numerical evidence using a simple model of
turbulent transport. This example is chosen for its lack of Lagrangian
structure, to stress the generality of the ideas. | nlin_CD |
Symmetrical emergence of extreme events at multiple regions in a damped
and driven velocity-dependent mechanical system: In this work, we report the emergence of extreme events in a damped and
driven velocity-dependent mechanical system. We observe that the extreme events
emerge at multiple points. We further notice that the extreme events occur
symmetrically in both positive and negative values at all the points of
emergence. We statistically confirm the emergence of extreme events by plotting
the probability distribution function of peaks and interevent intervals. We
also determine the mechanism behind the emergence of extreme events at all the
points and classify these points into two categories depending on the region at
which the extreme events emerge. Finally, we plot the two parameter diagram in
order to have a complete overview of the system. | nlin_CD |
Discrete Symmetry and Stability in Hamiltonian Dynamics: In this tutorial we address the existence and stability of periodic and
quasiperiodic orbits in N degree of freedom Hamiltonian systems and their
connection with discrete symmetries. Of primary importance in our study are the
nonlinear normal modes (NNMs), i.e periodic solutions which represent
continuations of the system's linear normal modes in the nonlinear regime. We
examine the existence of such solutions and discuss different methods for
constructing them and studying their stability under fixed and periodic
boundary conditions. In the periodic case, we employ group theoretical concepts
to identify a special type of NNMs called one-dimensional "bushes". We describe
how to use linear combinations such NNMs to construct s(>1)-dimensional bushes
of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit
the symmetries of the linearized equations to simplify the study of their
destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we
review a number of interesting results, which have appeared in the recent
literature. We then turn to an analytical and numerical construction of
quasiperiodic orbits, which does not depend on the symmetries or boundary
conditions. We demonstrate that the well-known "paradox" of FPU recurrences may
be explained in terms of the exponential localization of the energies Eq of
NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,....
Thus, we show that the stability of these low-dimensional manifolds called
q-tori is related to the persistence or FPU recurrences at low energies.
Finally, we discuss a novel approach to the stability of orbits of conservative
systems, the GALIk, k=2,...,2N, by means of which one can determine accurately
and efficiently the destabilization of q-tori, leading to the breakdown of
recurrences and the equipartition of energy, at high values of the total energy
E. | nlin_CD |
Trapping enhanced by noise in nonhyperbolic and hyperbolic chaotic
scattering: The noise-enhanced trapping is a surprising phenomenon that has already been
studied in chaotic scattering problems where the noise affects the physical
variables but not the parameters of the system. Following this research, in
this work we provide strong numerical evidence to show that an additional
mechanism that enhances the trapping arises when the noise influences the
energy of the system. For this purpose, we have included a source of Gaussian
white noise in the H\'enon-Heiles system, which is a paradigmatic example of
open Hamiltonian system. For a particular value of the noise intensity, some
trajectories decrease their energy due to the stochastic fluctuations. This
drop in energy allows the particles to spend very long transients in the
scattering region, increasing their average escape times. This result, together
with the previously studied mechanisms, points out the generality of the
noise-enhanced trapping in chaotic scattering problems. | nlin_CD |
Bifurcations of a neural network model with symmetry: We analyze a family of clustered excitatory-inhibitory neural networks and
the underlying bifurcation structures that arise because of permutation
symmetries in the network as the global coupling strength $g$ is varied. We
primarily consider two network topologies: an all-to-all connected network
which excludes self-connections, and a network in which the excitatory cells
are broken into clusters of equal size. Although in both cases the bifurcation
structure is determined by symmetries in the system, the behavior of the two
systems is qualitatively different. In the all-to-all connected network, the
system undergoes Hopf bifurcations leading to periodic orbit solutions;
notably, for large $g$, there is a single, stable periodic orbit solution and
no stable fixed points. By contrast, in the clustered network, there are no
Hopf bifurcations, and there is a family of stable fixed points for large $g$. | nlin_CD |
Quantum cat maps with spin 1/2: We derive a semiclassical trace formula for quantized chaotic transformations
of the torus coupled to a two-spinor precessing in a magnetic field. The trace
formula is applied to semiclassical correlation densities of the quantum map,
which, according to the conjecture of Bohigas, Giannoni and Schmit, are
expected to converge to those of the circular symplectic ensemble (CSE) of
random matrices. In particular, we show that the diagonal approximation of the
spectral form factor for small arguments agrees with the CSE prediction. The
results are confirmed by numerical investigations. | nlin_CD |
Extreme rotational events in a forced-damped nonlinear pendulum: Since Galileo's time, the pendulum has evolved into one of the most exciting
physical objects in mathematical modeling due to its vast range of applications
for studying various oscillatory dynamics, including bifurcations and chaos,
under various interests. This well-deserved focus aids in comprehending various
oscillatory physical phenomena that can be reduced to the equations of the
pendulum. The present article focuses on the rotational dynamics of the
two-dimensional forced damped pendulum under the influence of the ac and dc
torque. Interestingly, we are able to detect a range of the pendulum's length
for which the angular velocity exhibits a few intermittent extreme rotational
events that deviate significantly from a certain well-defined threshold. The
statistics of the return intervals between these extreme rotational events are
supported by our data to be spread exponentially. The numerical results show a
sudden increase in the size of the chaotic attractor due to interior crisis
which is the source of instability that is responsible for triggering large
amplitude events in our system. We also notice the occurrence of phase slips
with the appearance of extreme rotational events when phase difference between
the instantaneous phase of the system and the externally applied ac torque is
observed. | nlin_CD |
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