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Memristor Circuits for Simulating Nonlinear Dynamics and Their Periodic
Forcing: In this paper, we show that the dynamics of a wide variety of nonlinear
systems such as engineering, physical, chemical, biological, and ecological
systems, can be simulated or modeled by the dynamics of memristor circuits. It
has the advantage that we can apply nonlinear circuit theory to analyze the
dynamics of memristor circuits. Applying an external source to these memristor
circuits, they exhibit complex behavior, such as chaos and non-periodic
oscillation. If the memristor circuits have an integral invariant, they can
exhibit quasi-periodic or non-periodic behavior by the sinusoidal forcing.
Their behavior greatly depends on the initial conditions, the parameters, and
the maximum step size of the numerical integration. Furthermore, an overflow is
likely to occur due to the numerical instability in long-time simulations. In
order to generate a non-periodic oscillation, we have to choose the initial
conditions, the parameters, and the maximum step size, carefully. We also show
that we can reconstruct chaotic attractors by using the terminal voltage and
current of the memristor. Furthermore, in many memristor circuits, the active
memristor switches between passive and active modes of operation, depending on
its terminal voltage. We can measure its complexity order by defining the
binary coding for the operation modes. By using this coding, we show that in
the forced memristor Toda lattice equations, the memristor's operation modes
exhibit the higher complexity. Furthermore, in the memristor Chua circuit, the
memristor has the special operation modes. | nlin_CD |
Mode selectivity of dynamically induced conformation in many-body
chain-like bead-spring model: We consider conformation of a chain consisting of beads connected by stiff
springs, where the conformation is determined by the bending angles between the
consecutive two springs. A conformation is stabilized or destabilized not only
by a given bending potential but also the fast spring motion, and stabilization
by the spring motion depends on their excited normal modes. This stabilization
mechanism has been named the dynamically induced conformation in a previous
work on a three-body system. We extend analyses of the dynamically induced
conformation in many-body chain-like bead-spring systems. The normal modes of
the springs depend on the conformation, and the simple rule of the dynamical
stabilization is that the lowest eigenfrequency mode contributes to the
stabilization of the conformation. The high the eigenfrequency is, the more the
destabilization emerges. We verify theoretical predictions by performing
numerical simulations. | nlin_CD |
Recurrence-time statistics in non-Hamiltonian volume preserving maps and
flows: We analyze the recurrence-time statistics (RTS) in three-dimensional
non-Hamiltonian volume preserving systems (VPS): an extended standard map, and
a fluid model. The extended map is a standard map weakly coupled to an
extra-dimension which contains a deterministic regular, mixed (regular and
chaotic) or chaotic motion. The extra-dimension strongly enhances the trapping
times inducing plateaus and distinct algebraic and exponential decays in the
RTS plots. The combined analysis of the RTS with the classification of ordered
and chaotic regimes and scaling properties, allows us to describe the intricate
way trajectories penetrate the before impenetrable regular islands from the
uncoupled case. Essentially the plateaus found in the RTS are related to
trajectories that stay long times inside trapping tubes, not allowing
recurrences, and then penetrates diffusively the islands (from the uncoupled
case) by a diffusive motion along such tubes in the extra-dimension. All
asymptotic exponential decays for the RTS are related to an ordered regime
(quasi-regular motion) and a mixing dynamics is conjectured for the model.
These results are compared to the RTS of the standard map with dissipation or
noise, showing the peculiarities obtained by using three-dimensional VPS. We
also analyze the RTS for a fluid model and show remarkable similarities to the
RTS in the extended standard map problem. | nlin_CD |
Sublattice synchronization of chaotic networks with delayed couplings: Synchronization of chaotic units coupled by their time delayed variables are
investigated analytically. A new type of cooperative behavior is found:
sublattice synchronization. Although the units of one sublattice are not
directly coupled to each other, they completely synchronize without time delay.
The chaotic trajectories of different sublattices are only weakly correlated
but not related by generalized synchronization. Nevertheless, the trajectory of
one sublattice is predictable from the complete trajectory of the other one.
The spectra of Lyapunov exponents are calculated analytically in the limit of
infinite delay times, and phase diagrams are derived for different topologies. | nlin_CD |
Amplitude and phase dynamics in oscillators with distributed-delay
coupling: This paper studies the effects of distributed delay coupling on the dynamics
in a system of non-identical coupled Stuart-Landau oscillators. For uniform and
gamma delay distribution kernels, conditions for amplitude death are obtained
in terms of average frequency, frequency detuning and parameters of the
coupling, including coupling strength and phase, as well as the mean time delay
and the width of the delay distribution. To gain further insight into the
dynamics inside amplitude death regions, eigenvalues of the corresponding
characteristic equations are computed numerically. Oscillatory dynamics of the
system is also investigated using amplitude and phase representation. Various
branches of phase-locked solutions are identified, and their stability is
analysed for different types of delay distributions. | nlin_CD |
Variational Principles for Lagrangian Averaged Fluid Dynamics: The Lagrangian average (LA) of the ideal fluid equations preserves their
transport structure. This transport structure is responsible for the Kelvin
circulation theorem of the LA flow and, hence, for its convection of potential
vorticity and its conservation of helicity.
Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations. | nlin_CD |
Experimental investigation of nodal domains in the chaotic microwave
rough billiard: We present the results of experimental study of nodal domains of wave
functions (electric field distributions) lying in the regime of Shnirelman
ergodicity in the chaotic half-circular microwave rough billiard. Nodal domains
are regions where a wave function has a definite sign. The wave functions Psi_N
of the rough billiard were measured up to the level number N=435. In this way
the dependence of the number of nodal domains \aleph_N on the level number $N$
was found. We show that in the limit N->infty a least squares fit of the
experimental data reveals the asymptotic number of nodal domains aleph_N/N =
0.058 +- 0.006 that is close to the theoretical prediction aleph_N/N +- 0.062.
We also found that the distributions of the areas s of nodal domains and their
perimeters l have power behaviors n_s ~ s^{-tau} and n_l ~ l^{-tau'}, where
scaling exponents are equal to \tau = 1.99 +- 0.14 and \tau'=2.13 +- 0.23,
respectively. These results are in a good agreement with the predictions of
percolation theory. Finally, we demonstrate that for higher level numbers N =
220-435 the signed area distribution oscillates around the theoretical limit
Sigma_{A} = 0.0386 N^{-1}. | nlin_CD |
On scaling and statistical geometry in passive scalar turbulence: We show that the statistics of a turbulent passive scalar at scales larger
than the pumping may exhibit multiscaling due to a weaker mechanism than the
presence of statistical conservation laws. We develop a general formalism to
give explicit predictions for the large scale scaling exponents in the case of
the Kraichnan model and discuss their geometric origin at small and large
scale. | nlin_CD |
A consistent approach for the treatment of Fermi acceleration in
time-dependent billiards: The standard description of Fermi acceleration, developing in a class of
time-dependent billiards, is given in terms of a diffusion process taking place
in momentum space. Within this framework the evolution of the probability
density function (PDF) of the magnitude of particle velocities as a function of
the number of collisions $n$ is determined by the Fokker-Planck equation (FPE).
In the literature the FPE is constructed by identifying the transport
coefficients with the ensemble averages of the change of the magnitude of
particle velocity and its square in the course of one collision. Although this
treatment leads to the correct solution after a sufficiently large number of
collisions has been reached, the transient part of the evolution of the PDF is
not described. Moreover, in the case of the Fermi-Ulam model (FUM), if a
stadanrd simplification is employed, the solution of the FPE is even
inconsistent with the values of the transport coefficients used for its
derivation. The goal of our work is to provide a self-consistent methodology
for the treatment of Fermi acceleration in time-dependent billiards. The
proposed approach obviates any assumptions for the continuity of the random
process and the existence of the limits formally defining the transport
coefficients of the FPE. Specifically, we suggest, instead of the calculation
of ensemble averages, the derivation of the one-step transition probability
function and the use of the Chapman-Kolmogorov forward equation. This approach
is generic and can be applied to any time-dependent billiard for the treatment
of Fermi-acceleration. As a first step, we apply this methodology to the FUM,
being the archetype of time-dependent billiards to exhibit Fermi acceleration. | nlin_CD |
Universality in spectral statistics of "open" quantum graphs: The unitary evolution maps in closed chaotic quantum graphs are known to have
universal spectral correlations, as predicted by random matrix theory. In
chaotic graphs with absorption the quantum maps become non-unitary. We show
that their spectral statistics exhibit universality at the "soft" edges of the
spectrum. The same spectral behavior is observed in many classical non-unitary
ensembles of random matrices with rotationally invariant measures. | nlin_CD |
Two-cluster regular states, chimeras and hyperchaos in a system of
globally coupled phase oscillators with inertia: In this work, two-cluster modes are studied in a system of globally coupled
Kuramoto-Sakaguchi phase oscillators with inertia. It is shown that these
regimes can be of two types: with a constant intercluster phase difference
rotating at the same frequency (according to the analysis, such regimes are
always unstable) and with a periodically changing (taking into account the
multiplicity of $2\pi$) phase mismatch. The issues of existence and stability,
emergence and destruction of two-cluster modes are studied depending on the
parameters: effective mass (responsible for inertial processes in the model
system under consideration) and phase shift in the coupling function. The
analytical results are confirmed and supplemented by numerical simulation of
the rotators (second order) interacting globally through the mean field. | nlin_CD |
Circular, elliptic and oval billiards in a gravitational field: We consider classical dynamical properties of a particle in a constant
gravitational force and making specular reflections with circular, elliptic or
oval boundaries. The model and collision map are described and a detailed study
of the energy regimes is made. The linear stability of fixed points is studied,
yielding exact analytical expressions for parameter values at which a
period-doubling bifurcation occurs. The dynamics is apparently ergodic at
certain energies in all three models, in contrast to the regularity of the
circular and elliptic billiard dynamics in the field-free case. This finding is
confirmed using a sensitive test involving Lyapunov weighted dynamics. In the
last part of the paper a time dependence is introduced in the billiard
boundary, where it is shown that for the circular billiard the average velocity
saturates for zero gravitational force but in the presence of gravitational it
increases with a very slow growth rate, which may be explained using Arnold
diffusion. For the oval billiard, where chaos is present in the static case,
the particle has an unlimited velocity growth with an exponent of approximately
1/6. | nlin_CD |
Regions of multistability in some low-dimensional logistic models with
excitation type coupling: A naive model of many networked logistic maps with an excitation type
coupling [Neural Networks, vol. 20, 102--108 (2007)], which is an extension of
other low dimensional models, has been recently proposed to mimic the
waking-sleeping bistability found in brain systems. Although the dynamics of
large and complex aggregates of elementary components can not be understood nor
extrapolated from the properties of a few components, some patterns of behavior
could be conserved independently of the topology and of the number of coupled
units. Following this insight, we have collected several of those systems where
a few logistic maps are coupled under a similar mutual excitation scheme. The
regions of bi- and multistability of these systems are sketched and reported. | nlin_CD |
Synchronization induced by intermittent versus partial drives in chaotic
systems: We show that the synchronized states of two systems of identical chaotic maps
subject to either, a common drive that acts with a probability p in time or to
the same drive acting on a fraction p of the maps, are similar. The
synchronization behavior of both systems can be inferred by considering the
dynamics of a single chaotic map driven with a probability p. The synchronized
states for these systems are characterized on their common space of parameters.
Our results show that the presence of a common external drive for all times is
not essential for reaching synchronization in a system of chaotic oscillators,
nor is the simultaneous sharing of the drive by all the elements in the system.
Rather, a crucial condition for achieving synchronization is the sharing of
some minimal, average information by the elements in the system over long
times. | nlin_CD |
Statistics of quantum recurrences in the Hilbert space: This paper has been withdrawn by the authors due to a crucial error. | nlin_CD |
Parameter Estimation in Models with Complex Dynamics: Mathematical models of real life phenomena are highly nonlinear involving
multiple parameters and often exhibiting complex dynamics. Experimental data
sets are typically small and noisy, rendering estimation of parameters from
such data unreliable and difficult. This paper presents a study of the Bayesian
posterior distribution for unknown parameters of two chaotic discrete dynamical
systems conditioned on observations of the system. The study shows how the
qualitative properties of the posterior are affected by the intrinsic noise
present in the data, the representation of this noise in the parameter
estimation process, and the length of the data-set. The results indicate that
increasing length of dataset does not significantly increase the precision of
the estimate, and this is true for both periodic and chaotic data. On the other
hand, increasing precision of the measurements leads to significant increase in
precision of the estimated parameter in case of periodic data, but not in the
case of chaotic data. These results are highly useful in designing laboratory
and field-based studies in biology in general, and ecology and conservation in
particular. | nlin_CD |
Experimental realization of a highly secure chaos communication under
strong channel noise: A one-way coupled spatiotemporally chaotic map lattice is used to contruct
cryptosystem. With the combinatorial applications of both chaotic computations
and conventional algebraic operations, our system has optimal cryptographic
properties much better than the separative applications of known chaotic and
conventional methods. We have realized experiments to pratice duplex voice
secure communications in realistic Wired Public Switched Telephone Network by
applying our chaotic system and the system of Advanced Encryption Standard
(AES), respectively, for cryptography. Our system can work stably against
strong channel noise when AES fails to work. | nlin_CD |
A Tube Dynamics Perspective Governing Stability Transitions: An Example
Based on Snap-through Buckling: The equilibrium configuration of an engineering structure, able to withstand
a certain loading condition, is usually associated with a local minimum of the
underlying potential energy. However, in the nonlinear context, there may be
other equilibria present, and this brings with it the possibility of a
transition to an alternative (remote) minimum. That is, given a sufficient
disturbance, the structure might buckle, perhaps suddenly, to another shape.
This paper considers the dynamic mechanisms under which such transitions
(typically via saddle points) occur. A two-mode Hamiltonian is developed for a
shallow arch/buckled beam. The resulting form of the potential energy---two
stable wells connected by rank-1 saddle points---shows an analogy with
resonance transitions in celestial mechanics or molecular reconfigurations in
chemistry, whereas here the transition corresponds to switching between two
stable structural configurations. Then, from Hamilton's equations, the
analytical equilibria are determined and linearization of the equations of
motion about the saddle is obtained. After computing the eigenvalues and
eigenvectors of the coefficient matrix associated with the linearization, a
symplectic transformation is given which puts the Hamiltonian into normal form
and simplifies the equations, allowing us to use the conceptual framework known
as tube dynamics. The flow in the equilibrium region of phase space as well as
the invariant manifold tubes in position space are discussed. Also, we account
for the addition of damping in the tube dynamics framework, which leads to a
richer set of behaviors in transition dynamics than previously explored. | nlin_CD |
Optimal Phase Description of Chaotic Oscillators: We introduce an optimal phase description of chaotic oscillations by
generalizing the concept of isochrones. On chaotic attractors possessing a
general phase description, we define the optimal isophases as Poincar\'e
surfaces showing return times as constant as possible. The dynamics of the
resultant optimal phase is maximally decoupled of the amplitude dynamics, and
provides a proper description of phase resetting of chaotic oscillations. The
method is illustrated with the R\"ossler and Lorenz systems. | nlin_CD |
A stochastic model of cascades in 2D turbulence: The dual cascade of energy and enstrophy in 2D turbulence cannot easily be
understood in terms of an analog to the Richardson-Kolmogorov scenario
describing the energy cascade in 3D turbulence. The coherent up- and downscale
fluxes points to non-locality of interactions in spectral space, and thus the
specific spatial structure of the flow could be important. Shell models, which
lack spacial structure and have only local interactions in spectral space,
indeed fail in reproducing the correct scaling for the inverse cascade of
energy. In order to exclude the possibility that non-locality of interactions
in spectral space is crucial for the dual cascade, we introduce a stochastic
spectral model of the cascades which is local in spectral space and which shows
the correct scaling for both the direct enstrophy - and the inverse energy
cascade. | nlin_CD |
SICNNs with Li-Yorke Chaotic Outputs on a Time Scale: In the present study, we investigate the existence of Li-Yorke chaos in the
dynamics of shunting inhibitory cellular neural networks (SICNNs) on time
scales. It is rigorously proved by taking advantage of external inputs that the
outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported
by simulations, and the controllability of chaos on the time scale is
demonstrated by means of the Pyragas control technique. This is the first time
in the literature that the existence as well as the control of chaos are
provided for neural networks on time scales. | nlin_CD |
Wave Functions and Energy Spectra in Rational Billiards Are Determined
Completely by Their Periods: The rational billiards (RB) are classically pseudointegrable, i.e. their
trajectories in the phase space lie on multi-tori. Each such a multi-torus can
be unfolded into elementary polygon pattern (EPP). A rational billiards Riemann
surface (RBRS) corresponding to each RB is then an infinite mosaic made by a
periodic distribution of EPP. Periods of RBRS are directly related to periodic
orbits of RB. It is shown that any stationary solutions (SS) to the
Schr\"odinger equation (SE) in RB can be extended on the whole RBRS. The
extended stationary wave functions (ESS) are then periodic on RBRS with its
periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet
or the the Neumann ones or their mixture) consistent with EPP one can find so
called stationary pre-solutions (SPS) of the Schr\"odinger equation defined on
RBRS and respecting its periodic structure together with their energy spectra.
Using SPS one can easily construct SS of RB for most boundary conditions on it
by a trivial algebra over SPS. It proves therefore that the energy spectra
defined by the boundary conditions for SS corresponding to each RB are totally
determined by $2g$ independent periods of RBRS being homogeneous functions of
these periods. RBRS can be constructed exclusively due to the rationality of
the polygon billiards considered. Therefore the approach developed in the
present paper can be seen as a new way in obtaining SS to SE in RB. SPS can be
constructed explicitly for a class of RB which EPP can be decomposed into a set
of periodic orbit channel (POC) parallel to each other (POCDRB). For such a
class of RB the respective RBRS can be built as a standard multi-sheeted
Riemann surface with a periodic structure. For POCDRB a discussion of the
existence of the superscar states (SSS) can be done thoroughly. | nlin_CD |
Can One Hear the Shape of a Graph?: We show that the spectrum of the Schrodinger operator on a finite, metric
graph determines uniquely the connectivity matrix and the bond lengths,
provided that the lengths are non-commensurate and the connectivity is simple
(no parallel bonds between vertices and no loops connecting a vertex to
itself). That is, one can hear the shape of the graph! We also consider a
related inversion problem: A compact graph can be converted into a scattering
system by attaching to its vertices leads to infinity. We show that the
scattering phase determines uniquely the compact part of the graph, under
similar conditions as above. | nlin_CD |
Aerodynamics at the Particle Level: This paper is intended to clarify some of the rather well-known aerodynamic
phenomena. It is also intended to pique the interest of the layman as well as
the professional. All aerodynamic forces on a surface are caused by collisions
of fluid particles with the surface. While the standard approach to fluid
dynamics, which is founded on the fluid approximation, is effective in
providing a means of calculating various behavior and properties, it begs the
question of causality. The determination of the causes of many of the most
important aerodynamic effects requires a microscopic examination of the fluid
and of the surface with which it interacts. The Kutta-Joukowski theorem is
investigated from first physical principles. It is noted that the circulation
does not arise alone as a physical phenomenon, e.g. air doesn't flow forward
under a wing, but must be added to the translation flow. The circulation term
is necessary to take into account the vertical deflection of the air flow by a
wing. Various aerodynamic devices are discussed, e.g. rocket engine exhaust
diffuser and the perfume atomizer. The section on slurries discusses
pyroclastic flow as occurs in violent volcanic eruptions. | nlin_CD |
A new chaotic attractor in a basic multi-strain epidemiological model
with temporary cross-immunity: An epidemic multi-strain model with temporary cross-immunity shows chaos,
even in a previously unexpected parameter region. Especially dengue fever
models with strong enhanced infectivity on secondary infection have previously
shown deterministic chaos motivated by experimental findings of
antibody-dependent-enhancement (ADE). Including temporary cross-immunity in
such models, which is common knowledge among field researchers in dengue, we
find a deterministically chaotic attractor in the more realistic parameter
region of reduced infectivity on secondary infection (''inverse ADE'' parameter
region). This is realistic for dengue fever since on second infection people
are more likely to be hospitalized, hence do not contribute to the force of
infection as much as people with first infection.
Our finding has wider implications beyond dengue in any multi-strain
epidemiological systems with altered infectivity upon secondary infection,
since we can relax the condition of rather high infectivity on secondary
infection previously required for deterministic chaos. For dengue the finding
of wide ranges of chaotic attractors open new ways to analysis of existing data
sets. | nlin_CD |
Multistable jittering in oscillators with pulsatile delayed feedback: Oscillatory systems with time-delayed pulsatile feedback appear in various
applied and theoretical research areas, and received a growing interest in the
last years. For such systems, we report a remarkable scenario of
destabilization of a periodic regular spiking regime. In the bifurcation point
numerous regimes with non-equal interspike intervals emerge simultaneously. We
show that this bifurcation is triggered by the steepness of the oscillator's
phase resetting curve and that the number of the emerging, so-called
"jittering" regimes grows exponentially with the delay value. Although this
appears as highly degenerate from a dynamical systems viewpoint, the
"multi-jitter" bifurcation occurs robustly in a large class of systems. We
observe it not only in a paradigmatic phase-reduced model, but also in a
simulated Hodgkin-Huxley neuron model and in an experiment with an electronic
circuit. | nlin_CD |
Enstrophy dissipation in freely evolving two-dimensional turbulence: Freely decaying two-dimensional Navier--Stokes turbulence is studied. The
conservation of vorticity by advective nonlinearities renders a class of
Casimirs that decays under viscous effects. A rigorous constraint on the
palinstrophy production by nonlinear transfer is derived, and an upper bound
for the enstrophy dissipation is obtained. This bound depends only on the
decaying Casimirs, thus allowing the enstrophy dissipation to be bounded from
above in terms of initial data of the flows. An upper bound for the enstrophy
dissipation wavenumber is derived and the new result is compared with the
classical dissipation wavenumber. | nlin_CD |
A higher-dimensional generalization of the Lozi map: Bifurcations and
dynamics: We generalize the two dimensional Lozi map in order to systematically obtain
piece-wise continuous maps in three and higher dimensions. Similar to
higher-dimensional generalizations of the related Henon map, these
higher-dimensional Lozi maps support hyperchaotic dynamics. We carry out a
bifurcation analysis and investigate the dynamics through both numerical and
analytical means. The analysis is extended to a sequence of approximations that
smooth the discontinuity in the Lozi map. | nlin_CD |
Bifurcation without parameters in a chaotic system with a memristive
element: We investigate the effect of memory on a chaotic system experimentally and
theoretically. For this purpose, we use Chua's oscillator as an electrical
model system showing chaotic dynamics extended by a memory element in form of a
double-barrier memristive device. The device consists of
Au/NbO$_\text{x}$/Al$_\text{2}$O$_\text{3}$/Al/Nb layers and exhibits strong
analog-type resistive changes depending on the history of the charge flow. In
the extended system strong changes in the dynamics of chaotic oscillations are
observable. The otherwise fluctuating amplitudes of the Chua system are
disrupted by transient silent states. After developing a model for Chua's
oscillator with a memristive device, the numerical treatment reveals the
underling dynamics as driven by the slow-fast dynamics of the memory element.
Furthermore, the stabilizing and destabilizing dynamic bifurcations are
identified that are passed by the system during its chaotic behavior. | nlin_CD |
A saddle in a corner - a model of collinear triatomic chemical reactions: A geometrical model which captures the main ingredients governing atom-diatom
collinear chemical reactions is proposed. This model is neither near-integrable
nor hyperbolic, yet it is amenable to analysis using a combination of the
recently developed tools for studying systems with steep potentials and the
study of the phase space structure near a center-saddle equilibrium. The
nontrivial dependence of the reaction rates on parameters, initial conditions
and energy is thus qualitatively explained. Conditions under which the phase
space transition state theory assumptions are satisfied and conditions under
which these fail are derived. | nlin_CD |
Unstable dimension variability, heterodimensional cycles, and blenders
in the border-collision normal form: Chaotic attractors commonly contain periodic solutions with unstable
manifolds of different dimensions. This allows for a zoo of dynamical phenomena
not possible for hyperbolic attractors. The purpose of this Letter is to
demonstrate these phenomena in the border-collision normal form. This is a
continuous, piecewise-linear family of maps that is physically relevant as it
captures the dynamics created in border-collision bifurcations in diverse
applications. Since the maps are piecewise-linear they are relatively amenable
to an exact analysis and we are able to explicitly identify parameter values
for heterodimensional cycles and blenders. For a one-parameter subfamily we
identify bifurcations involved in a transition through unstable dimension
variability. This is facilitated by being able to compute periodic solutions
quickly and accurately, and the piecewise-linear form should provide a useful
test-bed for further study. | nlin_CD |
Acceleration statistics of heavy particles in turbulence: We present the results of direct numerical simulations of heavy particle
transport in homogeneous, isotropic, fully developed turbulence, up to
resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up
to 120 million particles with Stokes numbers, $St$, in the range from 0.16 to
3.5 we are able to characterize in full detail the statistics of particle
acceleration. We show that: ({\it i}) The root-mean-squared acceleration
$a_{\rm rms}$ sharply falls off from the fluid tracer value already at quite
small Stokes numbers; ({\it ii}) At a given $St$ the normalised acceleration
$a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently
with the trend observed for fluid tracers; ({\it iii}) The tails of the
probability density function of the normalised acceleration $a/a_{\rm rms}$
decrease with $St$. Two concurrent mechanisms lead to the above results:
preferential concentration of particles, very effective at small $St$, and
filtering induced by the particle response time, that takes over at larger
$St$. | nlin_CD |
On the fractal dimension of the Duffing attractor: The box counting dimension $d_C$ and the correlation dimension $d_G$ change
with the number of numerically generated points forming the attractor. At a
sufficiently large number of points the fractal dimension tends to a finite
value. The obtained values are $d_C\approx 1.43$ and $d_G\approx 1.38$. | nlin_CD |
Nonlinear elastic polymers in random flow: Polymer stretching in random smooth flows is investigated within the
framework of the FENE dumbbell model. The advecting flow is Gaussian and
short-correlated in time. The stationary probability density function of
polymer extension is derived exactly. The characteristic time needed for the
system to attain the stationary regime is computed as a function of the
Weissenberg number and the maximum length of polymers. The transient relaxation
to the stationary regime is predicted to be exceptionally slow in the proximity
of the coil-stretch transition. | nlin_CD |
Analysis of the chaos dynamics in(Xn,Xn+1)plane: in the last decade, studies of chaotic system are more often used for
classical choatic system than for quantum chaotic system, there are many ways
of observing the chaotic system such us analyzing the frequency with Fourier
transform or analyzing initial condition distance with Liapunov Exponent, this
paper explains dynamic chaotic process by observing trajectory of dynamic
system in (Xn,Xn+1) | nlin_CD |
Coherence resonance in an unijunction transistor relaxation oscillator: The phenomenon of coherence resonance (CR) is investigated in an unijunction
transistor relaxation oscillator (UJT-RO) and quantified by estimating the
normal variance (NV). Depending upon the measuring points two types of NV
curves have been obtained. We have observed that the degradations in coherency
at higher noise amplitudes in our system is probably the result of direct
interference of coherent oscillations and the stochastic perturbation.
Degradation of coherency may be minimal if this direct interference of noise
and coherent oscillations is eliminated. | nlin_CD |
A non subjective approach to the GP algorithm for analysing noisy time
series: We present an adaptation of the standard Grassberger-Proccacia (GP) algorithm
for estimating the Correlation Dimension of a time series in a non subjective
manner. The validity and accuracy of this approach is tested using different
types of time series, such as, those from standard chaotic systems, pure white
and colored noise and chaotic systems added with noise. The effectiveness of
the scheme in analysing noisy time series, particularly those involving colored
noise, is investigated. An interesting result we have obtained is that, for the
same percentage of noise addition, data with colored noise is more
distinguishable from the corresponding surrogates, than data with white noise.
As examples for real life applications, analysis of data from an astrophysical
X-ray object and human brain EEG, are presented. | nlin_CD |
Chaotic maps and flows: Exact Riemann-Siegel lookalike for spectral
fluctuations: To treat the spectral statistics of quantum maps and flows that are fully
chaotic classically, we use the rigorous Riemann-Siegel lookalike available for
the spectral determinant of unitary time evolution operators $F$. Concentrating
on dynamics without time reversal invariance we get the exact two-point
correlator of the spectral density for finite dimension $N$ of the matrix
representative of $F$, as phenomenologically given by random matrix theory. In
the limit $N\to\infty$ the correlator of the Gaussian unitary ensemble is
recovered. Previously conjectured cancellations of contributions of
pseudo-orbits with periods beyond half the Heisenberg time are shown to be
implied by the Riemann-Siegel lookalike. | nlin_CD |
Chaotic fluctuation of temperature on environmental interface exchanging
energy by visible and infrared radiation, convection and conduction: The concept of environmental interface is defined and analyzed from the point
of view of the possible source of non-standard behaviour. The energy balance
equation is written for the interface where all kinds of energy transfer occur.
It is shown that under certain conditions, the discrete version of the equation
for the temperature time rate turns in to the well-known logistic equation and
the conditions for chaotic behaviour are studied. They are determined by the
Lyapunov exponent. The realistic situation when the coefficients of the
equation vary with time, is studied for the Earth-environment general system. | nlin_CD |
On study of nonlinear viscoelastic behavior of red blood cell membrane: The linear viscoelastic behavior of the red blood cell membrane of mammal and
human was studied in previous works proposing different experimental methods to
determine their viscoelastic parameters. In the present work the nonlinear
component of dynamic viscosity of the red blood cell membrane by nonlinear time
series analysis is used. For such aim, it obtained time series of test in vitro
of samples of humans and rats red blood cells using the Erythrodeformeter in
oscillating regime. The signal filtrate suppresses any linear behavior as well
as represented by a system of linear ordinary differential equations. The test
shown as much in humans as in rats resonance frequencies associated to an
attractor of unknown nature independently of excitation in the physiological
range. The preliminary studies shown that attractor could be correspond to a
complex form bull. These results allow to extend the present knowledge on
dynamic of the cellular membrane to similar stimulus which happens in the blood
circulation and it will allows to make better models of the same one. | nlin_CD |
Ratchet transport and periodic structures in parameter space: Ratchet models are prominent candidates to describe the transport phenomenum
in nature in the absence of external bias. This work analyzes the parameter
space of a discrete ratchet model and gives direct connections between chaotic
domains and a family of isoperiodic stable structures with the ratchet current.
The isoperiodic structures appear along preferred direction in the parameter
space giving a guide to follow the current, which usually increases inside the
structures but is independent of the corresponding period. One of such
structures has the shrimp-shaped form which is known to be an universal
structure in the parameter space of dissipative systems. Currents in parameter
space provide a direct measure of the momentum asymmetry of the multistable and
chaotic attractors times the size of the corresponding basin of attraction.
Transport structures are shown to exist in the parameter space of the Langevin
equation with an external oscillating force. | nlin_CD |
Non-integrability of flail triple pendulum: We consider a special type of triple pendulum with two pendula attached to
end mass of another one. Although we consider this system in the absence of the
gravity, a quick analysis of of Poincar\'e cross sections shows that it is not
integrable. We give an analytic proof of this fact analysing properties the of
differential Galois group of variational equation along certain particular
solutions of the system. | nlin_CD |
Hidden Chaos: When a medium composed of microscopic elements is subjected to a high
intensity field, the individual behaviors of microscopic elements can become
chaotic. In such cases it is important to consider the effects of this
irregularity at microscopical level onto the macroscopic behavior of the
medium. We show that the macroscopic field produced by a large group of chaotic
scatterers can remain regular, due to the partial or complete phase coherence
of the scattering elements and the incoherence of the chaotic components of
their responses. Thus when only macroscopic fields are observed, one may be
unaware of chaotic microscopical motion, as it appears to be hidden from the
observer. The coupling among the elements may lead to partial chaos
synchronization, which exposes the chaotic nature of the system making the
oscillations of macroscopic fields more irregular. | nlin_CD |
Onset of Synchronization in the Disordered Hamiltonian Mean Field Model: We study the Hamiltonian Mean Field (HMF) model of coupled Hamiltonian rotors
with a heterogeneous distribution of moments of inertia and coupling strengths.
We show that when the parameters of the rotors are heterogeneous, finite size
fluctuations can greatly modify the coupling strength at which the incoherent
state loses stability by inducing correlations between the momenta and
parameters of the rotors. When the distribution of initial frequencies of the
oscillators is sufficiently narrow, an analytical expression for the
modification in critical coupling strength is obtained that confirms numerical
simulations. We find that heterogeneity in the moments of inertia tends to
stabilize the incoherent state, while heterogeneity in the coupling strengths
tends to destabilize the incoherent state. Numerical simulations show that
these effects disappear for a wide, bimodal frequency distribution. | nlin_CD |
Strong and Weak Chaos in Networks of Semiconductor Lasers with
Time-delayed Couplings: Nonlinear networks with time-delayed couplings may show strong and weak
chaos, depending on the scaling of their Lyapunov exponent with the delay time.
We study strong and weak chaos for semiconductor lasers, either with
time-delayed self-feedback or for small networks. We examine the dependence on
the pump current and consider the question whether strong and weak chaos can be
identified from the shape of the intensity trace, the auto-correlations and the
external cavity modes. The concept of the sub-Lyapunov exponent $\lambda_0$ is
generalized to the case of two time-scale separated delays in the system. We
give the first experimental evidence of strong and weak chaos in a network of
lasers which supports the sequence 'weak to strong to weak chaos' upon
monotonically increasing the coupling strength. Finally, we discuss strong and
weak chaos for networks with several distinct sub-Lyapunov exponents and
comment on the dependence of the sub-Lyapunov exponent on the number of a
laser's inputs in a network. | nlin_CD |
Emergence of patterns in driven and in autonomous spatiotemporal systems: The relationship between a driven extended system and an autonomous
spatiotemporal system is investigated in the context of coupled map lattice
models. Specifically, a locally coupled map lattice subjected to an external
drive is compared to a coupled map system with similar local couplings plus a
global interaction. It is shown that, under some conditions, the emergent
patterns in both systems are analogous. Based on the knowledge of the dynamical
responses of the driven lattice, we present a method that allows the prediction
of parameter values for the emergence of ordered spatiotemporal patterns in a
class of coupled map systems having local coupling and general forms of global
interactions. | nlin_CD |
Chaotic dynamics of three-dimensional Hénon maps that originate from a
homoclinic bifurcation: We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has
a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers
$(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma)$, where
$0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a
three-parameter family, $g_{\eps}$, of diffeomorphisms close to $g_0$, there
exist infinitely many open regions near $\eps =0$ where the corresponding
normal form of the first return map to a neighborhood of a homoclinic point is
a three-dimensional H\'enon-like map. This map possesses, in some parameter
regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that
this homoclinic bifurcation leads to a strange attractor. We also discuss the
place that these three-dimensional H\'enon maps occupy in the class of
quadratic volume-preserving diffeomorphisms. | nlin_CD |
Synchronization of time delay systems using variable delay with reset
for enhanced security in communication: We have introduced a mechanism for synchronizing chaotic systems by one way
coupling with a variable delay that is reset at finite intervals. Here we
extend this method to time delay systems and suggest a new cryptosystem based
on this. We present the stability analysis as applied to time delay systems and
supplement this by numerical simulations in a standard time delay system like
Mackey Glass system. We extend the theory to multi- delay systems and propose a
bi channel scheme for the implementation of the scheme for communication with
enhanced security. We show that since the synchronizing channel carries
information from transmitter only at intervals of reset time, it is not
susceptible to reconstruction. The message channel being separate can be made
complex by linear combination of transmitter variable at different delay times
using mutiple delay systems. This method has the additional advantage that it
can be adjusted to be delay or anticipatory in synchronization and these
provide two additional basic keys that are independent of system delay. | nlin_CD |
Strong Evidence of Normal Heat Conduction in a one-Dimensional Quantum
System: We investigate how the normal energy transport is realized in one-dimensional
quantum systems using a quantum spin system. The direct investigation of local
energy distribution under thermal gradient is made using the quantum master
equation, and the mixing properties and the convergence of the Green-Kubo
formula are investigated when the number of spin increases. We find that the
autocorrelation function in the Green-Kubo formula decays as $\sim t^{-1.5}$ to
a finite value which vanishes rapidly with the increase of the system size. As
a result, the Green-Kubo formula converges to a finite value in the
thermodynamic limit. These facts strongly support the realization of Fourier
heat law in a quantum system. | nlin_CD |
Topology of Chaotic Mixing Patterns: A stirring device consisting of a periodic motion of rods induces a mapping
of the fluid domain to itself, which can be regarded as a homeomorphism of a
punctured surface. Having the rods undergo a topologically-complex motion
guarantees at least a minimum amount of stretching of material lines, which is
important for chaotic mixing. We use topological considerations to describe the
nature of the injection of unmixed material into a central mixing region, which
takes place at injection cusps. A topological index formula allow us to predict
the possible types of unstable foliations that can arise for a fixed number of
rods. | nlin_CD |
Mechanism of synchronization in a random dynamical system: The mechanism of synchronization in the random Zaslavsky map is investigated.
From the error dynamics of two particles, the structure of phase space was
analyzed, and a transcritical bifurcation between a saddle and a stable fixed
point was found. We have verified the structure of on-off intermittency in
terms of a biased random walk. Furthermore, for the generalized case of the
ensemble of particles, \emph{a modified definition} of the size of a snapshot
attractor was exploited to establish the link with a random walk. As a result,
the structure of on-off intermittency in the ensemble of particles was
explicitly revealed near the transition. | nlin_CD |
An analysis of the periodically forced PP04 climate model, using the
theory of non-smooth dynamical systems: In this paper we perform a careful analysis of the forced PP04 model for
climate change, in particular the behaviour of the ice-ages. This system models
the transition from a glacial to an inter-glacial state through a sudden
release of oceanic Carbon Dioxide into the atmosphere. This process can be cast
in terms of a Filippov dynamical system, with a discontinuous change in its
dynamics related to the Carbon Dioxide release. By using techniques from the
theory of non-smooth dynamical systems, we give an analysis of this model in
the cases of both no insolation forcing and also periodic insolation forcing.
This reveals a rich, and novel, dynamical structure to the solutions of the
PP04 model. In particular we see synchronised periodic solutions with subtle
regions of existence which depend on the amplitude and frequency of the
forcing. The orbits can be created/destroyed in both smooth and discontinuity
induced bifurcations. We study both the orbits and the transitions between them
and make comparisons with actual climate dynamics. | nlin_CD |
Control of Fractional-Order Chua's System: This paper deals with feedback control of fractional-order Chua's system. The
fractional-order Chua's system with total order less than three which exhibit
chaos as well as other nonlinear behavior and theory for control of chaotic
systems using sampled data are presented. Numerical experimental example is
shown to verify the theoretical results. | nlin_CD |
Detection of fixed points in spatiotemporal signals by clustering method: We present a method to determine fixed points in spatiotemporal signals. A
144-dimensioanl simulated signal, similar to a Kueppers-Lortz instability, is
analyzed and its fixed points are reconstructed. | nlin_CD |
Coupled Hénon Map, Part I: Topological Horseshoes and Uniform
Hyperbolicity: We derive a sufficient condition for topological horseshoe and uniform
hyperbolicity of a 4-dimensional symplectic map, which is introduced by
coupling the two 2-dimensional H\'enon maps via linear terms. The coupled
H\'enon map thus constructed can be viewed as a simple map modeling the
horseshoe in higher dimensions. We show that there are two different types of
horseshoes, each of which is realized around different anti-integrable limits
in the parameter regime. | nlin_CD |
Eddy diffusivity in convective hydromagnetic systems: An eigenvalue equation, for linear instability modes involving large scales
in a convective hydromagnetic system, is derived in the framework of multiscale
analysis. We consider a horizontal layer with electrically conducting
boundaries, kept at fixed temperatures and with free surface boundary
conditions for the velocity field; periodicity in horizontal directions is
assumed. The steady states must be stable to short (fast) scale perturbations
and possess symmetry about the vertical axis, allowing instabilities involving
large (slow) scales to develop. We expand the modes and their growth rates in
power series in the scale separation parameter and obtain a hierarchy of
equations, which are solved numerically. Second order solvability condition
yields a closed equation for the leading terms of the asymptotic expansions and
respective growth rate, whose origin is in the (combined) eddy diffusivity
phenomenon. For about 10% of randomly generated steady convective hydromagnetic
regimes, negative eddy diffusivity is found. | nlin_CD |
Turbulence on hyperbolic plane: the fate of inverse cascade: We describe ideal incompressible hydrodynamics on the hyperbolic plane which
is an infinite surface of constant negative curvature. We derive equations of
motion, general symmetries and conservation laws, and then consider turbulence
with the energy density linearly increasing with time due to action of
small-scale forcing. In a flat space, such energy growth is due to an inverse
cascade, which builds a constant part of the velocity autocorrelation function
proportional to time and expanding in scales, while the moments of the velocity
difference saturate during a time depending on the distance. For the curved
space, we analyze the long-time long-distance scaling limit, that lives in a
degenerate conical geometry, and find that the energy-containing mode linearly
growing with time is not constant in space. The shape of the velocity
correlation function indicates that the energy builds up in vortical rings of
arbitrary diameter but of width comparable to the curvature radius of the
hyperbolic plane. The energy current across scales does not increase linearly
with the scale, as in a flat space, but reaches a maximum around the curvature
radius. That means that the energy flux through scales decreases at larger
scales so that the energy is transferred in a non-cascade way, that is the
inverse cascade spills over to all larger scales where the energy pumped into
the system is cumulated in the rings. The time-saturated part of the spectral
density of velocity fluctuations contains a finite energy per unit area, unlike
in the flat space where the time-saturated spectrum behaves as k^{-5/3}. | nlin_CD |
Quantum and Wave Dynamical Chaos in Superconducting Microwave Billiards: Experiments with superconducting microwave cavities have been performed in
our laboratory for more than two decades. The purpose of the present article is
to recapitulate some of the highlights achieved. We briefly review (i) results
obtained with flat, cylindrical microwave resonators, so-called microwave
billiards, concerning the universal fluctuation properties of the eigenvalues
of classically chaotic systems with no, a threefold and a broken symmetry; (ii)
summarize our findings concerning the wave-dynamical chaos in three-dimensional
microwave cavities; (iii) present a new approach for the understanding of the
phenomenon of dynamical tunneling which was developed on the basis of
experiments that were performed recently with unprecedented precision, and
finally, (iv) give an insight into an ongoing project, where we investigate
universal properties of (artificial) graphene with superconducting microwave
photonic crystals that are enclosed in a microwave resonator, i.e., so-called
Dirac billiards. | nlin_CD |
Synchronization of a class of master-slave non-autonomous chaotic
systems with parameter mismatch via sinusoidal feedback control: In this paper we investigate a master-slave synchronization scheme of two
n-dimensional non-autonomous chaotic systems coupled by sinusoidal state error
feedback control, where parameter mismatch exists between the external harmonic
excitation of master system and that of slave one. A concept of synchronization
with error bound is introduced due to parameter mismatch, and then the bounds
of synchronization error are estimated analytically. Some synchronization
criteria are firstly obtained in the form of matrix inequalities by the
Lyapunov direct method, and then simplified into some algebraic inequalities by
the Gerschgorin disc theorem. The relationship between the estimated
synchronization error bound and system parameters reveals that the
synchronization error can be controlled as small as possible by increasing the
coupling strength or decreasing the magnitude of mismatch. A three-dimensional
gyrostat system is chosen as an example to verify the effectiveness of these
criteria, and the estimated synchronization error bounds are compared with the
numerical error bounds. Both the theoretical and numerical results show that
the present sinusoidal state error feedback control is effective for the
synchronization. Numerical examples verify that the present control is robust
against amplitude or phase mismatch. | nlin_CD |
Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins: The field of quantum chaos originated in the study of spectral statistics for
interacting many-body systems, but this heritage was almost forgotten when
single-particle systems moved into the focus. In recent years new interest
emerged in many-body aspects of quantum chaos. We study a chain of interacting,
kicked spins and carry out a semiclassical analysis that is capable of
identifying all kinds of genuin many-body periodic orbits. We show that the
collective many-body periodic orbits can fully dominate the spectra in certain
cases. | nlin_CD |
Extended Prigozhin theorem: method for universal characterization of
complex system evolution: Evolution of arbitrary stochastic system was considered in frame of phase
transition description. Concept of Reynolds parameter of hydrodynamic motion
was extended to arbitrary complex system. Basic phase parameter was expressed
through power of energy, injected into system and power of energy, dissipated
through internal nonlinear mechanisms. It was found out that basic phase
parameter as control parameter must be delimited for two types of system -
accelerator and decelerator. It was suggested to select zero state entropy on
through condition of zero value for entropy production. Zero state introduces
universal principle of disorder characterization. On basis of self organization
theorem we have derived relations for entropy production behavior in the
vicinity stationary state of system. Advantage of these relations in comparison
to classical Prigozhin theorem is versatility of their application to arbitrary
nonlinear systems. It was found out that extended Prigozhin theorem introduces
two relations for accelerator and decelerator correspondingly, which remarks
their quantitative difference. At the same time classic Prigozhin theorem makes
possible description of linear decelerator only. For unstable motion it
corresponds to strange attractor. | nlin_CD |
A Numerical Study of a Simple Stochastic/Deterministic Model of
Cycle-to-Cycle Combustion Fluctuations in Spark Ignition Engines: We examine a simple, fuel-air, model of combustion in a spark ignition (si)
engine with indirect injection. In our two fluid model, variations of fuel mass
burned in cycle sequences appear due to stochastic fluctuations of a fuel feed
amount. We have shown that a small amplitude of these fluctuations affects
considerably the stability of a combustion process strongly depending on the
quality of air-fuel mixture. The largest influence was found in the limit of a
lean combustion. The possible effect of nonlinearities in the combustion
process has been also discussed. | nlin_CD |
Major open problems in chaos theory and nonlinear dynamics: Nowadays, chaos theory and nonlinear dynamics lack research focuses. Here we
mention a few major open problems: 1. an effective description of chaos and
turbulence, 2. rough dependence on initial data, 3. arrow of time, 4. the
paradox of enrichment, 5. the paradox of pesticides, 6. the paradox of
plankton. | nlin_CD |
Semiclassical Study on Tunneling Processes via Complex-Domain Chaos: We investigate the semiclassical mechanism of tunneling process in
non-integrable systems. The significant role of complex-phase-space chaos in
the description of the tunneling process is elucidated by studying a simple
scattering map model. Behaviors of tunneling orbits are encoded into symbolic
sequences based on the structure of complex homoclinic tanglement. By means of
the symbolic coding, the phase space itineraries of tunneling orbits are
related with the amounts of imaginary parts of actions gained by the orbits, so
that the systematic search of significant tunneling orbits becomes possible. | nlin_CD |
Long-time saturation of the Loschmidt echo in quantum chaotic billiards: The Loschmidt echo (LE) (or fidelity) quantifies the sensitivity of the time
evolution of a quantum system with respect to a perturbation of the
Hamiltonian. In a typical chaotic system the LE has been previously argued to
exhibit a long-time saturation at a value inversely proportional to the
effective size of the Hilbert space of the system. However, until now no
quantitative results have been known and, in particular, no explicit expression
for the proportionality constant has been proposed. In this paper we perform a
quantitative analysis of the phenomenon of the LE saturation and provide the
analytical expression for its long-time saturation value for a semiclassical
particle in a two-dimensional chaotic billiard. We further perform extensive
(fully quantum mechanical) numerical calculations of the LE saturation value
and find the numerical results to support the semiclassical theory. | nlin_CD |
Random-Matrix Ensembles for Semi-Separable Systems: Many models for chaotic systems consist of joining two integrable systems
with incompatible constants of motion. The quantum counterparts of such models
have a propagator which factorizes into two integrable parts. Each part can be
diagonalized. The two eigenvector bases are related by an orthogonal (or
unitary) transformation. We construct a random matrix ensemble that mimics this
situation and consists of a product of a diagonal, an orthogonal, another
diagonal and the transposed orthogonal matrix. The diagonal phases are chosen
at random and the orthogonal matrix from Haar's measure. We derive asymptotic
results (dimension N -> \infty) using Wick contractions. A new approximation
for the group integration yields the next order in 1/N. We obtain a finite
correction to the circular orthogonal ensemble, important in the long-range
part of spectral correlations. | nlin_CD |
The Euler Equations and Non-Local Conservative Riccati Equations: We present an infinite dimensional family of of exact solutions of the
incompressible three-dimensional Euler equations. These solutions, proposed by
Gibbon and Ohkitani, have infinite kinetic energy and blow up in finite time. | nlin_CD |
Synchronization of pairwise-coupled, identical, relaxation oscillators
based on metal-insulator phase transition devices: A Model Study: Computing with networks of synchronous oscillators has attracted wide-spread
attention as novel materials and device topologies have enabled realization of
compact, scalable and low-power coupled oscillatory systems. Of particular
interest are compact and low-power relaxation oscillators that have been
recently demonstrated using MIT (metal- insulator-transition) devices using
properties of correlated oxides. This paper presents an analysis of the
dynamics and synchronization of a system of two such identical coupled
relaxation oscillators implemented with MIT devices. We focus on two
implementations of the oscillator: (a) a D-D configuration where complementary
MIT devices (D) are connected in series to provide oscillations and (b) a D-R
configuration where it is composed of a resistor (R) in series with a
voltage-triggered state changing MIT device (D). The MIT device acts like a
hysteresis resistor with different resistances in the two different states. The
synchronization dynamics of such a system has been analyzed with purely charge
based coupling using a resistive (Rc) and a capacitive (Cc) element in
parallel. It is shown that in a D-D configuration symmetric, identical and
capacitively coupled relaxation oscillator system synchronizes to an anti-phase
locking state, whereas when coupled resistively the system locks in phase.
Further, we demonstrate that for certain range of values of Rc and Cc, a
bistable system is possible which can have potential applications in
associative computing. In D-R configuration, we demonstrate the existence of
rich dynamics including non-monotonic flows and complex phase relationship
governed by the ratios of the coupling impedance. Finally, the developed
theoretical formulations have been shown to explain experimentally measured
waveforms of such pairwise coupled relaxation oscillators. | nlin_CD |
Positive-entropy Hamiltonian systems on Nilmanifolds via Scattering: Let $\Sigma$ be a compact quotient of $T_4$, the Lie group of $4 \times 4$
upper triangular matrices with unity along the diagonal. The Lie algebra $t_4$
of $T_4$ has the standard basis $\{X_{ij}\}$ of matrices with $0$ everywhere
but in the $(i,j)$ entry, which is unity. Let $g$ be the Carnot metric, a
sub-riemannian metric, on $T_4$ for which $X_{i,i+1}$, $(i=1,2,3)$, is an
orthonormal basis. Montgomery, Shapiro and Stolin showed that the geodesic flow
of $g$ is algebraically non-integrable. This note proves that the geodesic flow
of that Carnot metric on $T \Sigma$ has positive topological entropy and is
real-analytically non-integrable. It extends earlier work by Butler and
Gelfreich. | nlin_CD |
A Study of the forced Van der Pol generalized oscillator with
renormalization group method: In this paper the equation of forced Van der Pol generalized oscillator is
examined with renormalization group method. A brief recall of the
renormalization group technique is done. We have applied this method to the
equation of forced Van der Pol generalized oscillator to search for its
asymptotic solution and its renormalization group equation. The analysis of the
numerical simulation graph is done; the method's efficiency is pointed out. | nlin_CD |
One-dimensional lattice of oscillators coupled through power-law
interactions: Continuum limit and dynamics of spatial Fourier modes: We study synchronization in a system of phase-only oscillators residing on
the sites of a one-dimensional periodic lattice. The oscillators interact with
a strength that decays as a power law of the separation along the lattice
length and is normalized by a size-dependent constant. The exponent $\alpha$ of
the power law is taken in the range $0 \le \alpha <1$. The oscillator frequency
distribution is symmetric about its mean (taken to be zero), and is
non-increasing on $[0,\infty)$. In the continuum limit, the local density of
oscillators evolves in time following the continuity equation that expresses
the conservation of the number of oscillators of each frequency under the
dynamics. This equation admits as a stationary solution the unsynchronized
state uniform both in phase and over the space of the lattice. We perform a
linear stability analysis of this state to show that when it is unstable,
different spatial Fourier modes of fluctuations have different stability
thresholds beyond which they grow exponentially in time with rates that depend
on the Fourier modes. However, numerical simulations show that at long times,
all the non-zero Fourier modes decay in time, while only the zero Fourier mode
(i.e., the "mean-field" mode) grows in time, thereby dominating the instability
process and driving the system to a synchronized state. Our theoretical
analysis is supported by extensive numerical simulations. | nlin_CD |
On the universality of anomalous scaling exponents of structure
functions in turbulent flows: All previous experiments in open turbulent flows (e.g. downstream of grids,
jet and atmospheric boundary layer) have produced quantitatively consistent
values for the scaling exponents of velocity structure functions. The only
measurement in closed turbulent flow (von K\'arm\'an swirling flow) using
Taylor-hypothesis, however, produced scaling exponents that are significantly
smaller, suggesting that the universality of these exponents are broken with
respect to change of large scale geometry of the flow. Here, we report
measurements of longitudinal structure functions of velocity in a von
K\'arm\'an setup without the use of Taylor-hypothesis. The measurements are
made using Stereo Particle Image Velocimetry at 4 different ranges of spatial
scales, in order to observe a combined inertial subrange spanning roughly one
and a half order of magnitude. We found scaling exponents (up to 9th order)
that are consistent with values from open turbulent flows, suggesting that they
might be in fact universal. | nlin_CD |
Semiclassical Accuracy in Phase Space for Regular and Chaotic Dynamics: A phase-space semiclassical approximation valid to $O(\hbar)$ at short times
is used to compare semiclassical accuracy for long-time and stationary
observables in chaotic, stable, and mixed systems. Given the same level of
semiclassical accuracy for the short time behavior, the squared semiclassical
error in the chaotic system grows linearly in time, in contrast with quadratic
growth in the classically stable system. In the chaotic system, the relative
squared error at the Heisenberg time scales linearly with $\hbar_{\rm eff}$,
allowing for unambiguous semiclassical determination of the eigenvalues and
wave functions in the high-energy limit, while in the stable case the
eigenvalue error always remains of the order of a mean level spacing. For a
mixed classical phase space, eigenvalues associated with the chaotic sea can be
semiclassically computed with greater accuracy than the ones associated with
stable islands. | nlin_CD |
Modeling Time Series Data of Real Systems: Dynamics of complex systems is studied by first considering a chaotic time
series generated by Lorenz equations and adding noise to it. The trend (smooth
behavior) is separated from fluctuations at different scales using wavelet
analysis and a prediction method proposed by Lorenz is applied to make out of
sample predictions at different regions of the time series. The prediction
capability of this method is studied by considering several improvements over
this method. We then apply this approach to a real financial time series. The
smooth time series is modeled using techniques of non linear dynamics. Our
results for predictions suggest that the modified Lorenz method gives better
predictions compared to those from the original Lorenz method. Fluctuations are
analyzed using probabilistic considerations. | nlin_CD |
Statistical theory of reversals in two-dimensional confined turbulent
flows: It is shown that the Truncated Euler Equations, i.e. a finite set of ordinary
differential equations for the amplitude of the large-scale modes, can
correctly describe the complex transitional dynamics that occur within the
turbulent regime of a confined 2D Navier-Stokes flow with bottom friction and a
spatially periodic forcing. In particular, the random reversals of the large
scale circulation on the turbulent background involve bifurcations of the
probability distribution function of the large-scale circulation velocity that
are described by the related microcanonical distribution which displays
transitions from gaussian to bimodal and broken ergodicity. A minimal 13-mode
model reproduces these results. | nlin_CD |
Additive Equivalence in Turbulent Drag Reduction by Flexible and Rodlike
Polymers: We address the "Additive Equivalence" discovered by Virk and coworkers: drag
reduction affected by flexible and rigid rodlike polymers added to turbulent
wall-bounded flows is limited from above by a very similar Maximum Drag
Reduction (MDR) asymptote. Considering the equations of motion of rodlike
polymers in wall-bounded turbulent ensembles, we show that although the
microscopic mechanism of attaining the MDR is very different, the macroscopic
theory is isomorphic, rationalizing the interesting experimental observations. | nlin_CD |
The role of dissipation in flexural wave turbulence: from experimental
spectrum to Kolmogorov-Zakharov spectrum: The Weak Turbulence Theory has been applied to waves in thin elastic plates
obeying the F\"oppl-Von K\'arm\'an dynamical equations. Subsequent experiments
have shown a strong discrepancy between the theoretical predictions and the
measurements. Both the dynamical equations and the Weak Turbulence Theory
treatment require some restrictive hypotheses. Here a direct numerical
simulation of the F\"oppl-Von K\'arm\'an equations is performed and reproduces
qualitatively and quantitatively the experimental results when the
experimentally measured damping rate of waves $\gamma_\mathbf{k}= a + bk^2$ is
used. This confirms that the F\"oppl-Von K\'arm\'an equations are a valid
theoretical framework to describe our experiments. When we progressively tune
the dissipation so that to localize it at the smallest scales, we observe a
gradual transition between the experimental spectrum and the
Kolmogorov-Zakharov prediction. Thus it is shown dissipation has a major
influence on the scaling properties stationary solutions of weakly non linear
wave turbulence. | nlin_CD |
Properties of synchronization in the systems of non-identical coupled
van der Pol and van der Pol - Duffing oscillators. Broadband synchronization: The particular properties of dynamics are discussed for the dissipatively
coupled van der Pol oscillators, non-identical in values of parameters
controlling the Hopf bifurcation. Possibility of a special synchronization
regime in an infinitively long band between oscillation death and quasiperiodic
areas is shown for such system. Features of the bifurcation picture are
discussed for different values of the control parameters and for the case of
additional Duffing-type nonlinearity. Analysis of the abridged equations is
presented. | nlin_CD |
$1/f^α$ noise and integrable systems: An innovative test for detecting quantum chaos based on the analysis of the
spectral fluctuations regarded as a time series has been recently proposed.
According to this test, the fluctuations of a fully chaotic system should
exhibit 1/f noise, whereas for an integrable system this noise should obey the
1/f^2 power law. In this letter, we show that there is a family of well-known
integrable systems, namely spin chains of Haldane-Shastry type, whose spectral
fluctuations decay instead as 1/f^4. We present a simple theoretical
justification of this fact, and propose an alternative characterization of
quantum chaos versus integrability formulated directly in terms of the power
spectrum of the spacings of the unfolded spectrum. | nlin_CD |
Meeting time distributions in Bernoulli systems: Meeting time is defined as the time for which two orbits approach each other
within distance $\epsilon$ in phase space. We show that the distribution of the
meeting time is exponential in $(p_1,...,p_k)$-Bernoulli systems. In the limit
of $\epsilon\to0$, the distribution converges to exp(-\alpha\tau), where $\tau$
is the meeting time normalized by the average. The exponent is shown to be
$\alpha=\sum_{l=1}^{k}p_l(1-p_l)$ for the Bernoulli systems. | nlin_CD |
The autocorrelation function for spectral determinants of quantum graphs: The paper considers the spectral determinant of quantum graph families with
chaotic classical limit and no symmetries. The secular coefficients of the
spectral determinant are found to follow distributions with zero mean and
variance approaching a constant in the limit of large network size. This
constant is in general different from the random matrix result and depends on
the classical limit. A closed expression for this system dependent constant is
given here explicitly in terms of the spectrum of an underlying Markov process. | nlin_CD |
Fractal Properties of Anomalous Diffusion in Intermittent Maps: An intermittent nonlinear map generating subdiffusion is investigated.
Computer simulations show that the generalized diffusion coefficient of this
map has a fractal, discontinuous dependence on control parameters. An amended
continuous time random walk theory well approximates the coarse behavior of
this quantity in terms of a continuous function. This theory also reproduces a
full suppression of the strength of diffusion, which occurs at the dynamical
phase transition from one type of diffusive behavior to another. Similarly, the
probability density function of this map exhibits a nontrivial fine structure
while its coarse functional form is governed by a time fractional diffusion
equation. A more detailed understanding of the irregular structure of the
generalized diffusion coefficient is provided by an anomalous Taylor-Green-Kubo
formula establishing a relation to de Rham-type fractal functions. | nlin_CD |
Controlling double ionization of atoms in intense bichromatic laser
pulses: We consider the classical dynamics of a two-electron system subjected to an
intense bichromatic linearly polarized laser pulse. By varying the parameters
of the field, such as the phase lag and the relative amplitude between the two
colors of the field, we observe several trends from the statistical analysis of
a large ensemble of trajectories initially in the ground state energy of the
helium atom: High sensitivity of the sequential double ionization component,
low sensitivity of the intensities where nonsequential double ionization occurs
while the corresponding yields can vary drastically. All these trends hold
irrespective of which parameter is varied: the phase lag or the relative
amplitude. We rationalize these observations by an analysis of the phase space
structures which drive the dynamics of this system and determine the extent of
double ionization. These trends turn out to be mainly regulated by the dynamics
of the inner electron. | nlin_CD |
Applied Symbolic Vector Dynamics of Coupled Map Lattice: Symbolic dynamics, which partitions an infinite number of finite-length
trajectories into a finite number of trajectory sets, describes the dynamics of
a system in a simplified and coarse-grained way with a limited number of
symbols. The study of symbolic dynamics in 1D chaotic map has been further
developed and is named as the applied symbolic dynamics. In this paper, we will
study the applied symbolic vector dynamics of CML. Based on the original
contribution proposed in Refs.[6], we will study the ergodic property of CML.
We will analyze the relation between admissibility condition and control
parameters, and then give a coupling coefficient estimation method based on the
ergodic property. Both theoretical and experimental results show that we
provide a natural analytical technique for understanding turbulences in CML.
Many of our findings could be expanded to a wider range of application. | nlin_CD |
Perturbation approach to multifractal dimensions for certain critical
random matrix ensembles: Fractal dimensions of eigenfunctions for various critical random matrix
ensembles are investigated in perturbation series in the regimes of strong and
weak multifractality. In both regimes we obtain expressions similar to those of
the critical banded random matrix ensemble extensively discussed in the
literature. For certain ensembles, the leading-order term for weak
multifractality can be calculated within standard perturbation theory. For
other models such a direct approach requires modifications which are briefly
discussed. Our analytical formulas are in good agreement with numerical
calculations. | nlin_CD |
Exploring noise-induced chaos and complexity in a red blood cell system: We investigate dynamical changes and its corresponding phase space complexity
in a stochastic red blood cell system. The system is obtained by incorporating
power noise with the associated sinusoidal flow. Both chaotic and non-chaotic
dynamics of sinusoidal flow in red blood cell are identified by 0-1 test.
Furthermore, dynamical complexity of the sinusoidal flow in the system is
investigated by heterogeneous recurrence based entropy. The numerical
simulation is performed to quantify the existence of chaotic dynamics and
complexity for the sinusoidal blood flow. | nlin_CD |
Universal Fractional Map and Cascade of Bifurcations Type Attractors: We modified the way in which the Universal Map is obtained in the regular
dynamics to derive the Universal $\alpha$-Family of Maps depending on a single
parameter $\alpha > 0$ which is the order of the fractional derivative in the
nonlinear fractional differential equation describing a system experiencing
periodic kicks. We consider two particular $\alpha$-families corresponding to
the Standard and Logistic Maps. For fractional $\alpha<2$ in the area of
parameter values of the transition through the period doubling cascade of
bifurcations from regular to chaotic motion in regular dynamics corresponding
fractional systems demonstrate a new type of attractors - cascade of
bifurcations type trajectories. | nlin_CD |
Towards a semiclassical understanding of chaotic single- and
many-particle quantum dynamics at post-Heisenberg time scales: Despite considerable progress during the last decades in devising a
semiclassical theory for classically chaotic quantum systems a quantitative
semiclassical understanding of their dynamics at late times (beyond the
so-called Heisenberg time $T_H$) is still missing. This challenge,
corresponding to resolving spectral structures on energy scales below the mean
level spacing, is intimately related to the quest for semiclassically restoring
quantum unitarity, which is reflected in real-valued spectral determinants.
Guided through insights for quantum graphs we devise a periodic-orbit
resummation procedure for quantum chaotic systems invoking periodic-orbit self
encounters as the structuring element of a hierarchical phase space dynamics.
We propose a way to purely semiclassically construct real spectral determinants
based on two major underlying mechanisms: (i) Complementary contributions to
the spectral determinant from regrouped pseudo orbits of duration $T < T_H$ and
$T_H-T$ are complex conjugate to each other. (ii) Contributions from long
periodic orbits involving multiple traversals along shorter orbits cancel out.
We furthermore discuss implications for interacting $N$-particle quantum
systems with a chaotic classical large-$N$ limit that have recently attracted
interest in the context of many-body quantum chaos. | nlin_CD |
Computer Assisted 'Proof' of the Global Existence of Periodic Orbits in
the Rössler System: The numerical optimized shooting method for finding periodic orbits in
nonlinear dynamical systems was employed to determine the existence of periodic
orbits in the well-known R\"ossler system. By optimizing the period $T$ and the
three system parameters, $a$, $b$ and $c$, simultaneously, it was found that,
for any initial condition $(x_0,y_0,z_0) \in \Re^3$, there exists at least one
set of optimized parameters corresponding to a periodic orbit passing through $
(x_0,y_0,z_0)$. After a discussion of this result it was concluded that its
analytical proof may present an interesting new mathematical challenge. | nlin_CD |
Frobenius-Perron Resonances for Maps with a Mixed Phase Space: Resonances of the time evolution (Frobenius-Perron) operator P for phase
space densities have recently been shown to play a key role for the
interrelations of classical, semiclassical and quantum dynamics. Efficient
methods to determine resonances are thus in demand, in particular for
Hamiltonian systems displaying a mix of chaotic and regular behavior. We
present a powerful method based on truncating P to a finite matrix which not
only allows to identify resonances but also the associated phase space
structures. It is demonstrated to work well for a prototypical dynamical
system. | nlin_CD |
Nonlinear resonances and multi-stability in simple neural circuits: This article describes a numerical procedure designed to tune the parameters
of periodically-driven dynamical systems to a state in which they exhibit rich
dynamical behavior. This is achieved by maximizing the diversity of subharmonic
solutions available to the system within a range of the parameters that define
the driving. The procedure is applied to a problem of interest in computational
neuroscience: a circuit composed of two interacting populations of neurons
under external periodic forcing. Depending on the parameters that define the
circuit, such as the weights of the connections between the populations, the
response of the circuit to the driving can be strikingly rich and diverse. The
procedure is employed to find circuits that, when driven by external input,
exhibit multiple stable patterns of periodic activity organized in complex
tuning diagrams and signatures of low dimensional chaos. | nlin_CD |
Effect of a sub and supra-threshold periodic forcing an excitable glow
discharge plasma near its bifurcation point: In this paper non-linear dynamics of a periodically forced excitable glow
discharge plasma has been studied. The experiments were performed in glow
discharge plasma where excitability was achieved for suitable discharge voltage
and gas pressure. The plasma was first perturbed by a sub-threshold sawtooth
periodic signal, and we obtained small sub-threshold oscillations. These
oscillations showed resonance when the frequency of the perturbation was around
the characteristic frequency the plasma, and hence may be useful to estimate
characteristic of an excitable system. On the other hand, when the perturbation
was supra-threshold, system showed frequency entrainments. We obtained harmonic
frequency entrainments for perturbation frequency greater than the
characteristic frequency of the system and for lesser than the characteristic
frequency, system showed only excitable behaviour. | nlin_CD |
Analytic Approach for Controlling Quantum States in Complex Systems: We examine random matrix systems driven by an external field in view of
optimal control theory (OCT). By numerically solving OCT equations, we can show
that there exists a smooth transition between two states called "moving bases"
which are dynamically related to initial and final states. In our previous work
[J. Phys. Soc. Jpn. 73 (2004) 3215-3216; Adv. Chem. Phys. 130A (2005) 435-458],
they were assumed to be orthogonal, but in this paper, we introduce orthogonal
moving bases. We can construct a Rabi-oscillation like representation of a
wavpacket using such moving bases, and derive an analytic optimal field as a
solution of the OCT equations. We also numerically show that the newly obtained
optimal field outperforms the previous one. | nlin_CD |
Chaotic mixing induced transitions in reaction-diffusion systems: We study the evolution of a localized perturbation in a chemical system with
multiple homogeneous steady states, in the presence of stirring by a fluid
flow. Two distinct regimes are found as the rate of stirring is varied relative
to the rate of the chemical reaction. When the stirring is fast localized
perturbations decay towards a spatially homogeneous state. When the stirring is
slow (or fast reaction) localized perturbations propagate by advection in form
of a filament with a roughly constant width and exponentially increasing
length. The width of the filament depends on the stirring rate and reaction
rate but is independent of the initial perturbation. We investigate this
problem numerically in both closed and open flow systems and explain the
results using a one-dimensional "mean-strain" model for the transverse profile
of the filament that captures the interplay between the propagation of the
reaction-diffusion front and the stretching due to chaotic advection. | nlin_CD |
Growth of the Wang-Casati-Prosen counter in an integrable billiard: This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev.
E vol. 89, 042918 (2014)] devoted to a study of ergodicity in two-dimensional
irrational right-triangular billiards. Numerical results presented there
suggest that these billiards are generally not ergodic. However, they become
ergodic when the billiard angle is equal to $\pi/2$ times a Liouvillian
irrational, a Liouvillian irrational, a class of irrational numbers which are
well approximated by rationals.
In particular, Wang et al. study a special integer counter that reflects the
irrational contribution to the velocity orientation; they conjecture that this
counter is localized in the generic case, but grows in the Liouvillian case. We
propose a generalization of the Wang-Casati-Prosen counter: this generalization
allows to include rational billiards into consideration. We show that in the
case of a $45^{\circ} \!\! : \! 45^{\circ} \!\! : \! 90^{\circ}$ billiard, the
counter grows indefinitely, consistent with the Liouvillian scenario suggested
by Wang et al. | nlin_CD |
The Burnett expansion of the periodic Lorentz gas: Recently, stretched exponential decay of multiple correlations in the
periodic Lorentz gas has been used to show the convergence of a series of
correlations which has the physical interpretation as the fourth order Burnett
coefficient, a generalisation of the diffusion coefficient. Here the result is
extended to include all higher order Burnett coefficients, and give a plausible
argument that the expansion constructed from the Burnett coefficients has a
finite radius of convergence. | nlin_CD |
Periodic orbit analysis of a system with continuous symmetry - a
tutorial: Dynamical systems with translational or rotational symmetry arise frequently
in studies of spatially extended physical systems, such as Navier-Stokes flows
on periodic domains. In these cases, it is natural to express the state of the
fluid in terms of a Fourier series truncated to a finite number of modes. Here,
we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar
to those that appear in fluid dynamics problems. A crucial step in the analysis
of such a system is symmetry reduction. We use the model to illustrate
different symmetry-reduction techniques. Its relative equilibria are
conveniently determined by rewriting the dynamics in terms of a
symmetry-invariant polynomial basis. However, for the analysis of its chaotic
dynamics, the `method of slices', which is applicable to very high-dimensional
problems, is preferable. We show that a Poincar\'e section taken on the `slice'
can be used to further reduce this flow to what is for all practical purposes a
unimodal map. This enables us to systematically determine all relative periodic
orbits and their symbolic dynamics up to any desired period. We then present
cycle averaging formulas adequate for systems with continuous symmetry and use
them to compute dynamical averages using relative periodic orbits. The
convergence of such computations is discussed. | nlin_CD |
Local evolution equations for non-Markovian processes: A Fokker-Planck equation approach for the treatment of non-Markovian
stochastic processes is proposed. The approach is based on the introduction of
fictitious trajectories sharing with the real ones their local structure and
initial conditions. Different statistical quantities are generated by different
construction rules for the trajectories, which coincide only in the Markovian
case. The merits and limitations of the approach are discussed and applications
to transport in ratchets and to anomalous diffusion are illustated. | nlin_CD |
Rectification of current in ac-driven nonlinear systems and symmetry
properties of the Boltzmann equation: We study rectification of a current of particles moving in a spatially
periodic potential under the influence of time-periodic forces with zero mean
value. If certain time-space symmetries are broken a non-zero directed current
of particles is possible. We investigate this phenomenon in the framework of
the kinetic Boltzmann equation. We find that the attractor of the Boltzmann
equation completely reflects the symmetries of the original one-particle
equation of motion. Especially, we analyse the limits of weak and strong
relaxation. The dc current increases by several orders of magnitude with
decreasing dissipation. | nlin_CD |
Chaotic time-dependent billiards: A billiard in the form of a stadium with periodically perturbed boundary is
considered. Two types of such billiards are studied: stadium with strong
chaotic properties and a near-rectangle billiard. Phase portraits of such
billiards are investigated. In the phase plane areas corresponding to decrease
and increase of the velocity of billiard particles are found. Average
velocities of the particle ensemble as functions of the number of collisions
are obtained. | nlin_CD |
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