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Stochastic suspensions of heavy particles: Turbulent suspensions of heavy particles in incompressible flows have gained
much attention in recent years. A large amount of work focused on the impact
that the inertia and the dissipative dynamics of the particles have on their
dynamical and statistical properties. Substantial progress followed from the
study of suspensions in model flows which, although much simpler, reproduce
most of the important mechanisms observed in real turbulence. This paper
presents recent developments made on the relative motion of a pair of particles
suspended in time-uncorrelated and spatially self-similar Gaussian flows. This
review is complemented by new results. By introducing a time-dependent Stokes
number, it is demonstrated that inertial particle relative dispersion recovers
asymptotically Richardson's diffusion associated to simple tracers. A
perturbative (homogeneization) technique is used in the small-Stokes-number
asymptotics and leads to interpreting first-order corrections to tracer
dynamics in terms of an effective drift. This expansion implies that the
correlation dimension deficit behaves linearly as a function of the Stokes
number. The validity and the accuracy of this prediction is confirmed by
numerical simulations. | nlin_CD |
Non-integrability of the dumbbell and point mass problem: This paper discusses a constrained gravitational three-body problem with two
of the point masses separated by a massless inflexible rod to form a dumbbell.
The non-integrability of this system is proven using differential Galois
theory. | nlin_CD |
Interplay of Delay and multiplexing: Impact on Cluster Synchronization: Communication delays and multiplexing are ubiquitous features of real-world
networked systems. We here introduce a simple model where these two features
are simultaneously present, and report the rich phe- nomenology which is
actually due to their interplay on cluster synchronization. A delay in one
layer has non trivial impacts on the collective dynamics of the other layers,
enhancing or suppressing synchronization. At the same time, multiplexing may
also enhance cluster synchronization of delayed layers. We elucidate several
non trivial (and anti-intuitive) scenarios, which are of interest and potential
application in various real-world systems, where introduction of a delay may
render synchronization of a layer robust against changes in the properties of
the other layers. | nlin_CD |
Suppressing noise-induced intensity pulsations in semiconductor lasers
by means of time-delayed feedback: We investigate the possibility to suppress noise-induced intensity pulsations
(relaxation oscillations) in semiconductor lasers by means of a time-delayed
feedback control scheme. This idea is first studied in a generic normal form
model, where we derive an analytic expression for the mean amplitude of the
oscillations and demonstrate that it can be strongly modulated by varying the
delay time. We then investigate the control scheme analytically and numerically
in a laser model of Lang-Kobayashi type and show that relaxation oscillations
excited by noise can be very efficiently suppressed via feedback from a
Fabry-Perot resonator. | nlin_CD |
The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on
Billiards: In this work we address the question of proving the stability of elliptic
2-periodic orbits for strictly convex billiards. Eventhough it is part of a
widely accepted belief that ellipticity implies stability, classical theorems
show that the certainty of stability relies upon more fine conditions. We
present a review of the main results and general theorems and describe the
procedure to fullfill the supplementary conditions for strictly convex
billiards. | nlin_CD |
Random matrix description of decaying quantum systems: This contribution describes a statistical model for decaying quantum systems
(e.g. photo-dissociation or -ionization). It takes the interference between
direct and indirect decay processes explicitely into account. The resulting
expressions for the partial decay amplitudes and the corresponding cross
sections may be considered a many-channel many-resonance generalization of
Fano's original work on resonance lineshapes [Phys. Rev 124, 1866 (1961)].
A statistical (random matrix) model is then introduced. It allows to describe
chaotic scattering systems with tunable couplings to the decay channels. We
focus on the autocorrelation function of the total (photo) cross section, and
we find that it depends on the same combination of parameters, as the
Fano-parameter distribution. These combinations are statistical variants of the
one-channel Fano parameter. It is thus possible to study Fano interference
(i.e. the interference between direct and indirect decay paths) on the basis of
the autocorrelation function, and thereby in the regime of overlapping
resonances. It allows us, to study the Fano interference in the limit of
strongly overlapping resonances, where we find a persisting effect on the level
of the weak localization correction. | nlin_CD |
Controlling Chaotic transport on Periodic Surfaces: We uncover and characterize different chaotic transport scenarios on perfect
periodic surfaces by controlling the chaotic dynamics of particles subjected to
periodic external forces in the absence of a ratchet effect. After identifying
relevant {\it symmetries} of chaotic solutions, analytical estimates in
parameter space for the occurrence of different transport scenarios are
provided and confirmed by numerical simulations. These scenarios are highly
sensitive to variations of the system's asymmetry parameters, including the
eccentricity of the periodic surface and the direction of dc and ac forces,
which could be useful for particle sorting purposes in those cases where chaos
is unavoidable. | nlin_CD |
Anomalous scaling of passively advected magnetic field in the presence
of strong anisotropy: Inertial-range scaling behavior of high-order (up to order N=51) structure
functions of a passively advected vector field has been analyzed in the
framework of the rapid-change model with strong small-scale anisotropy with the
aid of the renormalization group and the operator-product expansion. It has
been shown that in inertial range the leading terms of the structure functions
are coordinate independent, but powerlike corrections appear with the same
anomalous scaling exponents as for the passively advected scalar field. These
exponents depend on anisotropy parameters in such a way that a specific
hierarchy related to the degree of anisotropy is observed. Deviations from
power-law behavior like oscillations or logarithmic behavior in the corrections
to structure functions have not been found. | nlin_CD |
Knowledge-Based Learning of Nonlinear Dynamics and Chaos: Extracting predictive models from nonlinear systems is a central task in
scientific machine learning. One key problem is the reconciliation between
modern data-driven approaches and first principles. Despite rapid advances in
machine learning techniques, embedding domain knowledge into data-driven models
remains a challenge. In this work, we present a universal learning framework
for extracting predictive models from nonlinear systems based on observations.
Our framework can readily incorporate first principle knowledge because it
naturally models nonlinear systems as continuous-time systems. This both
improves the extracted models' extrapolation power and reduces the amount of
data needed for training. In addition, our framework has the advantages of
robustness to observational noise and applicability to irregularly sampled
data. We demonstrate the effectiveness of our scheme by learning predictive
models for a wide variety of systems including a stiff Van der Pol oscillator,
the Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz
system, different types of domain knowledge are incorporated to demonstrate the
strength of knowledge embedding in data-driven system identification. | nlin_CD |
Periodic orbits, basins of attraction and chaotic beats in two coupled
Kerr oscillators: Kerr oscillators are model systems which have practical applications in
nonlinear optics. Optical Kerr effect i.e. interaction of optical waves with
nonlinear medium with polarizability $\chi^{(3)}$ is the basic phenomenon
needed to explain for example the process of light transmission in fibers and
optical couplers. In this paper we analyze the two Kerr oscillators coupler and
we show that there is a possibility to control the dynamics of this system,
especially by switching its dynamics from periodic to chaotic motion and vice
versa. Moreover the switching between two different stable periodic states is
investigated. The stability of the system is described by the so-called maps of
Lyapunov exponents in parametric spaces. Comparison of basins of attractions
between two Kerr couplers and a single Kerr system is also presented. | nlin_CD |
A route to chaos in the Boros-Moll map: The Boros-Moll map appears as a subsystem of a Landen transformation
associated to certain rational integrals and its dynamics is related to the
convergence of them. In the paper, we study the dynamics of a one-parameter
family of maps which unfolds the Boros-Moll one, showing that the existence of
an unbounded invariant chaotic region in the Boros-Moll map is a peculiar
feature within the family. We relate this singularity with a specific property
of the critical lines that occurs only for this special case. In particular, we
explain how the unbounded chaotic region in the Boros-Moll map appears. Special
attention is devoted to explain the main contact/homoclinic bifurcations that
occur in the family. We also report some other bifurcation phenomena that
appear in the considered unfolding. | nlin_CD |
A Unified Approach to Attractor Reconstruction: In the analysis of complex, nonlinear time series, scientists in a variety of
disciplines have relied on a time delayed embedding of their data, i.e.
attractor reconstruction. The process has focused primarily on heuristic and
empirical arguments for selection of the key embedding parameters, delay and
embedding dimension. This approach has left several long-standing, but common
problems unresolved in which the standard approaches produce inferior results
or give no guidance at all. We view the current reconstruction process as
unnecessarily broken into separate problems. We propose an alternative approach
that views the problem of choosing all embedding parameters as being one and
the same problem addressable using a single statistical test formulated
directly from the reconstruction theorems. This allows for varying time delays
appropriate to the data and simultaneously helps decide on embedding dimension.
A second new statistic, undersampling, acts as a check against overly long time
delays and overly large embedding dimension. Our approach is more flexible than
those currently used, but is more directly connected with the mathematical
requirements of embedding. In addition, the statistics developed guide the user
by allowing optimization and warning when embedding parameters are chosen
beyond what the data can support. We demonstrate our approach on uni- and
multivariate data, data possessing multiple time scales, and chaotic data. This
unified approach resolves all the main issues in attractor reconstruction. | nlin_CD |
Synchronous Behavior of Coupled Systems with Discrete Time: The dynamics of one-way coupled systems with discrete time is considered. The
behavior of the coupled logistic maps is compared to the dynamics of maps
obtained using the Poincare sectioning procedure applied to the coupled
continuous-time systems in the phase synchronization regime. The behavior
(previously considered as asynchronous) of the coupled maps that appears when
the complete synchronization regime is broken as the coupling parameter
decreases, corresponds to the phase synchronization of flow systems, and should
be considered as a synchronous regime. A quantitative measure of the degree of
synchronism for the interacting systems with discrete time is proposed. | nlin_CD |
A review of linear response theory for general differentiable dynamical
systems: The classical theory of linear response applies to statistical mechanics
close to equilibrium. Away from equilibrium, one may describe the microscopic
time evolution by a general differentiable dynamical system, identify
nonequilibrium steady states (NESS), and study how these vary under
perturbations of the dynamics. Remarkably, it turns out that for uniformly
hyperbolic dynamical systems (those satisfying the "chaotic hypothesis"), the
linear response away from equilibrium is very similar to the linear response
close to equilibrium: the Kramers-Kronig dispersion relations hold, and the
fluctuation-dispersion theorem survives in a modified form (which takes into
account the oscillations around the "attractor" corresponding to the NESS). If
the chaotic hypothesis does not hold, two new phenomena may arise. The first is
a violation of linear response in the sense that the NESS does not depend
differentiably on parameters (but this nondifferentiability may be hard to see
experimentally). The second phenomenon is a violation of the dispersion
relations: the susceptibility has singularities in the upper half complex
plane. These "acausal" singularities are actually due to "energy
nonconservation": for a small periodic perturbation of the system, the
amplitude of the linear response is arbitrarily large. This means that the NESS
of the dynamical system under study is not "inert" but can give energy to the
outside world. An "active" NESS of this sort is very different from an
equilibrium state, and it would be interesting to see what happens for active
states to the Gallavotti-Cohen fluctuation theorem. | nlin_CD |
Chaos and Stochastic Models in Physics: Ontic and Epistemic Aspects: There is a persistent confusion about determinism and predictability. In
spite of the opinions of some eminent philosophers (e.g., Popper), it is
possible to understand that the two concepts are completely unrelated. In few
words we can say that determinism is ontic and has to do with how Nature
behaves, while predictability is epistemic and is related to what the human
beings are able to compute. An analysis of the Lyapunov exponents and the
Kolmogorov-Sinai entropy shows how deterministic chaos, although with an
epistemic character, is non subjective at all. This should clarify the role and
content of stochastic models in the description of the physical world. | nlin_CD |
Model-free measure of coupling from embedding principle: A model-free measure of coupling between dynamical variables is built from
time series embedding principle. The approach described does not require a
mathematical form for the dynamics to be assumed. The approach also does not
require density estimation which is an intractable problem in high dimensions.
The measure has strict asymptotic bounds and is robust to noise. The proposed
approach is used to demonstrate coupling between complex time series from the
finance world. | nlin_CD |
A quasi-periodic route to chaos in a parametrically driven nonlinear
medium: Small-sized systems exhibit a finite number of routes to chaos. However, in
extended systems, not all routes to complex spatiotemporal behavior have been
fully explored. Starting from the sine-Gordon model of parametrically driven
chain of damped nonlinear oscillators, we investigate a route to spatiotemporal
chaos emerging from standing waves. The route from the stationary to the
chaotic state proceeds through quasiperiodic dynamics. The standing wave
undergoes the onset of oscillatory instability, which subsequently exhibits a
different critical frequency, from which the complexity originates. A suitable
amplitude equation, valid close to the parametric resonance, makes it possible
to produce universe results. The respective phase-space structure and
bifurcation diagrams are produced in a numerical form. We characterize the
relevant dynamical regimes by means of the largest Lyapunov exponent, the power
spectrum, and the evolution of the total intensity of the wave field. | nlin_CD |
Jacobian deformation ellipsoid and Lyapunov stability analysis revisited: The stability analysis introduced by Lyapunov and extended by Oseledec is an
excellent tool to describe the character of nonlinear n-dimensional flows by n
global exponents if these flows are stable in time. However, there are two main
shortcomings: (a) The local exponents fail to indicate the origin of
instability where trajectories start to diverge. Instead, their time evolution
contains a much stronger chaos than the trajectories, which is only eliminated
by integrating over a long time. Therefore, shorter time intervals cannot be
characterized correctly, which would be essential to analyse changes of chaotic
character as in transients. (b) Moreover, although Oseledec uses an n
dimensional sphere around a point x to be transformed into an n dimensional
ellipse in first order, this local ellipse has yet not been evaluated. The aim
of this contribution is to eliminate these two shortcomings. Problem (a)
disappears if the Oseledec method is replaced by a frame with a 'constraint' as
performed by Rateitschak and Klages (RK) [Phys. Rev. E 65 036209 (2002)]. The
reasons why this method is better will be illustrated by comparing different
systems. In order to analyze shorter time intervals, integrals between
consecutive Poincare points will be evaluated. The local problems (b) will be
solved analytically by introducing the symmetric 'Jacobian deformation
ellipsoid' and its orthogonal submatrix, which enable to search in the full
phase space for extreme local separation exponents. These are close to the RK
exponents but need no time integration of the RK frame. Finally, four sets of
local exponents are compared: Oseledec frame, RK frame, Jacobian deformation
ellipsoid and its orthogonal submatrix. | nlin_CD |
Many Roads to Synchrony: Natural Time Scales and Their Algorithms: We consider two important time scales---the Markov and cryptic orders---that
monitor how an observer synchronizes to a finitary stochastic process. We show
how to compute these orders exactly and that they are most efficiently
calculated from the epsilon-machine, a process's minimal unifilar model.
Surprisingly, though the Markov order is a basic concept from stochastic
process theory, it is not a probabilistic property of a process. Rather, it is
a topological property and, moreover, it is not computable from any
finite-state model other than the epsilon-machine. Via an exhaustive survey, we
close by demonstrating that infinite Markov and infinite cryptic orders are a
dominant feature in the space of finite-memory processes. We draw out the roles
played in statistical mechanical spin systems by these two complementary length
scales. | nlin_CD |
Transition from anticipatory to lag synchronization via complete
synchronization in time-delay systems: The existence of anticipatory, complete and lag synchronization in a single
system having two different time-delays, that is feedback delay $\tau_1$ and
coupling delay $\tau_2$, is identified. The transition from anticipatory to
complete synchronization and from complete to lag synchronization as a function
of coupling delay $\tau_2$ with suitable stability condition is discussed. The
existence of anticipatory and lag synchronization is characterized both by the
minimum of similarity function and the transition from on-off intermittency to
periodic structure in laminar phase distribution. | nlin_CD |
Synchronization of hypernetworks of coupled dynamical systems: We consider synchronization of coupled dynamical systems when different types
of interactions are simultaneously present. We assume that a set of dynamical
systems are coupled through the connections of two or more distinct networks
(each of which corresponds to a distinct type of interaction), and we refer to
such a system as a hypernetwork. Applications include neural networks formed of
both electrical gap junctions and chemical synapses, the coordinated motion of
shoals of fishes communicating through both vision and flow sensing, and
hypernetworks of coupled chaotic oscillators. We first analyze the case of a
hypernetwork formed of $m=2$ networks. We look for necessary and sufficient
conditions for synchronization. We attempt at reducing the linear stability
problem in a master stability function form, i.e., at decoupling the effects of
the coupling functions from the structure of the networks. Unfortunately, we
are unable to obtain a reduction in a master stability function form for the
general case. However, we show that such a reduction is possible in three cases
of interest: (i) the Laplacian matrices associated with the two networks
commute; (ii) one of the two networks is unweighted and fully connected; (iii)
one of the two networks is such that the coupling strength from node $i$ to
node $j$ is a function of $j$ but not of $i$. Furthermore, we define a class of
networks such that if either one of the two coupling networks belongs to this
class, the reduction can be obtained independently of the other network. As an
example of interest, we study synchronization of a neural hypernetwork for
which the connections can be either chemical synapses or electrical gap
junctions. We propose a generalization of our stability results to the case of
hypernetworks formed of $m\geq 2$ networks. | nlin_CD |
Parameter Mismatches, Chaos Synchronization and Fast Dynamic Logic Gates: By using chaos synchronization between non-identical multiple time delay
semiconductor lasers with optoelectronic feedbacks, we demonstrate numerically
how fast dynamic logic gates can be constructed. The results may be helpful to
obtain a computational hardware with reconfigurable properties. | nlin_CD |
Application of largest Lyapunov exponent analysis on the studies of
dynamics under external forces: Dynamics of driven dissipative Frenkel-Kontorova model is examined by using
largest Lyapunov exponent computational technique. Obtained results show that
besides the usual way where behavior of the system in the presence of external
forces is studied by analyzing its dynamical response function, the largest
Lyapunov exponent analysis can represent a very convenient tool to examine
system dynamics. In the dc driven systems, the critical depinning force for
particular structure could be estimated by computing the largest Lyapunov
exponent. In the dc+ac driven systems, if the substrate potential is the
standard sinusoidal one, calculation of the largest Lyapunov exponent offers a
more sensitive way to detect the presence of Shapiro steps. When the amplitude
of the ac force is varied the behavior of the largest Lyapunov exponent in the
pinned regime completely reflects the behavior of Shapiro steps and the
critical depinning force, in particular, it represents the mirror image of the
amplitude dependence of critical depinning force. This points out an advantage
of this technique since by calculating the largest Lyapunov exponent in the
pinned regime we can get an insight into the dynamics of the system when
driving forces are applied. | nlin_CD |
Vibrational resonance in groundwater-dependent plant ecosystems: We report the phenomenon of vibrational resonance in a single species and a
two species models of groundwater-dependent plant ecosystems with a biharmonic
oscillation (with two widely different frequencies \omega and \Omega, \Omega >>
\omega) of the water table depth. In these two systems, the response amplitude
of the species biomass shows multiple resonances with different mechanisms. The
resonance occurs at both low- and high-frequencies of the biharmonic force. In
the single species bistable system, the resonance occurs at discrete values of
the amplitude g of the high-frequency component of the water table.
Furthermore, the best synchronization of biomass and its carrying capacity with
the biharmonic force occurs at the resonance. In the two species excitable and
time-delay model, the response amplitude (Q) profile shows several plateau
regions of resonance, where the period of evolution of the species biomass
remains the same and the value of Q is inversely proportional to it. The
response amplitude is highly sensitive to the time-delay parameter \tau and
shows two distinct sequences of resonance intervals with a decreasing amplitude
with \tau. | nlin_CD |
Anomalous transport and observable average in the standard map: The distribution of finite time observable averages and transport in low
dimensional Hamiltonian systems is studied. Finite time observable average
distributions are computed, from which an exponent $\alpha$ characteristic of
how the maximum of the distributions scales with time is extracted. To link
this exponent to transport properties, the characteristic exponent $\mu(q)$ of
the time evolution of the different moments of order $q$ related to transport
are computed. As a testbed for our study the standard map is used. The
stochasticity parameter $K$ is chosen so that either phase space is mixed with
a chaotic sea and islands of stability or with only a chaotic sea. Our
observations lead to a proposition of a law relating the slope in $q=0$ of the
function $\mu(q)$ with the exponent $\alpha$. | nlin_CD |
Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial: A new computational technique based on the symbolic description utilizing
kneading invariants is proposed and verified for explorations of dynamical and
parametric chaos in a few exemplary systems with the Lorenz attractor. The
technique allows for uncovering the stunning complexity and universality of
bi-parametric structures and detect their organizing centers - codimension-two
T-points and separating saddles in the kneading-based scans of the iconic
Lorenz equation from hydrodynamics, a normal model from mathematics, and a
laser model from nonlinear optics. | nlin_CD |
Hysteresis Models of Dynamic Mode Atomic Force Microscopes: Analysis and
Identification: A new class of models based on hysteresis functions is developed to describe
atomic force microscopes operating in dynamic mode. Such models are able to
account for dissipative phenomena in the tip-sample interaction which are
peculiar of this operation mode. The model analysis, which can be pursued using
frequency domain techniques, provides a clear insight of specific nonlinear
behaviours. Experiments show good agreement with the identified models. | nlin_CD |
Decoding Information by Following Parameter Modulation With Parameter
Adaptive Control: It has been proposed to realize secure communication using chaotic
synchronization via transmission of binary message encoded by parameter
modulation in the chaotic system. This paper considers the use of parameter
adaptive control techniques to extract the message, based on the assumptions
that we know the equation form of the chaotic system in the transmitter but do
not have access to the precise values of the parameters which are kept secret
as a secure set. In the case that a synchronizing system can be constructed
using parameter adaptive control by the transmitted signal and the
synchronization is robust to parameter mismatches, the parameter modulation can
be revealed and the message decoded without resorting to exact parameter values
in the secure set. A practical local Lyapunov function method for designing
parameter adaptive control rules based on originally synchronized systems is
presented. | nlin_CD |
Dynamics of Rotator Chain with Dissipative Boundary: We study the deterministic dynamics of rotator chain that subjected to purely
mechanical driving on the boundary by stability analysis and numerical
simulation. Globally synchronous rotation, clustered synchronous rotation, and
split synchronous rotation states are identified. In particular, we find that
the single-peaked variance distribution of angular momenta is the consequence
of the deterministic dynamics. As a result, the operational definition of
temperature used in the previous studies on rotator chain should be revisited. | nlin_CD |
Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling: Small lattices of $N$ nearest neighbor coupled excitable FitzHugh-Nagumo
systems, with time-delayed coupling are studied, and compared with systems of
FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of
equilibria in N=2 case are studied analytically, and it is then numerically
confirmed that the same bifurcations are relevant for the dynamics in the case
$N>2$. Bifurcations found include inverse and direct Hopf and fold limit cycle
bifurcations. Typical dynamics for different small time-lags and coupling
intensities could be excitable with a single globally stable equilibrium,
asymptotic oscillatory with symmetric limit cycle, bi-stable with stable
equilibrium and a symmetric limit cycle, and again coherent oscillatory but
non-symmetric and phase-shifted. For an intermediate range of time-lags inverse
sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of
oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo
oscillators with the same type of coupling. | nlin_CD |
Non-identical multiplexing promotes chimera states: We present the emergence of chimeras, a state referring to coexistence of
partly coherent, partly incoherent dynamics in networks of identical
oscillators, in a multiplex network consisting of two non-identical layers
which are interconnected. We demonstrate that the parameter range displaying
the chimera state in the homogeneous first layer of the multiplex networks can
be tuned by changing the link density or connection architecture of the same
nodes in the second layer. We focus on the impact of the interconnected second
layer on the enlargement or shrinking of the coupling regime for which chimeras
are displayed in the homogeneous first layer. We find that a denser homogeneous
second layer promotes chimera in a sparse first layer, where chimeras do not
occur in isolation. Furthermore, while a dense connection density is required
for the second layer if it is homogeneous, this is not true if the second layer
is inhomogeneous. We demonstrate that a sparse inhomogeneous second layer which
is common in real-world complex systems can promote chimera states in a sparse
homogeneous first layer. | nlin_CD |
Hydrodynamic superradiance in wave-mediated cooperative tunneling: Superradiance and subradiance occur in quantum optics when the emission rate
of photons from multiple atoms is enhanced and diminished, respectively, owing
to interaction between neighboring atoms. We here demonstrate a classical
analog thereof in a theoretical model of droplets walking on a vibrating bath.
Two droplets are confined to identical two-level systems, a pair of wells
between which the drops may tunnel, joined by an intervening coupling cavity.
The resulting classical superradiance is rationalized in terms of the system's
non-Markovian, pilot-wave dynamics. | nlin_CD |
Random Wandering Around Homoclinic-like Manifolds in Symplectic Map
Chain: We present a method to construct a symplecticity preserving renormalization
group map of a chain of weakly nonlinear symplectic maps and obtain a general
reduced symplectic map describing its long-time behaviour. It is found that the
modulational instability in the reduced map triggers random wandering of orbits
around some homoclinic-like manifolds, which is understood as the Bernoulli
shifts. | nlin_CD |
Almost Periodicity in Chaos: Periodicity plays a significant role in the chaos theory from the beginning
since the skeleton of chaos can consist of infinitely many unstable periodic
motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and
the one obtained through period-doubling cascade [3]. Countable number of
periodic orbits exist in any neighborhood of a structurally stable Poincar\'{e}
homoclinic orbit, which can be considered as a criterion for the presence of
complex dynamics [4]-[6]. It was certified by Shilnikov [7] and Seifert [8]
that it is possible to replace periodic solutions by Poisson stable or almost
periodic motions in a chaotic attractor. Despite the fact that the idea of
replacing periodic solutions by other types of regular motions is attractive,
very few results have been obtained on the subject. The present study
contributes to the chaos theory in that direction.
In this paper, we take into account chaos both through a cascade of almost
periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke
definition is modified by replacing infinitely many periodic motions with
almost periodic ones, which are separated from the motions of the scrambled
set. The theoretical results are valid for systems with arbitrary high
dimensions. Formation of the chaos is exemplified by means of unidirectionally
coupled Duffing oscillators. The controllability of the extended chaos is
demonstrated numerically by means of the Ott-Grebogi-Yorke [9] control
technique. In particular, the stabilization of tori is illustrated. | nlin_CD |
Linear and fractal diffusion coefficients in a family of one dimensional
chaotic maps: We analyse deterministic diffusion in a simple, one-dimensional setting
consisting of a family of four parameter dependent, chaotic maps defined over
the real line. When iterated under these maps, a probability density function
spreads out and one can define a diffusion coefficient. We look at how the
diffusion coefficient varies across the family of maps and under parameter
variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated
in terms of generalised Takagi functions, we derive exact, fully analytical
expressions for the diffusion coefficients. Typically, for simple maps these
quantities are fractal functions of control parameters. However, our family of
four maps exhibits both fractal and linear behavior. We explain these different
structures by looking at the topology of the Markov partitions and the ergodic
properties of the maps. | nlin_CD |
Power spectrum analysis and missing level statistics of microwave graphs
with violated time reversal invariance: We present experimental studies of the power spectrum and other fluctuation
properties in the spectra of microwave networks simulating chaotic quantum
graphs with violated time reversal in- variance. On the basis of our data sets
we demonstrate that the power spectrum in combination with other long-range and
also short-range spectral fluctuations provides a powerful tool for the
identification of the symmetries and the determination of the fraction of
missing levels. Such a pro- cedure is indispensable for the evaluation of the
fluctuation properties in the spectra of real physical systems like, e.g.,
nuclei or molecules, where one has to deal with the problem of missing levels. | nlin_CD |
Boundary crisis and suppression of Fermi acceleration in a dissipative
two dimensional non-integrable time-dependent billiard: Some dynamical properties for a dissipative time-dependent oval-shaped
billiard are studied. The system is described in terms of a four-dimensional
nonlinear mapping. Dissipation is introduced via inelastic collisions of the
particle with the boundary, thus implying that the particle has a fractional
loss of energy upon collision. The dissipation causes profound modifications in
the dynamics of the particle as well as in the phase space of the non
dissipative system. In particular, inelastic collisions can be assumed as an
efficient mechanism to suppress Fermi acceleration of the particle. The
dissipation also creates attractors in the system, including chaotic. We show
that a slightly modification of the intensity of the damping coefficient yields
a drastic and sudden destruction of the chaotic attractor, thus leading the
system to experience a boundary crisis. We have characterized such a boundary
crisis via a collision of the chaotic attractor with its own basin of
attraction and confirmed that inelastic collisions do indeed suppress Fermi
acceleration in two-dimensional time dependent billiards. | nlin_CD |
An Analytical Study on the Synchronization of Strange Non-Chaotic
Attractors: In this paper we present an analytical study on the synchronization dynamics
observed in unidirectionally-coupled quasiperiodically-forced systems that
exhibit Strange Non-chaotic Attractors (SNA) in their dynamics. The SNA
dynamics observed in the uncoupled system is studied analytically through phase
portraits and poincare maps. A difference system is obtained by coupling the
state equations of similar piecewise linear regions of the drive and response
systems. The mechanism of synchronization of the coupled system is realized
through the bifurcation of the eigenvalues in one of the piecewise linear
regions of the difference system. The analytical solutions obtained for the
normalized state equations in each piecewise linear region of the difference
system has been used to explain the synchronization dynamics though phase
portraits and timeseries analysis. The stability of the synchronized state is
confirmed through the Master Stability Function. An explicit analytical
solution explaining the synchronization of SNAs is reported in the literature
for the first time. | nlin_CD |
Impulse-induced optimum control of chaos in dissipative driven systems: Taming chaos arising from dissipative non-autonomous nonlinear systems by
applying additional harmonic excitations is a reliable and widely used
procedure nowadays. But the suppressory effectiveness of generic non-harmonic
periodic excitations continues to be a significant challenge both to our
theoretical understanding and in practical applications. Here we show how the
effectiveness of generic suppressory excitations is optimally enhanced when the
impulse transmitted by them (time integral over two consecutive zeros) is
judiciously controlled in a not obvious way. This is demonstrated
experimentally by means of an analog version of a universal model, and
confirmed numerically by simulations of such a damped driven system including
the presence of noise. Our theoretical analysis shows that the controlling
effect of varying the impulse is due to a correlative variation of the energy
transmitted by the suppressory excitation. | nlin_CD |
Few-Freedom Turbulence: The results of numerical experiments on the structure of chaotic attractors
in the Khalatnikov - Kroyter model of two freedoms are presented. This model
was developed for a qualitative description of the wave turbulence of the
second sound in helium. The attractor dimension, size, and the maximal Lyapunov
exponent in dependence on the single dimensionless parameter $F$ of the model
are found and discussed. The principal parameter $F$ is similar to the Reynolds
number in hydrodynamic turbulence. We were able to discern four different
attractors characterized by a specific critical value of the parameter
($F=F_{cr}$), such that the attractor exists for $F>F_{cr}$ only. A simple
empirical relation for this dependence on the argument ($F-F_{cr}$) is
presented which turns out to be universal for different attractors with respect
to the dimension and dimensionless Lyapunov exponents. Yet, it differs as to
the size of attractor. In the main region of our studies the dependence of all
dimensionless characteristics of the chaotic attractor on parameter $F$ is very
slow (logarithmic) which is qualitatively different as compared to that of a
multi-freedom attractor, e.g., in hydrodynamic turbulence (a power law).
However, at very large $F\sim 10^7$ the transition to a power-law dependence
has been finally found, similar to the multi-freedom attractor. Some unsolved
problems and open questions are also discussed. | nlin_CD |
Bailout Embeddings, Targeting of KAM Orbits, and the Control of
Hamiltonian Chaos: We present a novel technique, which we term bailout embedding, that can be
used to target orbits having particular properties out of all orbits in a flow
or map. We explicitly construct a bailout embedding for Hamiltonian systems so
as to target KAM orbits. We show how the bailout dynamics is able to lock onto
extremely small KAM islands in an ergodic sea. | nlin_CD |
Lorenz cycle for the Lorenz attractor: In this note we study energetics of Lorenz-63 system through its Lie-Poisson
structure. | nlin_CD |
Designer dynamics through chaotic traps: Controlling complex behavior in
driven nonlinear systems: Control schemes for dynamical systems typically involve stabilizing unstable
periodic orbits. In this paper we introduce a new paradigm of control that
involves `trapping' the dynamics arbitrarily close to any desired trajectory.
This is achieved by a state-dependent dynamical selection of the input signal
applied to the driven nonlinear system. An emergent property of the trapping
process is that the signal changes in a chaotic sequence: a manifestation of
chaos-induced order. The simplicity of the control scheme makes it easily
implementable in experimental systems. | nlin_CD |
Synchronization in discrete-time networks with general pairwise coupling: We consider complete synchronization of identical maps coupled through a
general interaction function and in a general network topology where the edges
may be directed and may carry both positive and negative weights. We define
mixed transverse exponents and derive sufficient conditions for local complete
synchronization. The general non-diffusive coupling scheme can lead to new
synchronous behavior, in networks of identical units, that cannot be produced
by single units in isolation. In particular, we show that synchronous chaos can
emerge in networks of simple units. Conversely, in networks of chaotic units
simple synchronous dynamics can emerge; that is, chaos can be suppressed
through synchrony. | nlin_CD |
Non-permanent form solutions in the Hamiltonian formulation of surface
water waves: Using the KAM method, we exhibit some solutions of a finite-dimensional
approximation of the Zakharov Hamiltonian formulation of gravity water waves,
which are spatially periodic, quasi-periodic in time, and not permanent form
travelling waves. For this Hamiltonian, which is the total energy of the waves,
the canonical variables are some complex quantities an and a*n, which are
linear combinations of the Fourier components of the free surface elevation and
the velocity potential evaluated at the surface. We expose the method for the
case of a system with a finite number of degrees of freedom, the Zufiria model,
with only 3 modes interacting. | nlin_CD |
About universality of lifetime statistics in quantum chaotic scattering: The statistics of the resonance widths and the behavior of the survival
probability is studied in a particular model of quantum chaotic scattering (a
particle in a periodic potential subject to static and time-periodic forces)
introduced earlier in Ref.[5,6]. The coarse-grained distribution of the
resonance widths is shown to be in good agreement with the prediction of Random
Matrix Theory (RMT). The behavior of the survival probability shows, however,
some deviation from RMT. | nlin_CD |
Spatial Patterns in Chemically and Biologically Reacting Flows: We present here a number of processes, inspired by concepts in Nonlinear
Dynamics such as chaotic advection and excitability, that can be useful to
understand generic behaviors in chemical or biological systems in fluid flows.
Emphasis is put on the description of observed plankton patchiness in the sea.
The linearly decaying tracer, and excitable kinetics in a chaotic flow are
mainly the models described. Finally, some warnings are given about the
difficulties in modeling discrete individuals (such as planktonic organisms) in
terms of continuous concentration fields. | nlin_CD |
A Diagrammatic Representation of Phase Portraits and Bifurcation
Diagrams of Two-Dimensional Dynamical Systems: We treat the problem of characterizing in a systematic way the qualitative
features of two-dimensional dynamical systems. To that end, we construct a
representation of the topological features of phase portraits by means of
diagrams that discard their quantitative information. All codimension 1
bifurcations are naturally embodied in the possible ways of transitioning
smoothly between diagrams. We introduce a representation of bifurcation curves
in parameter space that guides the proposition of bifurcation diagrams
compatible with partial information about the system. | nlin_CD |
Lyapunov exponent and natural invariant density determination of chaotic
maps: An iterative maximum entropy ansatz: We apply the maximum entropy principle to construct the natural invariant
density and Lyapunov exponent of one-dimensional chaotic maps. Using a novel
function reconstruction technique that is based on the solution of Hausdorff
moment problem via maximizing Shannon entropy, we estimate the invariant
density and the Lyapunov exponent of nonlinear maps in one-dimension from a
knowledge of finite number of moments. The accuracy and the stability of the
algorithm are illustrated by comparing our results to a number of nonlinear
maps for which the exact analytical results are available. Furthermore, we also
consider a very complex example for which no exact analytical result for
invariant density is available. A comparison of our results to those available
in the literature is also discussed. | nlin_CD |
Stability of helical tubes conveying fluid: We study the linear stability of elastic collapsible tubes conveying fluid,
when the equilibrium configuration of the tube is helical. A particular case of
such tubes, commonly encountered in applications, is represented by quarter- or
semi-circular tubular joints used at pipe's turning points. The stability
theory for pipes with non-straight equilibrium configurations, especially for
collapsible tubes, allowing dynamical change of the cross-section, has been
elusive as it is difficult to accurately develop the dynamic description via
traditional methods. We develop a methodology for studying the
three-dimensional dynamics of collapsible tubes based on the geometric
variational approach. We show that the linear stability theory based on this
approach allows for a complete treatment for arbitrary three-dimensional
helical configurations of collapsible tubes by reduction to an equation with
constant coefficients. We discuss new results on stability loss of straight
tubes caused by the cross-sectional area change. Finally, we develop a
numerical algorithm for computation of the linear stability using our theory
and present the results of numerical studies for both straight and helical
tubes. | nlin_CD |
First experimental observation of generalized synchronization phenomena
in microwave oscillators: In this Letter we report for the first time on the experimental observation
of the generalized synchronization regime in the microwave electronic systems,
namely, in the multicavity klystron generators. A new approach devoted to the
generalized synchronization detection has been developed. The experimental
observations are in the excellent agreement with the results of numerical
simulation. The observed phenomena gives a strong potential for new
applications requiring microwave chaotic signals. | nlin_CD |
Dimensional collapse and fractal attractors of a system with fluctuating
delay times: A frequently encountered situation in the study of delay systems is that the
length of the delay time changes with time, which is of relevance in many
fields such as optics, mechanical machining, biology or physiology. A
characteristic feature of such systems is that the dimension of the system
dynamics collapses due to the fluctuations of delay times. In consequence, the
support of the long-trajectory attractors of this kind of systems is found
being fractal in contrast to the fuzzy attractors in most random systems. | nlin_CD |
Strong effect of weak diffusion on scalar turbulence at large scales: Passive scalar turbulence forced steadily is characterized by the velocity
correlation scale, $L$, injection scale, $l$, and diffusive scale, $r_d$. The
scales are well separated if the diffusivity is small, $r_d\ll l,L$, and one
normally says that effects of diffusion are confined to smaller scales, $r\ll
r_d$. However, if the velocity is single scale one finds that a weak dependence
of the scalar correlations on the molecular diffusivity persists to even larger
scales, e.g. $l\gg r\gg r_d$ \cite{95BCKL}. We consider the case of $L\gg l$
and report a counter-intuitive result -- the emergence of a new range of large
scales, $L\gg r\gg l^2/r_d$, where the diffusivity shows a strong effect on
scalar correlations. | nlin_CD |
Poincaré chaos and unpredictable functions: The results of this study are continuation of the research of Poincar\'e
chaos initiated in papers (Akhmet M, Fen MO. Commun Nonlinear Sci Numer Simulat
2016;40:1-5; Akhmet M, Fen MO. Turk J Math, doi:10.3906/mat-1603-51, accepted).
We focus on the construction of an unpredictable function, continuous on the
real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics
and the logistic map are obtained. By shaping the unpredictable function as
well as Poisson function we have performed the first step in the development of
the theory of unpredictable solutions for differential and discrete equations.
The results are preliminary ones for deep analysis of chaos existence in
differential and hybrid systems. Illustrative examples concerning unpredictable
solutions of differential equations are provided. | nlin_CD |
Genesis of d'Alembert's paradox and analytical elaboration of the drag
problem: We show that the issue of the drag exerted by an incompressible fluid on a
body in uniform motion has played a major role in the early development of
fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing
of the drag for a body of arbitrary shape; for this he exploited and
significantly extended existing ideas on decomposing the flow into thin
fillets; he did not however have a correct picture of the global structure of
the flow around a body. Borda in 1766 showed that the principle of live forces
implied the vanishing of the drag and should thus be inapplicable to the
problem. After having at first refused the possibility of a vanishing drag,
d'Alembert in 1768 established the paradox, but only for bodies with a
head-tail symmetry. A full understanding of the paradox, as due to the neglect
of viscous forces, had to wait until the work of Saint-Venant in 1846. | nlin_CD |
Rich-club network topology to minimize synchronization cost due to phase
difference among frequency-synchronized oscillators: Functions of some networks, such as power grids and large-scale brain
networks, rely on not only frequency synchronization, but also phase
synchronization. Nevertheless, even after the oscillators reach to
frequency-synchronized status, phase difference among oscillators often shows
non-zero constant values. Such phase difference potentially results in
inefficient transfer of power or information among oscillators, and avoid
proper and efficient functioning of the network. In the present study, we newly
define synchronization cost by the phase difference among the
frequency-synchronized oscillators, and investigate the optimal network
structure with the minimum synchronization cost through rewiring-based
optimization. By using the Kuramoto model, we demonstrate that the cost is
minimized in a network topology with rich-club organization, which comprises
the densely-connected center nodes and peripheral nodes connecting with the
center module. We also show that the network topology is characterized by its
bimodal degree distribution, which is quantified by Wolfson's polarization
index. Furthermore, we provide analytical interpretation on why the rich-club
network topology is related to the small amount of synchronization cost. | nlin_CD |
Quantum Graphs: A simple model for Chaotic Scattering: We connect quantum graphs with infinite leads, and turn them to scattering
systems. We show that they display all the features which characterize quantum
scattering systems with an underlying classical chaotic dynamics: typical
poles, delay time and conductance distributions, Ericson fluctuations, and when
considered statistically, the ensemble of scattering matrices reproduce quite
well the predictions of appropriately defined Random Matrix ensembles. The
underlying classical dynamics can be defined, and it provides important
parameters which are needed for the quantum theory. In particular, we derive
exact expressions for the scattering matrix, and an exact trace formula for the
density of resonances, in terms of classical orbits, analogous to the
semiclassical theory of chaotic scattering. We use this in order to investigate
the origin of the connection between Random Matrix Theory and the underlying
classical chaotic dynamics. Being an exact theory, and due to its relative
simplicity, it offers new insights into this problem which is at the fore-front
of the research in chaotic scattering and related fields. | nlin_CD |
Dissipative structures in a nonlinear dynamo: This paper considers magnetic field generation by a fluid flow in a system
referred to as the Archontis dynamo: a steady nonlinear magnetohydrodynamic
(MHD) state is driven by a prescribed body force. The field and flow become
almost equal and dissipation is concentrated in cigar-like structures centred
on straight-line separatrices. Numerical scaling laws for energy and
dissipation are given that extend previous calculations to smaller
diffusivities. The symmetries of the dynamo are set out, together with their
implications for the structure of field and flow along the separatrices. The
scaling of the cigar-like dissipative regions, as the square root of the
diffusivities, is explained by approximations near the separatrices. Rigorous
results on the existence and smoothness of solutions to the steady, forced MHD
equations are given. | nlin_CD |
How winding is the coast of Britain ? Conformal invariance of rocky
shorelines: We show that rocky shorelines with fractal dimension 4/3 are conformally
invariant curves by measuring the statistics of their winding angles from
global high-resolution data. Such coastlines are thus statistically equivalent
to the outer boundary of the random walk and of percolation clusters. A simple
model of coastal erosion gives an explanation for these results. Conformal
invariance allows also to predict the highly intermittent spatial distribution
of the flux of pollutant diffusing ashore. | nlin_CD |
Computing the multifractal spectrum from time series: An algorithmic
approach: We show that the existing methods for computing the f(\alpha) spectrum from a
time series can be improved by using a new algorithmic scheme. The scheme
relies on the basic idea that the smooth convex profile of a typical f(\alpha)
spectrum can be fitted with an analytic function involving a set of four
independent parameters. While the standard existing schemes [16, 18] generally
compute only an incomplete f(\alpha) spectrum (usually the top portion), we
show that this can be overcome by an algorithmic approach which is automated to
compute the Dq and f(\alpha) spectrum from a time series for any embedding
dimension. The scheme is first tested with the logistic attractor with known
f(\alpha) curve and subsequently applied to higher dimensional cases. We also
show that the scheme can be effectively adapted for analysing practcal time
series involving noise, with examples from two widely different real world
systems. Moreover, some preliminary results indicating that the set of four
independant parameters may be used as diagnostic measures is also included. | nlin_CD |
Detection of Generalized Synchronization using Echo State Networks: Generalized synchronization between coupled dynamical systems is a phenomenon
of relevance in applications that range from secure communications to
physiological modelling. Here we test the capabilities of reservoir computing
and, in particular, echo state networks for the detection of generalized
synchronization. A nonlinear dynamical system consisting of two coupled
R\"ossler chaotic attractors is used to generate temporal series consisting of
time-locked generalized synchronized sequences interleaved by unsynchronized
ones. Correctly tuned, echo state networks are able to efficiently discriminate
between unsynchronized and synchronized sequences. Compared to other
state-of-the-art techniques of synchronization detection, the online
capabilities of the proposed ESN based methodology make it a promising choice
for real-time applications aiming to monitor dynamical synchronization changes
in continuous signals. | nlin_CD |
Strange nonchaotic stars: The unprecedented light curves of the Kepler space telescope document how the
brightness of some stars pulsates at primary and secondary frequencies whose
ratios are near the golden mean, the most irrational number. A nonlinear
dynamical system driven by an irrational ratio of frequencies generically
exhibits a strange but nonchaotic attractor. For Kepler's "golden" stars, we
present evidence of the first observation of strange nonchaotic dynamics in
nature outside the laboratory. This discovery could aid the classification and
detailed modeling of variable stars. | nlin_CD |
Cryptanalysis of a one round chaos-based Substitution Permutation
Network: The interleaving of chaos and cryptography has been the aim of a large set of
works since the beginning of the nineties. Many encryption proposals have been
introduced to improve conventional cryptography. However, many proposals
possess serious problems according to the basic requirements for the secure
exchange of information. In this paper we highlight some of the main problems
of chaotic cryptography by means of the analysis of a very recent chaotic
cryptosystem based on a one round Substitution Permutation Network. More
specifically, we show that it is not possible to avoid the security problems of
that encryption architecture just by including a chaotic system as core of the
derived encryption system. | nlin_CD |
Pseudo-random number generator based on asymptotic deterministic
randomness: An approach to generate the pseudorandom-bit sequence from the asymptotic
deterministic randomness system is proposed in this Letter. We study the
characteristic of multi-value correspondence of the asymptotic deterministic
randomness constructed by the piecewise linear map and the noninvertible
nonlinearity transform, and then give the discretized systems in the finite
digitized state space. The statistic characteristics of the asymptotic
deterministic randomness are investigated numerically, such as stationary
probability density function and random-like behavior. Furthermore, we analyze
the dynamics of the symbolic sequence. Both theoretical and experimental
results show that the symbolic sequence of the asymptotic deterministic
randomness possesses very good cryptographic properties, which improve the
security of chaos based PRBGs and increase the resistance against entropy
attacks and symbolic dynamics attacks. | nlin_CD |
Interactions destroy dynamical localization with strong and weak chaos: Bose-Einstein condensates loaded into kicked optical lattices can be treated
as quantum kicked rotor systems. Noninteracting rotors show dynamical
localization in momentum space. The experimentally tunable condensate
interaction is included in a qualitative Gross-Pitaevskii type model based on
two-body interactions. We observe strong and weak chaos regimes of wave packet
spreading in momentum space. In the intermediate strong chaos regime the
condensate energy grows as $t^{1/2}$. In the asymptotic weak chaos case the
growth crosses over into a $t^{1/3}$ law. The results do not depend on the
details of the kicking. | nlin_CD |
Networked control systems: a perspective from chaos: In this paper, a nonlinear system aiming at reducing the signal transmission
rate in a networked control system is constructed by adding nonlinear
constraints to a linear feedback control system. Its stability is investigated
in detail. It turns out that this nonlinear system exhibits very interesting
dynamical behaviors: in addition to local stability, its trajectories may
converge to a non-origin equilibrium or be periodic or just be oscillatory.
Furthermore it exhibits sensitive dependence on initial conditions --- a sign
of chaos. Complicated bifurcation phenomena are exhibited by this system. After
that, control of the chaotic system is discussed. All these are studied under
scalar cases in detail. Some difficulties involved in the study of this type of
systems are analyzed. Finally an example is employed to reveal the
effectiveness of the scheme in the framework of networked control systems. | nlin_CD |
Dynamic synchronization of a time-evolving optical network of chaotic
oscillators: We present and experimentally demonstrate a technique for achieving and
maintaining a global state of identical synchrony of an arbitrary network of
chaotic oscillators even when the coupling strengths are unknown and
time-varying. At each node an adaptive synchronization algorithm dynamically
estimates the current strength of the net coupling signal to that node. We
experimentally demonstrate this scheme in a network of three bidirectionally
coupled chaotic optoelectronic feedback loops and we present numerical
simulations showing its application in larger networks. The stability of the
synchronous state for arbitrary coupling topologies is analyzed via a master
stability function approach. | nlin_CD |
Map representation of the time-delayed system in the presence of Delay
Time Modulation: an Application to the stability analysis: We introduce the map representation of a time-delayed system in the presence
of delay time modulation. Based on this representation, we find the method by
which to analyze the stability of that kind of a system. We apply this method
to a coupled chaotic system and discuss the results in comparison to the system
with a fixed delay time. | nlin_CD |
Simulation studies on the design of optimum PID controllers to suppress
chaotic oscillations in a family of Lorenz-like multi-wing attractors: Multi-wing chaotic attractors are highly complex nonlinear dynamical systems
with higher number of index-2 equilibrium points. Due to the presence of
several equilibrium points, randomness and hence the complexity of the state
time series for these multi-wing chaotic systems is much higher than that of
the conventional double-wing chaotic attractors. A real-coded Genetic Algorithm
(GA) based global optimization framework has been adopted in this paper as a
common template for designing optimum Proportional-Integral-Derivative (PID)
controllers in order to control the state trajectories of four different
multi-wing chaotic systems among the Lorenz family viz. Lu system, Chen system,
Rucklidge (or Shimizu Morioka) system and Sprott-1 system. Robustness of the
control scheme for different initial conditions of the multi-wing chaotic
systems has also been shown. | nlin_CD |
Riddling: Chimera's dilemma: We investigate the basin of attraction properties and its boundaries for
chimera states in a circulant network of H\'enon maps. Chimera states, for
which coherent and incoherent domains coexist, emerge as a consequence of the
coexistence of basin of attractions for each state. It is known that the
coexisting basins of attraction lead to a hysteretic behaviour in the diagrams
for the density of incoherent and coherent states as a function of a varying
parameter. Consequently, the distribution of chimera states can remain
invariant by a parameter change, as well as it can suffer subtle changes when
one of the basin ceases to exist. A similar phenomenon is observed when
perturbations are applied in the initial conditions. By means of the
uncertainty exponent, we characterise the basin boundaries between the coherent
and chimera states, and between the incoherent and chimera states, and uncover
fractal and riddled boundaries, respectively. This way, we show that the
density of chimera states can be not only moderately sensitive but also highly
sensitive to initial conditions. This chimera's dilemma is a consequence of the
fractal and riddled nature of the basins boundaries. | nlin_CD |
Effective stochastic model for chaos in the Fermi-Pasta-Ulam-Tsingou
chain: Understanding the interplay between different wave excitations, such as
phonons and localized solitons, is crucial for developing coarse-grained
descriptions of many-body, near-integrable systems. We treat the
Fermi-Pasta-Ulam-Tsingou (FPUT) non-linear chain and show numerically that at
short timescales, relevant to the largest Lyapunov exponent, it can be modeled
as a random perturbation of its integrable approximation -- the Toda chain. At
low energies, the separation between two trajectories that start at close
proximity is dictated by the interaction between few soliton modes and an
intrinsic, apparent bath representing a background of many radiative modes. It
is sufficient to consider only one randomly perturbed Toda soliton-like mode to
explain the power-law profiles reported in previous works, describing how the
Lyapunov exponent of large FPUT chains decreases with the energy density of the
system. | nlin_CD |
Using Synchronization for Prediction of High-Dimensional Chaotic
Dynamics: We experimentally observe the nonlinear dynamics of an optoelectronic
time-delayed feedback loop designed for chaotic communication using commercial
fiber optic links, and we simulate the system using delay differential
equations. We show that synchronization of a numerical model to experimental
measurements provides a new way to assimilate data and forecast the future of
this time-delayed high-dimensional system. For this system, which has a
feedback time delay of 22 ns, we show that one can predict the time series for
up to several delay periods, when the dynamics is about 15 dimensional. | nlin_CD |
Non-ergodicity and localization of invariant measure for two colliding
masses: We show evidence, based on extensive and carefully performed numerical
experiments, that the system of two elastic hard-point masses in one-dimension
is not ergodic for a generic mass ratio and consequently does not follow the
principle of energy equipartition. This system is equivalent to a right
triangular billiard. Remarkably, following the time-dependent probability
distribution in a suitably chosen velocity direction space, we find evidence of
exponential localization of invariant measure. For non-generic mass ratios
which correspond to billiard angles which are rational, or weak irrational
multiples of pi, the system is ergodic, in consistence with existing rigorous
results. | nlin_CD |
Radial disk heating by more than one spiral density wave: We consider a differentially rotating, 2D stellar disk perturbed by two
steady state spiral density waves moving at different patterns speeds. Our
investigation is based on direct numerical integration of initially circular
test-particle orbits. We examine a range of spiral strengths and spiral speeds
and show that stars in this time dependent gravitational field can be heated
(their random motions increased). This is particularly noticeable in the
simultaneous propagation of a 2-armed spiral density wave near the corotation
resonance (CR), and a weak 4-armed one near the inner and outer 4:1 Lindblad
resonances. In simulations with 2 spiral waves moving at different pattern
speeds we find: (1) the variance of the radial velocity, sigma_R^2, exceeds the
sum of the variances measured from simulations with each individual pattern;
(2) sigma_R^2 can grow with time throughout the entire simulation; (3)
sigma_R^2 is increased over a wider range of radii compared to that seen with
one spiral pattern; (4) particles diffuse radially in real space whereas they
don't when only one spiral density wave is present. Near the CR with the
stronger, 2-armed pattern, test particles are observed to migrate radially.
These effects take place at or near resonances of both spirals so we interpret
them as the result of stochastic motions. This provides a possible new
mechanism for increasing the stellar velocity dispersion in galactic disks. If
multiple spiral patterns are present in the Galaxy we predict that there should
be large variations in the stellar velocity dispersion as a function of radius. | nlin_CD |
Singular continuous spectra in a pseudo-integrable billiard: The pseudo-integrable barrier billiard invented by Hannay and McCraw [J.
Phys. A 23, 887 (1990)] -- rectangular billiard with line-segment barrier
placed on a symmetry axis -- is generalized. It is proven that the flow on
invariant surfaces of genus two exhibits a singular continuous spectral
component. | nlin_CD |
Localization Properties of Covariant Lyapunov Vectors: The Lyapunov exponent spectrum and covariant Lyapunov vectors are studied for
a quasi-one-dimensional system of hard disks as a function of density and
system size. We characterize the system using the angle distributions between
covariant vectors and the localization properties of both Gram-Schmidt and
covariant vectors. At low density there is a {\it kinetic regime} that has
simple scaling properties for the Lyapunov exponents and the average
localization for part of the spectrum. This regime shows strong localization in
a proportion of the first Gram-Schmidt and covariant vectors and this can be
understood as highly localized configurations dominating the vector. The
distribution of angles between neighbouring covariant vectors has
characteristic shapes depending upon the difference in vector number, which
vary over the continuous region of the spectrum. At dense gas or liquid like
densities the behaviour of the covariant vectors are quite different. The
possibility of tangencies between different components of the unstable manifold
and between the stable and unstable manifolds is explored but it appears that
exact tangencies do not occur for a generic chaotic trajectory. | nlin_CD |
Transport in Hamiltonian systems with slowly changing phase space
structure: Transport in Hamiltonian systems with weak chaotic perturbations has been
much studied in the past. In this paper, we introduce a new class of problems:
transport in Hamiltonian systems with slowly changing phase space structure
that are not order one perturbations of a given Hamiltonian. This class of
problems is very important for many applications, for instance in celestial
mechanics. As an example, we study a class of one-dimensional Hamiltonians that
depend explicitly on time and on stochastic external parameters. The variations
of the external parameters are responsible for a distortion of the phase space
structures: chaotic, weakly chaotic and regular sets change with time. We show
that theoretical predictions of transport rates can be made in the limit where
the variations of the stochastic parameters are very slow compared to the
Hamiltonian dynamics. Exact asymptotic results can be obtained in the classical
case where the Hamiltonian dynamics is integrable for fixed values of the
parameters. For the more interesting chaotic Hamiltonian dynamics case, we show
that two mechanisms contribute to the transport. For some range of the
parameter variations, one mechanism -called transport by migration together
with the mixing regions - is dominant. We are then able to model transport in
phase space by a Markov model, the local diffusion model, and to give
reasonably good transport estimates. | nlin_CD |
Poincare recurrences from the perspective of transient chaos: We obtain a description of the Poincar\'e recurrences of chaotic systems in
terms of the ergodic theory of transient chaos. It is based on the equivalence
between the recurrence time distribution and an escape time distribution
obtained by leaking the system and taking a special initial ensemble. This
ensemble is atypical in terms of the natural measure of the leaked system, the
conditionally invariant measure. Accordingly, for general initial ensembles,
the average recurrence and escape times are different. However, we show that
the decay rate of these distributions is always the same. Our results remain
valid for Hamiltonian systems with mixed phase space and validate a split of
the chaotic saddle in hyperbolic and non-hyperbolic components. | nlin_CD |
Quasiclassical Born-Oppenheimer approximations: We discuss several problems in quasiclassical physics for which approximate
solutions were recently obtained by a new method, and which can also be solved
by novel versions of the Born-Oppenheimer approximation. These cases include
the so-called bouncing ball modes, low angular momentum states in perturbed
circular billiards, resonant states in perturbed rectangular billiards, and
whispering gallery modes. Some rare, special eigenstates, concentrated close to
the edge or along a diagonal of a nearly rectangular billiard are found. This
kind of state has apparently previously escaped notice. | nlin_CD |
The effect of noise on a hyperbolic strange attractor in the system of
two coupled van der Pol oscillators: We study the effect of noise for a physically realizable flow system with a
hyperbolic chaotic attractor of the Smale - Williams type in the Poincare
cross-section [S.P. Kuznetsov, Phys. Rev. Lett. 95, 2005, 144101]. It is shown
numerically that slightly varying the initial conditions on the attractor one
can obtain a uniform approximation of a noisy orbit by the trajectory of the
system without noise, that is called as the "shadowing" trajectory. We propose
an algorithm for locating the shadowing trajectories in the system under
consideration. Using this algorithm, we show that the mean distance between a
noisy orbit and the approximating one does not depend essentially on the length
of the time interval of observation, but only on the noise intensity. This
dependance is nearly linear in a wide interval of the intensities of noise. It
is found out that for weak noise the Lyapunov exponents do not depend
noticeably on the noise intensity. However, in the case of a strong noise the
largest Lyapunov exponent decreases and even becomes negative indicating the
suppression of chaos by the external noise. | nlin_CD |
Entropic comparison of Landau-Zener and Demkov interactions in the phase
space of a quadrupole billiard: We investigate two types of avoided crossings in a chaotic billiard within
the framework of information theory. The Shannon entropy in the phase space for
the Landau--Zener interaction increases as the center of the avoided crossing
is approached. Meanwhile, that for the Demkov interaction decreases as the
center of avoided crossing is passed by with an increase in the deformation
parameter. This feature can provide a new indicator for scar formation. In
addition, it is found that the Fisher information of the Landau--Zener
interaction is significantly larger than that of the Demkov interaction. | nlin_CD |
Behavior of Dynamical Systems in the Regime of Transient Chaos: The transient chaos regime in a two-dimensional system with discrete time
(Eno map) is considered. It is demonstrated that a time series corresponding to
this regime differs from a chaotic series constructed for close values of the
control parameters by the presence of "nonregular" regions, the number of which
increases with the critical parameter. A possible mechanism of this effect is
discussed. | nlin_CD |
The largest Lyapunov exponent as a tool for detecting relative changes
in the particle positions: Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable
substrate potential is examined by analyzing response function, the largest
Lyapunov exponent and Poincar\'{e} sections for two neighboring particles. The
obtained results show that the largest Lyapunov exponent, besides being used
for investigating integral quantities, can be used for detecting microchanges
in chain configuration of both damped Frenkel-Kontorova model with inertial
term and its strictly overdamped limit. Slight changes in relative positions of
the particles are registered through jumps of the largest Lyapunov exponent in
the pinning regime. The occurrence of such jumps is highly dependent on type of
commensurate structure and deformation of substrate potential. The obtained
results also show that the minimal force required to initiate collective motion
of the chain is not dependent on the number of Lyapunov exponent jumps in the
pinning regime. These jumps are also registered in the sliding regime, where
they are a consequence of a more complex structure of largest Lyapunov exponent
on the step. | nlin_CD |
Zeta Function Zeros, Powers of Primes, and Quantum Chaos: We present a numerical study of Riemann's formula for the oscillating part of
the density of the primes and their powers. The formula is comprised of an
infinite series of oscillatory terms, one for each zero of the zeta function on
the critical line and was derived by Riemann in his paper on primes assuming
the Riemann hypothesis. We show that high resolution spectral lines can be
generated by the truncated series at all powers of primes and demonstrate
explicitly that the relative line intensities are correct. We then derive a
Gaussian sum rule for Riemann's formula. This is used to analyze the numerical
convergence of the truncated series. The connections to quantum chaos and
semiclassical physics are discussed. | nlin_CD |
Determinism, Complexity, and Predictability in Computer Performance: Computers are deterministic dynamical systems (CHAOS 19:033124, 2009). Among
other things, that implies that one should be able to use deterministic
forecast rules to predict their behavior. That statement is sometimes-but not
always-true. The memory and processor loads of some simple programs are easy to
predict, for example, but those of more-complex programs like compilers are
not. The goal of this paper is to determine why that is the case. We conjecture
that, in practice, complexity can effectively overwhelm the predictive power of
deterministic forecast models. To explore that, we build models of a number of
performance traces from different programs running on different Intel-based
computers. We then calculate the permutation entropy-a temporal entropy metric
that uses ordinal analysis-of those traces and correlate those values against
the prediction success | nlin_CD |
Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron
models: We have identified the occurrence of chimera states for various coupling
schemes in networks of two-dimensional and three-dimensional Hindmarsh-Rose
oscillators, which represent realistic models of neuronal ensembles. This
result, together with recent studies on multiple chimera states in nonlocally
coupled FitzHugh-Nagumo oscillators, provide strong evidence that the
phenomenon of chimeras may indeed be relevant in neuroscience applications.
Moreover, our work verifies the existence of chimera states in coupled bistable
elements, whereas to date chimeras were known to arise in models possessing a
single stable limit cycle. Finally, we have identified an interesting class of
mixed oscillatory states, in which desynchronized neurons are uniformly
interspersed among the remaining ones that are either stationary or oscillate
in synchronized motion. | nlin_CD |
Bichromatically driven double well: parametric perspective of the
strong-field control landscape reveals the influence of chaotic states: The aim of this work is to understand the influence of chaotic states in
control problems involving strong fields. Towards this end, we numerically
construct and study the strong field control landscape of a bichromatically
driven double well. A novel measure based on correlating the overlap
intensities between Floquet states and an initial phase space coherent state
with the parametric motion of the quasienergies is used to construct and
interpret the landscape features. "Walls" of no control, robust under
variations of the relative phase between the fields, are seen on the control
landscape and associated with multilevel interactions involving chaotic Floquet
states. | nlin_CD |
Chaotic flow and efficient mixing in a micro-channel with a polymer
solution: Microscopic flows are almost universally linear, laminar and stationary
because Reynolds number, $Re$, is usually very small. That impedes mixing in
micro-fluidic devices, which sometimes limits their performance. Here we show
that truly chaotic flow can be generated in a smooth micro-channel of a uniform
width at arbitrarily low $Re$, if a small amount of flexible polymers is added
to the working liquid. The chaotic flow regime is characterized by randomly
fluctuating three-dimensional velocity field and significant growth of the flow
resistance. Although the size of the polymer molecules extended in the flow may
become comparable with the micro-channel width, the flow behavior is fully
compatible with that in a table-top channel in the regime of elastic
turbulence. The chaotic flow leads to quite efficient mixing, which is almost
diffusion independent. For macromolecules, mixing time in this microscopic flow
can be three to four orders of magnitude shorter than due to molecular
diffusion. | nlin_CD |
Measures of Anisotropy and the Universal Properties of Turbulence: Local isotropy, or the statistical isotropy of small scales, is one of the
basic assumptions underlying Kolmogorov's theory of universality of small-scale
turbulent motion. While, until the mid-seventies or so, local isotropy was
accepted as a plausible approximation at high enough Reynolds numbers, various
empirical observations that have accumulated since then suggest that local
isotropy may not obtain at any Reynolds number. These notes examine in some
detail the isotropic and anisotropic contributions to structure functions by
considering their SO(3) decomposition. Viewed in terms of the relative
importance of the isotropic part to the anisotropic parts of structure
functions, the basic conclusion is that the isotropic part dominates the small
scales at least up to order 6. This follows from the fact that, at least up to
that order, there exists a hierarchy of increasingly larger power-law
exponents, corresponding to increasingly higher-order anisotropic sectors of
the SO(3) decomposition. The numerical values of the exponents deduced from
experiment suggest that the anisotropic parts in each order roll off less
sharply than previously thought by dimensional considerations, but they do so
nevertheless. | nlin_CD |
A non-autonomous flow system with Plykin type attractor: A non-autonomous flow system is introduced with an attractor of Plykin type
that may serve as a base for elaboration of real systems and devices
demonstrating the structurally stable chaotic dynamics. The starting point is a
map on a two-dimensional sphere, consisting of four stages of continuous
geometrically evident transformations. The computations indicate that in a
certain parameter range the map has a uniformly hyperbolic attractor. It may be
represented on a plane by means of a stereographic projection. Accounting
structural stability, a modification of the model is undertaken to obtain a set
of two non-autonomous differential equations of the first order with smooth
coefficients. As follows from computations, it has the Plykin type attractor in
the Poincar\'{e} cross-section. | nlin_CD |
Dependence of heat transport on the strength and shear rate of
prescribed circulating flows: We study numerically the dependence of heat transport on the maximum velocity
and shear rate of physical circulating flows, which are prescribed to have the
key characteristics of the large-scale mean flow observed in turbulent
convection. When the side-boundary thermal layer is thinner than the viscous
boundary layer, the Nusselt number (Nu), which measures the heat transport,
scales with the normalized shear rate to an exponent 1/3. On the other hand,
when the side-boundary thermal layer is thicker, the dependence of Nu on the
Peclet number, which measures the maximum velocity, or the normalized shear
rate when the viscous boundary layer thickness is fixed, is generally not a
power law. Scaling behavior is obtained only in an asymptotic regime. The
relevance of our results to the problem of heat transport in turbulent
convection is also discussed. | nlin_CD |
On near integrability of some impact systems: A class of Hamiltonian impact systems exhibiting smooth near integrable
behavior is presented. The underlying unperturbed model investigated is an
integrable, separable, 2 degrees of freedom mechanical impact system with
effectively bounded energy level sets and a single straight wall which
preserves the separable structure. Singularities in the system appear either as
trajectories with tangent impacts or as singularities in the underlying
Hamiltonian structure (e.g. separatrices). It is shown that away from these
singularities, a small perturbation from the integrable structure results in
smooth near integrable behavior. Such a perturbation may occur from a small
deformation or tilt of the wall which breaks the separability upon impact, the
addition of a small regular perturbation to the system, or the combination of
both. In some simple cases explicit formulae to the leading order term in the
near integrable return map are derived. Near integrability is also shown to
persist when the hard billiard boundary is replaced by a singular, smooth,
steep potential, thus extending the near-integrability results beyond the scope
of regular perturbations. These systems constitute an additional class of
examples of near integrable impact systems, beyond the traditional one
dimensional oscillating billiards, nearly elliptic billiards, and the
near-integrable behavior near the boundary of convex smooth billiards with or
without magnetic field. | nlin_CD |
Delay sober up drunkers: Control of diffusion in random walkers: Time delay in general leads to instability in some systems, while a specific
feedback with delay can control fluctuated motion in nonlinear deterministic
systems to a stable state. In this paper, we consider a non-stationary
stochastic process, i.e., a random walk and observe its diffusion phenomenon
with time delayed feedback. Surprisingly, the diffusion coefficient decreases
with increasing the delay time. We analytically illustrate this suppression of
diffusion by using stochastic delay differential equations and justify the
feasibility of this suppression by applying the time-delay feedback to a
molecular dynamics model. | nlin_CD |
Modeling Kelvin Wave Cascades in Superfluid Helium: We study two different types of simplified models for Kelvin wave turbulence
on quantized vortex lines in superfluids near zero temperature. Our first model
is obtained from a truncated expansion of the Local Induction Approximation
(Truncated-LIA) and it is shown to possess the same scalings and the essential
behaviour as the full Biot-Savart model, being much simpler than the latter
and, therefore, more amenable to theoretical and numerical investigations. The
Truncated-LIA model supports six-wave interactions and dual cascades, which are
clearly demonstrated via the direct numerical simulation of this model in the
present paper. In particular, our simulations confirm presence of the weak
turbulence regime and the theoretically predicted spectra for the direct energy
cascade and the inverse wave action cascade. The second type of model we study,
the Differential Approximation Model (DAM), takes a further drastic
simplification by assuming locality of interactions in $k$-space via a
differential closure that preserves the main scalings of the Kelvin wave
dynamics. DAMs are even more amenable to study and they form a useful tool by
providing simple analytical solutions in the cases when extra physical effects
are present, e.g. forcing by reconnections, friction dissipation and phonon
radiation. We study these models numerically and test their theoretical
predictions, in particular the formation of the stationary spectra, and the
closeness of the numerics for the higher-order DAM to the analytical
predictions for the lower-order DAM . | nlin_CD |
Sensitivity of long periodic orbits of chaotic systems: The properties of long, numerically-determined periodic orbits of two
low-dimensional chaotic systems, the Lorenz equations and the
Kuramoto-Sivashinsky system in a minimal-domain configuration, are examined.
The primary question is to establish whether the sensitivity of period averaged
quantities with respect to parameter perturbations computed over long orbits
can be used as a sufficiently good proxy for the response of the chaotic state
to finite-amplitude parameter perturbations. To address this question, an
inventory of thousands of orbits at least two orders of magnitude longer than
the shortest admissible cycles is constructed. The expectation of period
averages, Floquet exponents and sensitivities over such set is then obtained.
It is shown that all these quantities converge to a limiting value as the orbit
period is increased. However, while period averages and Floquet exponents
appear to converge to analogous quantities computed from chaotic trajectories,
the limiting value of the sensitivity is not necessarily consistent with the
response of the chaotic state, similar to observations made with other
shadowing algorithms. | nlin_CD |
Energy and potential enstrophy flux constraints in the two-layer
quasi-geostrophic model: We investigate an inequality constraining the energy and potential enstrophy
flux in the two-layer quasi-geostrophic model. This flux inequality is
unconditionally satisfied for the case of two-dimensional Navier-Stokes
turbulence. However, it is not obvious that it remains valid under the
multi-layer quasi-geostrophic model. The physical significance of this
inequality is that it decides whether any given model can reproduce the
Nastrom-Gage spectrum of the atmosphere, at least in terms of the total energy
spectrum. We derive the general form of the energy and potential enstrophy
dissipation rate spectra for a generalized multi-layer model. We then
specialize these results for the case of the two-layer quasi-geostrophic model
under dissipation configurations in which the dissipation terms for each layer
are dependent only on the streamfunction or potential vorticity of that layer.
We derive sufficient conditions for satisfying the flux inequality and discuss
the possibility of violating it under different conditions. | nlin_CD |
Ergodicity, mixing and recurrence in the three rotor problem: In the classical three rotor problem, three equal point masses move on a
circle subject to attractive cosine potentials of strength g. In the center of
mass frame, energy E is the only known conserved quantity. In earlier work
[Krishnaswami and Senapati, arXiv:1810.01317, Oct. 2018, arXiv:1811.05807, Nov.
2018], an order-chaos-order transition was discovered in this system along with
a band of global chaos for 5.33g < E < 5.6g. Here, we provide numerical
evidence for ergodicity and mixing in this band. The distributions of relative
angles and angular momenta along generic trajectories are shown to approach the
corresponding distributions over constant energy hypersurfaces (weighted by the
Liouville measure) as a power-law in time. Moreover, trajectories emanating
from a small volume are shown to become uniformly distributed over constant
energy hypersurfaces, indicating that the dynamics is mixing. Outside this
band, ergodicity and mixing fail, though the distributions of angular momenta
over constant energy hypersurfaces show interesting phase transitions from
Wignerian to bimodal with increasing energy. Finally, in the band of global
chaos, the distribution of recurrence times to finite size cells is found to
follow an exponential law with the mean recurrence time satisfying a scaling
law involving an exponent consistent with global chaos and ergodicity. | nlin_CD |
Intermittency and Universality in Fully-Developed Inviscid and
Weakly-Compressible Turbulent Flows: We performed high resolution numerical simulations of homogenous and
isotropic compressible turbulence, with an average 3D Mach number close to 0.3.
We study the statistical properties of intermittency for velocity, density and
entropy. For the velocity field, which is the primary quantity that can be
compared to the isotropic incompressible case, we find no statistical
differences in its behavior in the inertial range due either to the slight
compressibility or to the different dissipative mechanism. For the density
field, we find evidence of ``front-like'' structures, although no shocks are
produced by the simulation. | nlin_CD |
Dynamical tunneling and control: This article summarizes the recent work on the influence of dynamical
tunneling on the control of quantum systems. Specifically, two examples are
discussed. In the first, it is shown that the bichromatic control of tunneling
in a driven double well system is hampered by the phenomenon of chaos-assisted
tunneling. The bichromatic control landscape exhibits several regions
indicating lack of control with every such region involving chaos-assisted
tunneling. The second example illustrates the failure of controlling the
dissociation dynamics of a driven Morse oscillator due to the phenomenon of
resonance-assisted tunneling. In particular, attempts to control the
dissociation dynamics by rebuilding local phase space barriers are foiled due
to resonance-assisted tunneling. | nlin_CD |
Generalized Amplitude Truncation of Gaussian 1/f^alpha noise: We study a kind of filtering, an amplitude truncation with upper and lower
truncation levels x_max and x_min. This is a generalization of the simple
transformation y(t)=sgn[x(t)], for which a rigorous result was obtained
recently. So far numerical experiments have shown that a power law spectrum
1/f^alpha seems to be transformed again into a power law spectrum 1/f^beta
under rather general condition for the truncation levels. We examine the above
numerical results analytically. When 1<alpha<2 and x_max = -x_min = a, the
transformed spectrum is shown to be characterized by a certain corner frequency
f_c which divides the spectrum into two parts with different exponents. We
derive f_c depending on a as f_c sim a^(-2/(alpha-1)). It turns out that the
output signal should deviate from the power law spectrum when the truncation is
asymmetrical. We present a numerical example such that 1/f^2 noise converges to
1/f noise by applying the transformation y(t)=sgn[x(t)] repeatedly. | nlin_CD |
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