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Spectral Properties and Dynamical Tunneling in Constant-Width Billiards: We determine with unprecedented accuracy the lowest 900 eigenvalues of two
quantum constant-width billiards from resonance spectra measured with flat,
superconducting microwave resonators. While the classical dynamics of the
constant-width billiards is unidirectional, a change of the direction of motion
is possible in the corresponding quantum system via dynamical tunneling. This
becomes manifest in a splitting of the vast majority of resonances into
doublets of nearly degenerate ones. The fluctuation properties of the two
respective spectra are demonstrated to coincide with those of a random-matrix
model for systems with violated time-reversal invariance and a mixed dynamics.
Furthermore, we investigate tunneling in terms of the splittings of the doublet
partners. On the basis of the random-matrix model we derive an analytical
expression for the splitting distribution which is generally applicable to
systems exhibiting dynamical tunneling between two regions with (predominantly)
chaotic dynamics. | nlin_CD |
Possibilities of the Discrete Fourier Transform for Determining the
Order-Chaos Transition in a Dynamical System: This paper is devoted to a discussion of the Discrete Fourier Transform (DFT)
representation of a chaotic finite-duration sequence. This representation has
the advantage that is itself a finite-duration sequence corresponding to
samples equally spaced in the frequency domain. The Fast Fourier Transform
(FFT) algoritm allows us an effective computation, and it can be applied to a
relatively short time series. DFT representation requirements were analized and
applied for determining the order-chaos transition in a nonlinear system
described by the equation $x[n+1]=rx[n](1-x[n])$. Its effectiveness was
demonstrated by comparing the results with those obtained by calculating the
largest Lyapounov exponent for the time series set, obtained from the logistic
equation. | nlin_CD |
Numerical studies on the synchronization of a network of mutually
coupled simple chaotic systems: We present in this paper, the synchronization dynamics observed in a network
of mutually coupled simple chaotic systems. The network consisting of chaotic
systems arranged in a square matrix network is studied for their different
types of synchronization behavior. The chaotic attractors of the simple $2
\times 2$ matrix network exhibiting strange non-chaotic attractors in their
synchronization dynamics for smaller values of the coupling strength is
reported. Further, the existence of islands of unsynchronized and synchronized
states of strange non-chaotic attractors for smaller values of coupling
strength is observed. The process of complete synchronization observed in the
network with all the systems exhibiting strange non-chaotic behavior is
reported. The variation of the slope of the singular continuous spectra as a
function of the coupling strength confirming the strange non-chaotic state of
each of the system in the network is presented. The stability of complete
synchronization observed in the network is studied using the Master Stability
Function. | nlin_CD |
Control of spatiotemporal patterns in the Gray-Scott model: This paper studies the effects of a time-delayed feedback control on the
appearance and development of spatiotemporal patterns in a reaction-diffusion
system. Different types of control schemes are investigated, including
single-species, diagonal, and mixed control. This approach helps to unveil
different dynamical regimes, which arise from chaotic state or from traveling
waves. In the case of spatiotemporal chaos, the control can either stabilize
uniform steady states or lead to bistability between a trivial steady state and
a propagating traveling wave. Furthermore, when the basic state is a stable
traveling pulse, the control is able to advance stationary Turing patterns or
yield the above-mentioned bistability regime. In each case, the stability
boundary is found in the parameter space of the control strength and the time
delay, and numerical simulations suggest that diagonal control fails to control
the spatiotemporal chaos. | nlin_CD |
Universal spectral statistics in quantum graphs: We prove that the spectrum of an individual chaotic quantum graph shows
universal spectral correlations, as predicted by random--matrix theory. The
stability of these correlations with regard to non--universal corrections is
analyzed in terms of the linear operator governing the classical dynamics on
the graph. | nlin_CD |
Studies of dynamical localization in a finite-dimensional model of the
quantum kicked rotator: We review our recent works on the dynamical localization in the quantum
kicked rotator (QKR) and the related properties of the classical kicked rotator
(the standard map, SM). We introduce the Izrailev $N$-dimensional model of the
QKR and analyze the localization properties of the Floquet eigenstates [{\em
Phys. Rev. E} {\bf 87}, 062905 (2013)], and the statistical properties of the
quasienergy spectra. We survey normal and anomalous diffusion in the SM, and
the related accelerator modes [{\em Phys. Rev. E} {\bf 89}, 022905 (2014)]. We
analyze the statistical properties [{\em Phys. Rev. E} {\bf 91},042904 (2015)]
of the localization measure, and show that the reciprocal localization length
has an almost Gaussian distribution which has a finite variance even in the
limit of the infinitely dimensional model of the QKR, $N\rightarrow \infty$.
This sheds new light on the relation between the QKR and the Anderson
localization phenomenon in the one-dimensional tight-binding model. It explains
the so far mysterious strong fluctuations in the scaling properties of the QKR.
The reason is that the finite bandwidth approximation of the underlying
Hamilton dynamical system in the Shepelyansky picture [{\em Phys. Rev. Lett.}
{\bf 56}, 677 (1986)] does not apply rigorously. These results call for a more
refined theory of the localization length in the QKR and in similar Floquet
systems, where we must predict not only the mean value of the inverse of the
localization length but also its (Gaussian) distribution. We also numerically
analyze the related behavior of finite time Lyapunov exponents in the SM and of
the $2\times2$ transfer matrix formalism. | nlin_CD |
Instability statistics and mixing rates: We claim that looking at probability distributions of \emph{finite time}
largest Lyapunov exponents, and more precisely studying their large deviation
properties, yields an extremely powerful technique to get quantitative
estimates of polynomial decay rates of time correlations and Poincar\'e
recurrences in the -quite delicate- case of dynamical systems with weak chaotic
properties. | nlin_CD |
A semiquantal approach to finite systems of interacting particles: A novel approach is suggested for the statistical description of quantum
systems of interacting particles. The key point of this approach is that a
typical eigenstate in the energy representation (shape of eigenstates, SE) has
a well defined classical analog which can be easily obtained from the classical
equations of motion. Therefore, the occupation numbers for single-particle
states can be represented as a convolution of the classical SE with the quantum
occupation number operator for non-interacting particles. The latter takes into
account the wavefunctions symmetry and depends on the unperturbed energy
spectrum only. As a result, the distribution of occupation numbers $n_s$ can be
numerically found for a very large number of interacting particles. Using the
model of interacting spins we demonstrate that this approach gives a correct
description of $n_s$ even in a deep quantum region with few single-particle
orbitals. | nlin_CD |
Experimental observation of chimera and cluster states in a minimal
globally coupled network: A "chimera state" is a dynamical pattern that occurs in a network of coupled
identical oscillators when the symmetry of the oscillator population is broken
into synchronous and asynchronous parts. We report the experimental observation
of chimera and cluster states in a network of four globally coupled chaotic
opto-electronic oscillators. This is the minimal network that can support
chimera states, and our study provides new insight into the fundamental
mechanisms underlying their formation. We use a unified approach to determine
the stability of all the observed partially synchronous patterns, highlighting
the close relationship between chimera and cluster states as belonging to the
broader phenomenon of partial synchronization. Our approach is general in terms
of network size and connectivity. We also find that chimera states often appear
in regions of multistability between global, cluster, and desynchronized
states. | nlin_CD |
Leading off-diagonal approximation for the spectral form factor for
uniformly hyperbolic systems: We consider the semiclassical approximation to the spectral form factor
K(tau) for two-dimensional uniformly hyperbolic systems, and derive the first
off-diagonal correction for small tau. The result agrees with the tau^2-term of
the form factor for the GOE random matrix ensemble. | nlin_CD |
Sand stirred by chaotic advection: We study the spatial structure of a granular material, N particles subject to
inelastic mutual collisions, when it is stirred by a bidimensional smooth
chaotic flow. A simple dynamical model is introduced where four different time
scales are explicitly considered: i) the Stokes time, accounting for the
inertia of the particles, ii) the mean collision time among the grains, iii)
the typical time scale of the flow, and iv) the inverse of the Lyapunov
exponent of the chaotic flow, which gives a typical time for the separation of
two initially close parcels of fluid. Depending on the relative values of these
different times a complex scenario appears for the long-time steady spatial
distribution of particles, where clusters of particles may or not appear. | nlin_CD |
Relaxation and Diffusion in a Globally Coupled Hamiltonian System: The relation between relaxation and diffusion is investigated in a
Hamiltonian system of globally coupled rotators. Diffusion is anomalous if and
only if the system is going towards equilibrium. The anomaly in diffusion is
not anomalous diffusion taking a power-type function, but is a transient
anomaly due to non-stationarity. Contrary to previous claims, in
quasi-stationary states, diffusion can be explained by a stretched exponential
correlation function, whose stretching exponent is almost constant and
correlation time is linear as functions of degrees of freedom. The full time
evolution is characterized by varying stretching exponent and correlation time. | nlin_CD |
Hamiltonian Dynamics of Thermostated Systems: Two-Temperature
Heat-Conducting phi-4 Chains: We consider and compare four Hamiltonian formulations of thermostated
mechanics, three of them kinetic, and the other one configurational. Though all
four approaches ``work'' at equilibrium, their application to many-body
nonequilibrium simulations can fail to provide a proper flow of heat. All the
Hamiltonian formulations considered here are applied to the same prototypical
two-temperature "phi-4" model of a heat-conducting chain. This model
incorporates nearest-neighbor Hooke's-Law interactions plus a quartic tethering
potential. Physically correct results, obtained with the isokinetic Gaussian
and Nose-Hoover thermostats, are compared with two other Hamiltonian results.
The latter results, based on constrained Hamiltonian thermostats, fail
correctly to model the flow of heat. | nlin_CD |
Dynamics of impurities in a three-dimensional volume-preserving map: We study the dynamics of inertial particles in three dimensional
incompressible maps, as representations of volume preserving flows. The
impurity dynamics has been modeled, in the Lagrangian framework, by a
six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of
the base map is embedded in the particle dynamics governed by the map. The base
map considered for the present study is the Arnold-Beltrami-Childress (ABC)
map. We consider the behavior of the system both in the aerosol regime, where
the density of the particle is larger than that of the base flow, as well as
the bubble regime, where the particle density is less than that of the base
flow. The phase spaces in both the regimes show rich and complex dynamics with
three type of dynamical behaviors - chaotic structures, regular orbits and
hyperchaotic regions. In the one-action case, the aerosol regime is found to
have periodic attractors for certain values of the dissipation and inertia
parameters. For the aerosol regime of the two-action ABC map, an attractor
merging and widening crises is identified using the bifurcation diagram and the
spectrum of Lyapunov exponents. After the crisis an attractor with two parts is
seen, and trajectories hop between these parts with period 2. The bubble regime
of the embedded map shows strong hyperchaotic regions as well as crisis induced
intermittency with characteristic times between bursts that scale as a power
law behavior as a function of the dissipation parameter. Furthermore, we
observe riddled basin of attraction and unstable dimension variability in the
phase space in the bubble regime. The bubble regime in one-action shows similar
behavior. This study of a simple model of impurity dynamics may shed light upon
the transport properties of passive scalars in three dimensional flows. We also
compare our results with those seen earlier in two dimensional flows. | nlin_CD |
Quantum transport through partial barriers in higher-dimensional systems: Partial transport barriers in the chaotic sea of Hamiltonian systems
influence classical transport, as they allow for a small flux between chaotic
phase-space regions only. We establish for higher-dimensional systems that
quantum transport through such a partial barrier follows a universal transition
from quantum suppression to mimicking classical transport. The scaling
parameter involves the flux, the size of a Planck cell, and the localization
length due to dynamical localization along a resonance channel. This is
numerically demonstrated for coupled kicked rotors with a partial barrier that
generalizes a cantorus to higher dimensions. | nlin_CD |
Energy flux fluctuations in a finite volume of turbulent flow: The flux of turbulent kinetic energy from large to small spatial scales is
measured in a small domain B of varying size R. The probability distribution
function of the flux is obtained using a time-local version of Kolmogorov's
four-fifths law. The measurements, made at a moderate Reynolds number, show
frequent events where the flux is backscattered from small to large scales,
their frequency increasing as R is decreased. The observations are corroborated
by a numerical simulation based on the motion of many particles and on an
explicit form of the eddy damping. | nlin_CD |
Elucidating the escape dynamics of the four hill potential: The escape mechanism of the four hill potential is explored. A thorough
numerical investigation takes place in several types of two-dimensional planes
and also in a three-dimensional subspace of the entire four-dimensional phase
space in order to distinguish between bounded (ordered and chaotic) and
escaping orbits. The determination of the location of the basins of escape
toward the different escape channels and their correlations with the
corresponding escape time of the orbits is undoubtedly an issue of paramount
importance. It was found that in all examined cases all initial conditions
correspond to escaping orbits, while there is no numerical indication of stable
bounded motion, apart from some isolated unstable periodic orbits. Furthermore,
we monitor how the fractality evolves when the total orbital energy varies. The
larger escape periods have been measured for orbits with initial conditions in
the fractal basin boundaries, while the lowest escape rates belong to orbits
with initial conditions inside the basins of escape. We hope that our numerical
analysis will be useful for a further understanding of the escape dynamics of
orbits in open Hamiltonian systems with two degrees of freedom. | nlin_CD |
On two-dimensionalization of three-dimensional turbulence in shell
models: Applying a modified version of the Gledzer-Ohkitani-Yamada (GOY) shell model,
the signatures of so-called two-dimensionalization effect of three-dimensional
incompressible, homogeneous, isotropic fully developed unforced turbulence have
been studied and reproduced. Within the framework of shell models we have
obtained the following results: (i) progressive steepening of the energy
spectrum with increased strength of the rotation, and, (ii) depletion in the
energy flux of the forward forward cascade, sometimes leading to an inverse
cascade. The presence of extended self-similarity and self-similar PDFs for
longitudinal velocity differences are also presented for the rotating 3D
turbulence case. | nlin_CD |
Hamiltonian formulation of reduced Vlasov-Maxwell equations: The Hamiltonian formulation of the reduced Vlasov-Maxwell equations is
expressed in terms of the macroscopic fields D and H. These macroscopic fields
are themselves expressed in terms of the functional Lie-derivative generated by
the functional S with the Poisson bracket [.,.] for the exact Vlasov-Maxwell
equations. Hence, the polarization vector P= (D-E)/(4pi) and the magnetization
vector M=(B-H)/(4pi) are defined in terms of the expressions 4pi P=[S,E]+...
and 4pi M =-[S,B]+..., where lowest-order terms yield dipole contributions. | nlin_CD |
Possibility of using dual frequency to control chaotic oscillations of a
spherical bubble: Acoustic cavitation bubbles are known to exhibit highly nonlinear and
unpredictable chaotic dynamics. Their inevitable role in applications like
sonoluminescence, sonochemistry and medical procedures suggests that their
dynamics be controlled. Reducing chaotic oscillations could be the first step
in controlling the bubble dynamics by increasing the predictability of the
bubble response to an applied acoustic field. One way to achieve this concept
is to perturb the acoustic forcing. Recently, due to the improvements
associated with using dual frequency sources, this method has been the subject
of many studies which have proved its applicability and advantages. Due to this
reason, in this paper, the oscillations of a spherical bubble driven by a dual
frequency source, were studied and compared to the ones driven by a single
source. Results indicated that using dual frequency had a strong impact on
reducing the chaotic oscillations to regular ones. The governing parameters
influencing its dynamics are the secondary frequency and its phase difference
with the fundamental frequency. Also using dual frequency forcing may arm us by
the possibility of generating oscillations of desired amplitudes. To our
knowledge the investigation of the ability of using a dual frequency forcing to
control chaotic oscillations are presented for the first time in this paper. | nlin_CD |
Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a
microwave billiard with threefold symmetry: Recently it has been shown that time-reversal invariant systems with discrete
symmetries may display in certain irreducible subspaces spectral statistics
corresponding to the Gaussian unitary ensemble (GUE) rather than to the
expected orthogonal one (GOE). A Kramers type degeneracy is predicted in such
situations. We present results for a microwave billiard with a threefold
rotational symmetry and with the option to display or break a reflection
symmetry. This allows us to observe the change from GOE to GUE statistics for
one subset of levels. Since it was not possible to separate the three
subspectra reliably, the number variances for the superimposed spectra were
studied. The experimental results are compared with a theoretical and numerical
study considering the effects of level splitting and level loss. | nlin_CD |
The effect of symmetry breaking on the dynamics near a structurally
stable heteroclinic cycle between equilibria and a periodic orbit: The effect of small forced symmetry breaking on the dynamics near a
structurally stable heteroclinic cycle connecting two equilibria and a periodic
orbit is investigated. This type of system is known to exhibit complicated,
possibly chaotic dynamics including irregular switching of sign of various
phase space variables, but details of the mechanisms underlying the complicated
dynamics have not previously been investigated. We identify global bifurcations
that induce the onset of chaotic dynamics and switching near a heteroclinic
cycle of this type, and by construction and analysis of approximate return
maps, locate the global bifurcations in parameter space. We find there is a
threshold in the size of certain symmetry-breaking terms below which there can
be no persistent switching. Our results are illustrated by a numerical example. | nlin_CD |
Phase-flip bifurcation and synchronous transition in unidirectionally
coupled parametrically excited pendula: Phase-flip bifurcation plays an important role in the transition to
synchronization state in unidirectionally coupled parametrically excited
pendula. In coupled identical system it is the cause of complete
synchronization whereas in case of coupled non-identical system it causes
desynchronization. In coupled identical systems negativity of conditional
Lyapunov exponent is not always sufficient for complete synchronization. In
complete synchronization state the largest conditional Lyapunov exponent and
the second largest Lyapunov exponent are equal in magnitude and slope. | nlin_CD |
Spreading, Nonergodicity, and Selftrapping: a puzzle of interacting
disordered lattice waves: Localization of waves by disorder is a fundamental physical problem
encompassing a diverse spectrum of theoretical, experimental and numerical
studies in the context of metal-insulator transitions, the quantum Hall effect,
light propagation in photonic crystals, and dynamics of ultra-cold atoms in
optical arrays, to name just a few examples. Large intensity light can induce
nonlinear response, ultracold atomic gases can be tuned into an interacting
regime, which leads again to nonlinear wave equations on a mean field level.
The interplay between disorder and nonlinearity, their localizing and
delocalizing effects is currently an intriguing and challenging issue in the
field of lattice waves. In particular it leads to the prediction and
observation of two different regimes of destruction of Anderson localization -
asymptotic weak chaos, and intermediate strong chaos, separated by a crossover
condition on densities. On the other side approximate full quantum interacting
many body treatments were recently used to predict and obtain a novel many body
localization transition, and two distinct phases - a localization phase, and a
delocalization phase, both again separated by some typical density scale. We
will discuss selftrapping, nonergodicity and nonGibbsean phases which are
typical for such discrete models with particle number conservation and their
relation to the above crossover and transition physics. We will also discuss
potential connections to quantum many body theories. | nlin_CD |
Lyapunov spectra of periodic orbits for a many-particle system: The Lyapunov spectrum corresponding to a periodic orbit for a two dimensional
many particle system with hard core interactions is discussed. Noting that the
matrix to describe the tangent space dynamics has the block cyclic structure,
the calculation of the Lyapunov spectrum is attributed to the eigenvalue
problem of a 16x16 reduced matrices regardless of the number of particles. We
show that there is the thermodynamic limit of the Lyapunov spectrum in this
periodic orbit. The Lyapunov spectrum has a step structure, which is explained
by using symmetries of the reduced matrices. | nlin_CD |
Higher-order hbar corrections in the semiclassical quantization of
chaotic billiards: In the periodic orbit quantization of physical systems, usually only the
leading-order hbar contribution to the density of states is considered.
Therefore, by construction, the eigenvalues following from semiclassical trace
formulae generally agree with the exact quantum ones only to lowest order of
hbar. In different theoretical work the trace formulae have been extended to
higher orders of hbar. The problem remains, however, how to actually calculate
eigenvalues from the extended trace formulae since, even with hbar corrections
included, the periodic orbit sums still do not converge in the physical domain.
For lowest-order semiclassical trace formulae the convergence problem can be
elegantly, and universally, circumvented by application of the technique of
harmonic inversion. In this paper we show how, for general scaling chaotic
systems, also higher-order hbar corrections to the Gutzwiller formula can be
included in the harmonic inversion scheme, and demonstrate that corrected
semiclassical eigenvalues can be calculated despite the convergence problem.
The method is applied to the open three-disk scattering system, as a prototype
of a chaotic system. | nlin_CD |
Dynamical Localization in Quasi-Periodic Driven Systems: We investigate how the time dependence of the Hamiltonian determines the
occurrence of Dynamical Localization (DL) in driven quantum systems with two
incommensurate frequencies. If both frequencies are associated to impulsive
terms, DL is permanently destroyed. In this case, we show that the evolution is
similar to a decoherent case. On the other hand, if both frequencies are
associated to smooth driving functions, DL persists although on a time scale
longer than in the periodic case. When the driving function consists of a
series of pulses of duration $\sigma$, we show that the localization time
increases as $\sigma^{-2}$ as the impulsive limit, $\sigma\to 0$, is
approached. In the intermediate case, in which only one of the frequencies is
associated to an impulsive term in the Hamiltonian, a transition from a
localized to a delocalized dynamics takes place at a certain critical value of
the strength parameter. We provide an estimate for this critical value, based
on analytical considerations. We show how, in all cases, the frequency spectrum
of the dynamical response can be used to understand the global features of the
motion. All results are numerically checked. | nlin_CD |
On Noether's theorem for the Euler-Poincaré equation on the
diffeomorphism group with advected quantities: We show how Noether conservation laws can be obtained from the particle
relabelling symmetries in the Euler-Poincar\'e theory of ideal fluids with
advected quantities. All calculations can be performed without Lagrangian
variables, by using the Eulerian vector fields that generate the symmetries,
and we identify the time-evolution equation that these vector fields satisfy.
When advected quantities (such as advected scalars or densities) are present,
there is an additional constraint that the vector fields must leave the
advected quantities invariant. We show that if this constraint is satisfied
initially then it will be satisfied for all times. We then show how to solve
these constraint equations in various examples to obtain evolution equations
from the conservation laws. We also discuss some fluid conservation laws in the
Euler-Poincar\'e theory that do not arise from Noether symmetries, and explain
the relationship between the conservation laws obtained here, and the
Kelvin-Noether theorem given in Section 4 of Holm, Marsden and Ratiu, {\it Adv.
in Math.}, 1998. | nlin_CD |
Semiclassical matrix model for quantum chaotic transport with
time-reversal symmetry: We show that the semiclassical approach to chaotic quantum transport in the
presence of time-reversal symmetry can be described by a matrix model, i.e. a
matrix integral whose perturbative expansion satisfies the semiclassical
diagrammatic rules for the calculation of transport statistics. This approach
leads very naturally to the semiclassical derivation of universal predictions
from random matrix theory. | nlin_CD |
Classical chaos in atom-field systems: The relation between the onset of chaos and critical phenomena, like Quantum
Phase Transitions (QPT) and Excited-State Quantum Phase transitions (ESQPT), is
analyzed for atom-field systems. While it has been speculated that the onset of
hard chaos is associated with ESQPT based in the resonant case, the
off-resonant cases show clearly that both phenomena, ESQPT and chaos, respond
to different mechanisms. The results are supported in a detailed numerical
study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The
appearance of chaos is quantified calculating the largest Lyapunov exponent for
a wide sample of initial conditions in the whole available phase space for a
given energy. The percentage of the available phase space with chaotic
trajectories is evaluated as a function of energy and coupling between the
qubit and bosonic part, allowing to obtain maps in the space of coupling and
energy, where ergodic properties are observed in the model. Different sets of
Hamiltonian parameters are considered, including resonant and off-resonant
cases. | nlin_CD |
Statistical properties of time-reversible triangular maps of the square: Time reversal symmetric triangular maps of the unit square are introduced
with the property that the time evolution of one of their two variables is
determined by a piecewise expanding map of the unit interval. We study their
statistical properties and establish the conditions under which their
equilibrium measures have a product structure, i.e. factorises in a symmetric
form. When these conditions are not verified, the equilibrium measure does not
have a product form and therefore provides additional information on the
statistical properties of theses maps. This is the case of anti-symmetric cusp
maps, which have an intermittent fixed point and yet have uniform invariant
measures on the unit interval. We construct the invariant density of the
corresponding two-dimensional triangular map and prove that it exhibits a
singularity at the intermittent fixed point. | nlin_CD |
Using dimension reduction to improve outbreak predictability of
multistrain diseases: Multistrain diseases have multiple distinct coexisting serotypes (strains).
For some diseases, such as dengue fever, the serotypes interact by
antibody-dependent enhancement (ADE), in which infection with a single serotype
is asymptomatic, but contact with a second serotype leads to higher viral load
and greater infectivity. We present and analyze a dynamic compartmental model
for multiple serotypes exhibiting ADE. Using center manifold techniques, we
show how the dynamics rapidly collapses to a lower dimensional system. Using
the constructed reduced model, we can explain previously observed synchrony
between certain classes of primary and secondary infectives (Schwartz et al.,
Phys. Rev. E 72: 066201, 2005). Additionally, we show numerically that the
center manifold equations apply even to noisy systems. Both deterministic and
stochastic versions of the model enable prediction of asymptomatic individuals
that are difficult to track during an epidemic. We also show how this technique
may be applicable to other multistrain disease models, such as those with
cross-immunity. | nlin_CD |
Bottleneck crossover between classical and quantum superfluid turbulence: We consider superfluid turbulence near absolute zero of temperature generated
by classical means, e.g. towed grid or rotation but not by counterflow. We
argue that such turbulence consists of a {\em polarized} tangle of mutually
interacting vortex filaments with quantized vorticity. For this system we
predict and describe a bottleneck accumulation of the energy spectrum at the
classical-quantum crossover scale $\ell$. Demanding the same energy flux
through scales, the value of the energy at the crossover scale should exceed
the Kolmogorov-41 spectrum by a large factor $\ln^{10/3} (\ell/a_0)$ ($\ell$ is
the mean intervortex distance and $a_0$ is the vortex core radius) for the
classical and quantum spectra to be matched in value. One of the important
consequences of the bottleneck is that it causes the mean vortex line density
to be considerably higher that based on K41 alone, and this should be taken
into account in (re)interpretation of new (and old) experiments as well as in
further theoretical studies. | nlin_CD |
Using the Hadamard and related transforms for simplifying the spectrum
of the quantum baker's map: We rationalize the somewhat surprising efficacy of the Hadamard transform in
simplifying the eigenstates of the quantum baker's map, a paradigmatic model of
quantum chaos. This allows us to construct closely related, but new, transforms
that do significantly better, thus nearly solving for many states of the
quantum baker's map. These new transforms, which combine the standard Fourier
and Hadamard transforms in an interesting manner, are constructed from
eigenvectors of the shift permutation operator that are also simultaneous
eigenvectors of bit-flip (parity) and possess bit-reversal (time-reversal)
symmetry. | nlin_CD |
Is the subdominant part of the energy spectrum due to downscale energy
cascade hidden in quasi-geostrophic turbulence?: In systems governing two-dimensional turbulence, surface quasi-geostrophic
turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic
turbulence, etc., there often exist two conservative quadratic quantities, one
``energy''-like and one ``enstrophy''-like. In a finite inertial range there
are in general two spectral fluxes, one associated with each conserved
quantity. We derive here an inequality comparing the relative magnitudes of the
``energy'' and ``enstrophy'' fluxes for finite or infinitesimal dissipations,
and for hyper or hypo viscosities. When this inequality is satisfied, as is the
case of 2D turbulence,where the energy flux contribution to the energy spectrum
is small, the subdominant part will be effectively hidden. In sQG turbulence,
it is shown that the opposite is true: the downscale energy flux becomes the
dominant contribution to the energy spectrum. A combination of these two
behaviors appears to be the case in 2-layer QG turbulence, depending on the
baroclinicity of the system. | nlin_CD |
Chimera States are Fragile under Random Links: We study the dynamics of coupled systems, ranging from maps supporting
chaotic attractors to nonlinear differential equations yielding limit cycles,
under different coupling classes, connectivity ranges and initial states. Our
focus is the robustness of chimera states in the presence of a few time-varying
random links, and we demonstrate that chimera states are often destroyed,
yielding either spatiotemporal fixed points or spatiotemporal chaos, in the
presence of even a single dynamically changing random connection. We also study
the global impact of random links by exploring the Basin Stability of the
chimera state, and we find that the basin size of the chimera state rapidly
falls to zero under increasing fraction of random links. This indicates the
extreme fragility of chimera patterns under minimal spatial randomness in many
systems, significantly impacting the potential observability of chimera states
in naturally occurring scenarios. | nlin_CD |
A New Lorenz System Parameter Determination Method and Applications: This paper describes how to determine the parameter values of the chaotic
Lorenz system from one of its variables waveform. The geometrical properties of
the system are used firstly to reduce the parameter search space. Then, a
synchronization-based approach, with the help of the same geometrical
properties as coincidence criteria, is implemented to determine the parameter
values with the wanted accuracy. The method is not affected by a moderate
amount of noise in the waveform. As way of example of its effectiveness, the
method is applied to figure out directly from the ciphertext the secret keys of
two-channel chaotic cryptosystems using the variable $z$ as a synchronization
signal, based on the ultimate state projective chaos synchronization. | nlin_CD |
Scattering Experiments with Microwave Billiards at an Exceptional Point
under Broken Time Reversal Invariance: Scattering experiments with microwave cavities were performed and the effects
of broken time-reversal invariance (TRI), induced by means of a magnetized
ferrite placed inside the cavity, on an isolated doublet of nearly degenerate
resonances were investigated. All elements of the effective Hamiltonian of this
two-level system were extracted. As a function of two experimental parameters,
the doublet and also the associated eigenvectors could be tuned to coalesce at
a so-called exceptional point (EP). The behavior of the eigenvalues and
eigenvectors when encircling the EP in parameter space was studied, including
the geometric amplitude that builds up in the case of broken TRI. A
one-dimensional subspace of parameters was found where the differences of the
eigenvalues are either real or purely imaginary. There, the Hamiltonians were
found PT-invariant under the combined operation of parity (P) and time reversal
(T) in a generalized sense. The EP is the point of transition between both
regions. There a spontaneous breaking of PT occurs. | nlin_CD |
Comparing the basins of attraction for several methods in the circular
Sitnikov problem with spheroid primaries: The circular Sitnikov problem, where the two primary bodies are prolate or
oblate spheroids, is numerically investigated. In particular, the basins of
convergence on the complex plane are revealed by using a large collection of
numerical methods of several order. We consider four cases, regarding the value
of the oblateness coefficient which determines the nature of the roots
(attractors) of the system. For all cases we use the iterative schemes for
performing a thorough and systematic classification of the nodes on the complex
plane. The distribution of the iterations as well as the probability and their
correlations with the corresponding basins of convergence are also discussed.
Our numerical computations indicate that most of the iterative schemes provide
relatively similar convergence structures on the complex plane. However, there
are some numerical methods for which the corresponding basins of attraction are
extremely complicated with highly fractal basin boundaries. Moreover, it is
proved that the efficiency strongly varies between the numerical methods. | nlin_CD |
The influence of quantum field fluctuations on chaotic dynamics of
Yang-Mills system II. The role of the centrifugal term: We have considered SU(2)xU(1) gauge field theory describing electroweak
interactions. We have demonstrated that centrifugal term in model Hamiltonian
increases the region of regular dynamics of Yang-Mills and Higgs fields system
at low densities of energy. Also we have found analytically the approximate
relation for critical density of energy of the order to chaos transition on
centrifugal constant. It is necessary to note that mentioned increase of the
region of regular dynamics has linear dependance on the value of the
centrifugal constant. | nlin_CD |
Nonlinearity effects in the kicked oscillator: The quantum kicked oscillator is known to display a remarkable richness of
dynamical behaviour, from ballistic spreading to dynamical localization. Here
we investigate the effects of a Gross Pitaevskii nonlinearity on quantum
motion, and provide evidence that the qualitative features depend strongly on
the parameters of the system. | nlin_CD |
Geometrical approach for description of the mixed state in multi-well
potentials: We use so-called geometrical approach in description of transition from
regular motion to chaotic in Hamiltonian systems with potential energy surface
that has several local minima. Distinctive feature of such systems is
coexistence of different types of dynamics (regular or chaotic) in different
wells at the same energy Mixed state reveals unique opportunities in research
of quantum manifestations of classical stochasticity. Application of
traditional criteria for transition to chaos (resonance overlap criterion,
negative curvature criterion and stochastic layer destruction criterion) is
inefficient in case of potentials with complex topology. Geometrical approach
allows considering only configuration space but not phase space when
investigating stability. Trajectories are viewed as geodesics of configuration
space equipped with suitable metric. In this approach all information about
chaos and regularity consists in potential function. The aim of this work is to
determine what details of geometry of potential lead to chaos in Hamiltonian
systems using geometrical approach. Numerical calculations are executed for
potentials that are relevant with lowest umbilical catastrophes. | nlin_CD |
From a unstable periodic orbit to Lyapunov exponent and macroscopic
variable in a Hamiltonian lattice : Periodic orbit dependencies: We study which and how a periodic orbit in phase space links to both the
largest Lyapunov exponent and the expectation values of macroscopic variables
in a Hamiltonian system with many degrees of freedom. The model which we use in
this paper is the discrete nonlinear Schr\"odinger equation. Using a method
based on the modulational estimate of a periodic orbit, we predict the largest
Lyapunov exponent and the expectation value of a macroscopic variable. We show
that (i) the predicted largest Lyapunov exponent generally depends on the
periodic orbit which we employ, and (ii) the predicted expectation value of the
macroscopic variable does not depend on the periodic orbit at least in a high
energy regime. In addition, the physical meanings of these dependencies are
considered. | nlin_CD |
Locking-time and Information Capacity in CML with Statistical
Periodicity: In this work we address the statistical periodicity phenomenon on a coupled
map lattice. The study was done based on the asymptotic binary patterns. The
pattern multiplicity gives us the lattice information capacity, while the
entropy rate allows us to calculate the locking-time. Our results suggest that
the lattice has low locking-time and high capacity information when the
coupling is weak. This is the condition for the system to reproduce a kind of
behavior observed in neural networks. | nlin_CD |
Control technique for synchronization of selected nodes in directed
networks: In this Letter we propose a method to control a set of arbitrary nodes in a
directed network such that they follow a synchronous trajectory which is, in
general, not shared by the other units of the network. The problem is inspired
to those natural or artificial networks whose proper operating conditions are
associated to the presence of clusters of synchronous nodes. Our proposed
method is based on the introduction of distributed controllers that modify the
topology of the connections in order to generate outer symmetries in the nodes
to be controlled. An optimization problem for the selection of the controllers,
which includes as a special case the minimization of the number of the links
added or removed, is also formulated and an algorithm for its solution is
introduced. | nlin_CD |
Accelerator modes and their effect in the diffusion properties in the
kicked rotator: We highlight a few recent results on the effect of the diffusion process in
deterministic area preserving maps with noncompact phase space, namely the
standard map. In more detail, we focus on the anomalous diffusion arising due
to the accelerator modes, i.e. resonant-like features of the phase space which
are transported in a super-diffusive (ballistic) manner. Their presence affects
also the trajectories lying in the immediate neighborhood resulting in
anomalous (non-Gaussian) diffusion. We aim to shed some light on these special
properties of the phase space by utilizing the diffusion exponent (rate of
diffusion) and the momentum distribution in terms of the L\'evy stable
distributions, with the goal to reach an understanding of the global behaviour.
To this end we consider a rather large interval for the kick parameter ($0\le K
\le 70$) of the standard map where accelerator modes of different periodicity
exist. | nlin_CD |
The Complex Ginzburg-Landau Equation in the Presence of Walls and
Corners: We investigate the influence of walls and corners (with Dirichlet and Neumann
boundary conditions) in the evolution of twodimensional autooscillating fields
described by the complex Ginzburg-Landau equation. Analytical solutions are
found, and arguments provided, to show that Dirichlet walls introduce strong
selection mechanisms for the wave pattern. Corners between walls provide
additional synchronization mechanisms and associated selection criteria. The
numerical results fit well with the theoretical predictions in the parameter
range studied. | nlin_CD |
Synchronous motion in a devil's stick -- variation on a theme by Kapitza: The counter-intuitive rotational motion of the propeller of a devil's stick
when the agitator is rubbed against the pylon has long intrigued performers,
audiences and scientists. The apparently unrelated phenomenon of
self-stabilization of a Kapitza pendulum at the inverted position, once again
an amazing lecture demonstration, has on the other hand been subjected to years
of investigation and research. Here we show that the two systems are in fact
identical, and obtain a previously unreported, friction-stabilized
synchronously rotating state in the Kapitza pendulum, which in fact explains
the rotational motion of the Devil's stick. | nlin_CD |
Koopman Operator and Phase Space Partition of Chaotic Maps: Koopman operator describes evolution of observables in the phase space, which
could be used to extract characteristic dynamical features of a nonlinear
system. Here, we show that it is possible to carry out interesting symbolic
partitions based on properly constructed eigenfunctions of the operator for
chaotic maps. The partition boundaries are the extrema of these eigenfunctions,
the accuracy of which is improved by including more basis functions in the
numerical computation. The validity of this scheme is demonstrated in
well-known 1-d and 2-d maps. It seems no obstacle to extend the computation to
nonlinear systems of high dimensions, which provides a possible way of
dissecting complex dynamics. | nlin_CD |
When Darwin meets Lorenz: Evolving new chaotic attractors through
genetic programming: In this paper, we propose a novel methodology for automatically finding new
chaotic attractors through a computational intelligence technique known as
multi-gene genetic programming (MGGP). We apply this technique to the case of
the Lorenz attractor and evolve several new chaotic attractors based on the
basic Lorenz template. The MGGP algorithm automatically finds new nonlinear
expressions for the different state variables starting from the original Lorenz
system. The Lyapunov exponents of each of the attractors are calculated
numerically based on the time series of the state variables using time delay
embedding techniques. The MGGP algorithm tries to search the functional space
of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of
the evolved attractors. To demonstrate the potential of the proposed
methodology, we report over one hundred new chaotic attractor structures along
with their parameters, which are evolved from just the Lorenz system alone. | nlin_CD |
Stable Quantum Resonances in Atom Optics: A theory for stabilization of quantum resonances by a mechanism similar to
one leading to classical resonances in nonlinear systems is presented. It
explains recent surprising experimental results, obtained for cold Cesium atoms
when driven in the presence of gravity, and leads to further predictions. The
theory makes use of invariance properties of the system, that are similar to
those of solids, allowing for separation into independent kicked rotor
problems. The analysis relies on a fictitious classical limit where the small
parameter is {\em not} Planck's constant, but rather the detuning from the
frequency that is resonant in absence of gravity. | nlin_CD |
Detecting chaos in particle accelerators through the frequency map
analysis method: The motion of beams in particle accelerators is dominated by a plethora of
non-linear effects which can enhance chaotic motion and limit their
performance. The application of advanced non-linear dynamics methods for
detecting and correcting these effects and thereby increasing the region of
beam stability plays an essential role during the accelerator design phase but
also their operation. After describing the nature of non-linear effects and
their impact on performance parameters of different particle accelerator
categories, the theory of non-linear particle motion is outlined. The recent
developments on the methods employed for the analysis of chaotic beam motion
are detailed. In particular, the ability of the frequency map analysis method
to detect chaotic motion and guide the correction of non-linear effects is
demonstrated in particle tracking simulations but also experimental data. | nlin_CD |
Quasi-periodic oscillations in a network of four Rossler chaotic
oscillators: We consider a network of four non-identical chaotic Rossler oscillators. The
possibility is shown of appearance of two-, three- and four-frequency invariant
tori resulting from secondary quasi-periodic Hopf bifurcations and saddle-node
homoclinic bifurcations of tori. | nlin_CD |
A new diagnostic for the relative accuracy of Euler codes: A procedure is suggested for testing the resolution and comparing the
relative accuracy of numerical schemes for integration of the incompressible
Euler equations. | nlin_CD |
Breaking projective chaos synchronization secure communication using
filtering and generalized synchronization: This paper describes the security weaknesses of a recently proposed secure
communication method based on chaotic masking using projective synchronization
of two chaotic systems. We show that the system is insecure and how to break it
in two different ways, by high-pass filtering and by generalized
synchronization. | nlin_CD |
Dynamical equations for high-order structure functions, and a comparison
of a mean field theory with experiments in three-dimensional turbulence: Two recent publications [V. Yakhot, Phys. Rev. E {\bf 63}, 026307, (2001) and
R.J. Hill, J. Fluid Mech. {\bf 434}, 379, (2001)] derive, through two different
approaches that have the Navier-Stokes equations as the common starting point,
a set of steady-state dynamic equations for structure functions of arbitrary
order in hydrodynamic turbulence. These equations are not closed. Yakhot
proposed a "mean field theory" to close the equations for locally isotropic
turbulence, and obtained scaling exponents of structure functions and an
expression for the tails of the probability density function of transverse
velocity increments. At high Reynolds numbers, we present some relevant
experimental data on pressure and dissipation terms that are needed to provide
closure, as well as on aspects predicted by the theory. Comparison between the
theory and the data shows varying levels of agreement, and reveals gaps
inherent to the implementation of the theory. | nlin_CD |
Self-pulsing dynamics of ultrasound in a magnetoacoustic resonator: A theoretical model of parametric magnetostrictive generator of ultrasound is
considered, taking into account magnetic and magnetoacoustic nonlinearities.
The stability and temporal dynamics of the system is analized with standard
techniques revealing that, for a given set of parameters, the model presents a
homoclinic or saddle--loop bifurcation, which predicts that the ultrasound is
emitted in the form of pulses or spikes with arbitrarily low frequency. | nlin_CD |
Damping filter method for obtaining spatially localized solutions: Spatially localized structures are key components of turbulence and other
spatio-temporally chaotic systems. From a dynamical systems viewpoint, it is
desirable to obtain corresponding exact solutions, though their existence is
not guaranteed. A damping filter method is introduced to obtain variously
localized solutions, and adopted into two typical cases. This method introduces
a spatially selective damping effect to make a good guess at the exact
solution, and we can obtain an exact solution through a continuation with the
damping amplitude. First target is a steady solution to Swift-Hohenberg
equation, which is a representative of bi-stable systems in which localized
solutions coexist, and a model for span-wisely localized cases. Not only
solutions belonging to the well-known snaking branches but also those belonging
to an isolated branch known as "isolas" are found with a continuation paths
between them in phase space extended with the damping amplitude. This indicates
that this spatially selective excitation mechanism has an advantage in
searching spatially localized solutions. Second target is a spatially localized
traveling-wave solution to Kuramoto-Sivashinsky equation, which is a model for
stream-wisely localized cases. Since the spatially selective damping effect
breaks Galilean and translational invariances, the propagation velocity cannot
be determined uniquely while the damping is active, and a singularity arises
when these invariances are recovered. We demonstrate that this singularity can
be avoided by imposing a simple condition, and a localized traveling-wave
solution is obtained with a specific propagation speed. | nlin_CD |
Amplitude Mediated Chimera States with Active and Inactive Oscillators: The emergence and nature of amplitude mediated chimera states,
spatio-temporal patterns of co-existing coherent and incoherent regions, are
investigated for a globally coupled system of active and inactive
Ginzburg-Landau oscillators. The existence domain of such states is found to
shrink and shift in parametric space as the fraction of inactive oscillators is
increased. The role of inactive oscillators is found to be two fold - they get
activated to form a separate region of coherent oscillations and in addition
decrease the common collective frequency of the coherent regions by their
presence. The dynamical origin of these effects is delineated through a
detailed bifurcation analysis of a reduced model equation that is based on a
mean field approximation. Our results may have practical implications for the
robustness of such states in biological or physical systems where age related
deterioration in the functionality of components can occur. | nlin_CD |
Self-sustained irregular activity in an ensemble of neural oscillators: An ensemble of pulse-coupled phase-oscillators is thoroughly analysed in the
presence of a mean-field coupling and a dispersion of their natural
frequencies. In spite of the analogies with the Kuramoto setup, a much richer
scenario is observed. The "synchronised phase", which emerges upon increasing
the coupling strength, is characterized by highly-irregular fluctuations: a
time-series analysis reveals that the dynamics of the order parameter is indeed
high-dimensional. The complex dynamics appears to be the result of the
non-perturbative action of a suitably shaped phase-response curve. Such
mechanism differs from the often invoked balance between excitation and
inhibition and might provide an alternative basis to account for the
self-sustained brain activity in the resting state. The potential interest of
this dynamical regime is further strengthened by its (microscopic) linear
stability, which makes it quite suited for computational tasks. The overall
study has been performed by combining analytical and numerical studies,
starting from the linear stability analysis of the asynchronous regime, to
include the Fourier analysis of the Kuramoto order parameter, the computation
of various types of Lyapunov exponents, and a microscopic study of the
inter-spike intervals. | nlin_CD |
Discrete Shilnikov attractor and chaotic dynamics in the system of five
identical globally coupled phase oscillators with biharmonic coupling: We argue that a discrete Shilnikov attractor exists in the system of five
identical globally coupled phase oscillators with biharmonic coupling. We
explain the scenario that leads to birth of this kind of attractor and
numerically illustrate the sequence of bifurcations that supports our
statement. | nlin_CD |
Singular solutions for geodesic flows of Vlasov moments: The Vlasov equation for the collisionless evolution of the single-particle
probability distribution function (PDF) is a well-known example of coadjoint
motion. Remarkably, the property of coadjoint motion survives the process of
taking moments. That is, the evolution of the moments of the Vlasov PDF is also
coadjoint motion. We find that {\it geodesic} coadjoint motion of the Vlasov
moments with respect to powers of the single-particle momentum admits singular
(weak) solutions concentrated on embedded subspaces of physical space. The
motion and interactions of these embedded subspaces are governed by canonical
Hamiltonian equations for their geodesic evolution. | nlin_CD |
Interaction and chaotic dynamics of the classical hydrogen atom in an
electromagnetic field: Expressions for energy and angular momentum changes of the hydrogen atom due
to interaction with the electromagnetic field during the period of the electron
motion in the Coulomb field are derived. It is shown that only the energy
change for the motion between two subsequent passings of the pericenter is
related to the quasiclassical dipole matrix element for transitions between
excited states. | nlin_CD |
Swinging Atwood's Machine: Experimental and Theoretical Studies: A Swinging Atwood Machine (SAM) is built and some experimental results
concerning its dynamic behaviour are presented. Experiments clearly show that
pulleys play a role in the motion of the pendulum, since they can rotate and
have non-negligible radii and masses. Equations of motion must therefore take
into account the inertial momentum of the pulleys, as well as the winding of
the rope around them. Their influence is compared to previous studies. A
preliminary discussion of the role of dissipation is included. The theoretical
behaviour of the system with pulleys is illustrated numerically, and the
relevance of different parameters is highlighted. Finally, the integrability of
the dynamic system is studied, the main result being that the Machine with
pulleys is non-integrable. The status of the results on integrability of the
pulley-less Machine is also recalled. | nlin_CD |
Slim Fractals: The Geometry of Doubly Transient Chaos: Traditional studies of chaos in conservative and driven dissipative systems
have established a correspondence between sensitive dependence on initial
conditions and fractal basin boundaries, but much less is known about the
relation between geometry and dynamics in undriven dissipative systems. These
systems can exhibit a prevalent form of complex dynamics, dubbed doubly
transient chaos because not only typical trajectories but also the (otherwise
invariant) chaotic saddles are transient. This property, along with a manifest
lack of scale invariance, has hindered the study of the geometric properties of
basin boundaries in these systems--most remarkably, the very question of
whether they are fractal across all scales has yet to be answered. Here we
derive a general dynamical condition that answers this question, which we use
to demonstrate that the basin boundaries can indeed form a true fractal; in
fact, they do so generically in a broad class of transiently chaotic undriven
dissipative systems. Using physical examples, we demonstrate that the
boundaries typically form a slim fractal, which we define as a set whose
dimension at a given resolution decreases when the resolution is increased. To
properly characterize such sets, we introduce the notion of equivalent
dimension for quantifying their relation with sensitive dependence on initial
conditions at all scales. We show that slim fractal boundaries can exhibit
complex geometry even when they do not form a true fractal and fractal scaling
is observed only above a certain length scale at each boundary point. Thus, our
results reveal slim fractals as a geometrical hallmark of transient chaos in
undriven dissipative systems. | nlin_CD |
Role of chaos for the validity of statistical mechanics laws: diffusion
and conduction: Several years after the pioneering work by Fermi Pasta and Ulam, fundamental
questions about the link between dynamical and statistical properties remain
still open in modern statistical mechanics. Particularly controversial is the
role of deterministic chaos for the validity and consistency of statistical
approaches. This contribution reexamines such a debated issue taking
inspiration from the problem of diffusion and heat conduction in deterministic
systems. Is microscopic chaos a necessary ingredient to observe such
macroscopic phenomena? | nlin_CD |
Synchronization and time shifts of dynamical patterns for mutually
delay-coupled fiber ring lasers: A pair of coupled erbium doped fiber ring lasers is used to explore the
dynamics of coupled spatiotemporal systems. The lasers are mutually coupled
with a coupling delay less than the cavity round-trip time. We study
synchronization between the two lasers in the experiment and in a delay
differential equation model of the system. Because the lasers are internally
perturbed by spontaneous emission, we include a noise source in the model to
obtain stochastic realizations of the deterministic equations. Both amplitude
synchronization and phase synchronization are considered. We use the Hilbert
transform to define the phase variable and compute phase synchronization. We
find that synchronization increases with coupling strength in the experiment
and the model. When the time series from two lasers are time-shifted in either
direction by the delay time, approximately equal synchronization is frequently
observed, so that a clear leader and follower cannot be identified. We define
an algorithm to determine which laser leads the other when the synchronization
is sufficiently different with one direction of time shift, and statistics of
switches in leader and follower are studied. The frequency of switching between
leader and follower increases with coupling strength, as might be expected
since the lasers mutually influence each other more effectively with stronger
coupling. | nlin_CD |
Anticipating the dynamics of chaotic maps: We study the regime of anticipated synchronization in unidirectionally
coupled chaotic maps such that the slave map has its own output reinjected
after a certain delay. For a class of simple maps, we give analytic conditions
for the stability of the synchronized solution, and present results of
numerical simulations of coupled 1D Bernoulli-like maps and 2D Baker maps, that
agree well with the analytic predictions. | nlin_CD |
Coarse-grained description of a passive scalar: The issue of the parameterization of small-scale dynamics is addressed in the
context of passive-scalar turbulence. The basic idea of our strategy is to
identify dynamical equations for the coarse-grained scalar dynamics starting
from closed equations for two-point statistical indicators. With the aim of
performing a fully-analytical study, the Kraichnan advection model is
considered. The white-in-time character of the latter model indeed leads to
closed equations for the equal-time scalar correlation functions. The classical
closure problem however still arises if a standard filtering procedure is
applied to those equations in the spirit of the large-eddy-simulation strategy.
We show both how to perform exact closures and how to identify the
corresponding coarse-grained scalar evolution. | nlin_CD |
Turbulence in small-world networks: The transition to turbulence via spatiotemporal intermittency is investigated
in the context of coupled maps defined on small-world networks. The local
dynamics is given by the Chat\'e-Manneville minimal map previously used in
studies of spatiotemporal intermittency in ordered lattices. The critical
boundary separating laminar and turbulent regimes is calculated on the
parameter space of the system, given by the coupling strength and the rewiring
probability of the network. Windows of relaminarization are present in some
regions of the parameter space. New features arise in small-world networks; for
instance, the character of the transition to turbulence changes from second
order to a first order phase transition at some critical value of the rewiring
probability. A linear relation characterizing the change in the order of the
phase transition is found. The global quantity used as order parameter for the
transition also exhibits nontrivial collective behavior for some values of the
parameters. These models may describe several processes occurring in nonuniform
media where the degree of disorder can be continuously varied through a
parameter. | nlin_CD |
Is one dimensional return map sufficient to describe the chaotic
dynamics of a three dimensional system?: Study of continuous dynamical system through Poincare map is one of the most
popular topics in nonlinear analysis. This is done by taking intersections of
the orbit of flow by a hyper-plane parallel to one of the coordinate
hyper-planes of co-dimension one. Naturally for a 3D-attractor, the Poincare
map gives rise to 2D points, which can describe the dynamics of the attractor
properly. In a very special case, sometimes these 2D points are considered as
their 1D-projections to obtain a 1D map. However, this is an artificial way of
reducing the 2D map by dropping one of the variables. Sometimes it is found
that the two coordinates of the points on the Poincare section are functionally
related. This also reduces the 2D Poincare map to a 1D map. This reduction is
natural, and not artificial as mentioned above. In the present study, this
issue is being highlighted. In fact, we find out some examples, which show that
even this natural reduction of the 2D Poincare map is not always justified,
because the resultant 1D map may fail to generate the original dynamics. This
proves that to describe the dynamics of the 3D chaotic attractor, the minimum
dimension of the Poincare map must be two, in general. | nlin_CD |
Existence, uniqueness and analyticity of space-periodic solutions to the
regularised long-wave equation: We consider space-periodic evolutionary and travelling-wave solutions to the
regularised long-wave equation (RLWE) with damping and forcing. We establish
existence, uniqueness and smoothness of the evolutionary solutions for smooth
initial conditions, and global in time spatial analyticity of such solutions
for analytical initial conditions. The width of the analyticity strip decays at
most polynomially. We prove existence of travelling-wave solutions and
uniqueness of travelling waves of a sufficiently small norm. The importance of
damping is demonstrated by showing that the problem of finding travelling-wave
solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the
asymptotic convergence of the power series expansion of travelling waves for a
weak forcing. | nlin_CD |
Amplitude death in coupled robust-chaos oscillators: We investigate the synchronization behavior of a system of globally coupled,
continuous-time oscillators possessing robust chaos. The local dynamics
corresponds to the Shimizu-Morioka model where the occurrence of robust chaos
in a region of its parameter space has been recently discovered. We show that
the global coupling can drive the oscillators to synchronization into a fixed
point created by the coupling, resulting in amplitude death in the system. The
existence of robust chaos allows to introduce heterogeneity in the local
parameters, while guaranteeing the functioning of all the oscillators in a
chaotic mode. In this case, the system reaches a state of oscillation death,
with coexisting clusters of oscillators in different steady states. The
phenomena of amplitude death or oscillation death in coupled robust-chaos flows
could be employed as mechanisms for stabilization and control in systems that
require reliable operation under chaos. | nlin_CD |
Chimera states for directed networks: We demonstrate that chimera behavior can be observed in ensembles of phase
oscillators with unidirectional coupling. For a small network consisting of
only three identical oscillators (cyclic triple), tiny {\it chimera islands}
arise in the parameter space. They are surrounded by developed chaotic
switching behavior caused by a collision of rotating waves propagating in
opposite directions. For larger networks, as we show for hundred oscillators
(cyclic century), the islands merge into a single {\it chimera continent} which
incorporates the world of chimeras of different configuration. The phenomenon
inherits from networks with intermediate ranges of the unidirectional coupling
and it diminishes as it decreases. | nlin_CD |
Amplitude death in coupled chaotic oscillators: Amplitude death can occur in chaotic dynamical systems with time-delay
coupling, similar to the case of coupled limit cycles. The coupling leads to
stabilization of fixed points of the subsystems. This phenomenon is quite
general, and occurs for identical as well as nonidentical coupled chaotic
systems. Using the Lorenz and R\"ossler chaotic oscillators to construct
representative systems, various possible transitions from chaotic dynamics to
fixed points are discussed. | nlin_CD |
Core-halo instability in dynamical systems: This paper proves an instability theorem for dynamical systems. As one adds
interactions between subystems in a complex system, structured or random, a
threshold of connectivity is reached beyond which the overall dynamics
inevitably goes unstable. The threshold occurs at the point at which flows and
interactions between subsystems (`surface' effects) overwhelm internal
stabilizing dynamics (`volume' effects). The theorem is used to identify
instability thresholds in systems that possess a core-halo or core-periphery
structure, including the gravo-thermal catastrophe -- i.e., star collapse and
explosion -- and the interbank payment network. In the core-halo model, the
same dynamical instability underlies both gravitational and financial collapse. | nlin_CD |
The rhythm of coupled metronomes: Spontaneous synchronization of an ensemble of metronomes placed on a freely
rotating platform is studied experimentally and by computer simulations. A
striking in-phase synchronization is observed when the metronomes' beat
frequencies are fixed above a critical limit. Increasing the number of
metronomes placed on the disk leads to an observable decrease in the level of
the emerging synchronization. A realistic model with experimentally determined
parameters is considered in order to understand the observed results. The
conditions favoring the emergence of synchronization are investigated. It is
shown that the experimentally observed trends can be reproduced by assuming a
finite spread in the metronomes' natural frequencies. In the limit of large
numbers of metronomes, we show that synchronization emerges only above a
critical beat frequency value. | nlin_CD |
The Chaplygin sleigh with parametric excitation: chaotic dynamics and
nonholonomic acceleration: This paper is concerned with the Chaplygin sleigh with timevarying mass
distribution (parametric excitation). The focus is on the case where excitation
is induced by a material point that executes periodic oscillations in a
direction transverse to the plane of the knife edge of the sleigh. In this
case, the problem reduces to investigating a reduced system of two first-order
equations with periodic coefficients, which is similar to various nonlinear
parametric oscillators. Depending on the parameters in the reduced system, one
can observe different types of motion, including those accompanied by strange
attractors leading to a chaotic (diffusion) trajectory of the sleigh on the
plane. The problem of unbounded acceleration (an analog of Fermi acceleration)
of the sleigh is examined in detail. It is shown that such an acceleration
arises due to the position of the moving point relative to the line of action
of the nonholonomic constraint and the center of mass of the platform. Various
special cases of existence of tensor invariants are found. | nlin_CD |
Covariant Lyapunov vectors for rigid disk systems: We carry out extensive computer simulations to study the Lyapunov instability
of a two-dimensional hard disk system in a rectangular box with periodic
boundary conditions. The system is large enough to allow the formation of
Lyapunov modes parallel to the x axis of the box. The Oseledec splitting into
covariant subspaces of the tangent space is considered by computing the full
set of covariant perturbation vectors co-moving with the flow in tangent-space.
These vectors are shown to be transversal, but generally not orthogonal to each
other. Only the angle between covariant vectors associated with immediate
adjacent Lyapunov exponents in the Lyapunov spectrum may become small, but the
probability of this angle to vanish approaches zero. The stable and unstable
manifolds are transverse to each other and the system is hyperbolic. | nlin_CD |
Resonant eigenstates in quantum chaotic scattering: We study the spectrum of quantized open maps, as a model for the resonance
spectrum of quantum scattering systems. We are particularly interested in open
maps admitting a fractal repeller. Using the ``open baker's map'' as an
example, we numerically investigate the exponent appearing in the Fractal Weyl
law for the density of resonances; we show that this exponent is not related
with the ``information dimension'', but rather the Hausdorff dimension of the
repeller. We then consider the semiclassical measures associated with the
eigenstates: we prove that these measures are conditionally invariant with
respect to the classical dynamics. We then address the problem of classifying
semiclassical measures among conditionally invariant ones. For a solvable
model, the ``Walsh-quantized'' open baker's map, we manage to exhibit a family
of semiclassical measures with simple self-similar properties. | nlin_CD |
Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha
equations: We examine the effects of turbulence on elliptic instability of rotating
stratified incompressible flows, in the context of the Lagragian-averaged
Euler-Boussinesq-alpha, or \laeba, model of turbulence. We find that the \laeba
model alters the instability in a variety of ways for fixed Rossby number and
Brunt-V\"ais\"al\"a frequency. First, it alters the location of the instability
domains in the $(\gamma,\cos\theta)-$parameter plane, where $\theta$ is the
angle of incidence the Kelvin wave makes with the axis of rotation and $\gamma$
is the eccentricity of the elliptic flow, as well as the size of the associated
Lyapunov exponent. Second, the model shrinks the width of one instability band
while simultaneously increasing another. Third, the model introduces bands of
unstable eccentric flows when the Kelvin wave is two-dimensional. We introduce
two similarity variables--one is a ratio of the Brunt-V\"ais\"al\"a frequency
to the model parameter $\Upsilon_0 = 1+\alpha^2\beta^2$, and the other is the
ratio of the adjusted inverse Rossby number to the same model parameter. Here,
$\alpha$ is the turbulence correlation length, and $\beta$ is the Kelvin wave
number. We show that by adjusting the Rossby number and Brunt-V\"ais\"al\"a
frequency so that the similarity variables remain constant for a given value of
$\Upsilon_0$, turbulence has little effect on elliptic instability for small
eccentricities $(\gamma \ll 1)$. For moderate and large eccentricities,
however, we see drastic changes of the unstable Arnold tongues due to the
\laeba model. | nlin_CD |
The Universality of Dynamic Multiscaling in Homogeneous, Isotropic
Turbulence: We systematise the study of dynamic multiscaling of time-dependent structure
functions in different models of passive-scalar and fluid turbulence. We show
that, by suitably normalising these structure functions, we can eliminate their
dependence on the origin of time at which we start our measurements and that
these normalised structure functions yield the same linear bridge relations
that relate the dynamic-multiscaling and equal-time exponents for statistically
steady turbulence. We show analytically, for both the Kraichnan Model of
passive-scalar turbulence and its shell model analogue, and numerically, for
the GOY shell model of fluid turbulence and a shell model for passive-scalar
turbulence, that these exponents and bridge relations are the same for
statistically steady and decaying turbulence. Thus we provide strong evidence
for dynamic universality, i.e., dynamic-multiscaling exponents do not depend on
whether the turbulence decays or is statistically steady. | nlin_CD |
Statistical properties of a dissipative kicked system: critical
exponents and scaling invariance: A new universal {\it empirical} function that depends on a single critical
exponent (acceleration exponent) is proposed to describe the scaling behavior
in a dissipative kicked rotator. The scaling formalism is used to describe two
regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For
case (i) the model exhibits a route to chaos known as period doubling and the
Feigenbaum constant along the bifurcations is obtained. When weak dissipation
is considered the average action as well as its standard deviation are
described using scaling arguments with critical exponents. The universal {\it
empirical} function describes remarkably well a phase transition from limited
to unlimited growth of the average action. | nlin_CD |
An analytic approximation to the Diffusion Coefficient for the periodic
Lorentz Gas: An approximate stochastic model for the topological dynamics of the periodic
triangular Lorentz gas is constructed. The model, together with an extremum
principle, is used to find a closed form approximation to the diffusion
coefficient as a function of the lattice spacing. This approximation is
superior to the popular Machta and Zwanzig result and agrees well with a range
of numerical estimates. | nlin_CD |
Averaging Theory for Non-linear Oscillators: I have first discussed how averaging theory can be an effective tool in
solving weakly non-linear oscillators. Then I have applied this technique for a
Van der Pol oscillator and extended the stability criterion of a Van der Pol
oscillator for any integer n(odd or even). | nlin_CD |
Level spacings and periodic orbits: Starting from a semiclassical quantization condition based on the trace
formula, we derive a periodic-orbit formula for the distribution of spacings of
eigenvalues with k intermediate levels. Numerical tests verify the validity of
this representation for the nearest-neighbor level spacing (k=0). In a second
part, we present an asymptotic evaluation for large spacings, where consistency
with random matrix theory is achieved for large k. We also discuss the relation
with the method of Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472] for
two-point correlations. | nlin_CD |
Anomalous transport of a classical wave-particle entity in a tilted
potential: A classical wave-particle entity in the form of a millimetric walking droplet
can emerge on the free surface of a vertically vibrating liquid bath. Such
wave-particle entities have been shown to exhibit hydrodynamic analogs of
quantum systems. Using an idealized theoretical model of this wave-particle
entity in a tilted potential, we explore its transport behavior. The
integro-differential equation of motion governing the dynamics of the
wave-particle entity transforms to a Lorenz-like system of ordinary
differential equations (ODEs) that drives the particle's velocity. Several
anomalous transport regimes such as absolute negative mobility (ANM),
differential negative mobility (DNM) and lock-in regions corresponding to
force-independent mobility, are observed. These observations motivate
experiments in the hydrodynamic walking-droplet system for the experimental
realizations of anomalous transport phenomena. | nlin_CD |
Numerical aspects of eigenvalue and eigenfunction computations for
chaotic quantum systems: We give an introduction to some of the numerical aspects in quantum chaos.
The classical dynamics of two--dimensional area--preserving maps on the torus
is illustrated using the standard map and a perturbed cat map. The quantization
of area--preserving maps given by their generating function is discussed and
for the computation of the eigenvalues a computer program in Python is
presented. We illustrate the eigenvalue distribution for two types of perturbed
cat maps, one leading to COE and the other to CUE statistics. For the
eigenfunctions of quantum maps we study the distribution of the eigenvectors
and compare them with the corresponding random matrix distributions. The Husimi
representation allows for a direct comparison of the localization of the
eigenstates in phase space with the corresponding classical structures.
Examples for a perturbed cat map and the standard map with different parameters
are shown. Billiard systems and the corresponding quantum billiards are another
important class of systems (which are also relevant to applications, for
example in mesoscopic physics). We provide a detailed exposition of the
boundary integral method, which is one important method to determine the
eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several
methods to determine the eigenvalues from the Fredholm equation and illustrate
them for the stadium billiard. The occurrence of spurious solutions is
discussed in detail and illustrated for the circular billiard, the stadium
billiard, and the annular sector billiard. We emphasize the role of the normal
derivative function to compute the normalization of eigenfunctions, momentum
representations or autocorrelation functions in a very efficient and direct
way. Some examples for these quantities are given and discussed. | nlin_CD |
From synchronization to multistability in two coupled quadratic maps: The phenomenology of a system of two coupled quadratic maps is studied both
analytically and numerically. Conditions for synchronization are given and the
bifurcations of periodic orbits from this regime are identified. In addition,
we show that an arbitrarily large number of distinct stable periodic orbits may
be obtained when the maps parameter is at the Feigenbaum period-doubling
accumulation point. An estimate is given for the coupling strength needed to
obtain any given number of stable orbits. | nlin_CD |
Piezoelectric energy harvesting from colored fat-tailed fluctuations: An
electronic analogy: Aiming to optimize piezoelectric energy harvesting from strongly colored
fat-tailed fluctuations, we have recently studied the performance of a
monostable inertial device under a noise whose statistics depends on a
parameter $q$ (bounded for $q<1$, Gaussian for $q=1$, fat-tailed for $q>1$). We
have studied the interplay between the potential shape (interpolating between
square-well and harmonic-like behaviors) and the noise's statistics and
spectrum, and showed that its output power grows as $q$ increases above 1. We
now report a real experiment on an electronic analog of the proposed system,
which sheds light on its operating principle. | nlin_CD |
Bifurcations as dissociation mechanism in bichromatically driven
diatomic molecules: We discuss the influence of periodic orbits on the dissociation of a model
diatomic molecule driven by a strong bichromatic laser fields. Through the
stability of periodic orbits we analyze the dissociation probability when
parameters like the two amplitudes and the phase lag between the laser fields,
are varied. We find that qualitative features of dissociation can be reproduced
by considering a small set of short periodic orbits. The good agreement with
direct simulations demonstrates the importance of bifurcations of short
periodic orbits in the dissociation dynamics of diatomic molecules. | nlin_CD |
Action differences between fixed points and accurate heteroclinic orbits: A general relation is derived for the action difference between two fixed
points and a phase space area bounded by the irreducible component of a
heteroclinic tangle. The determination of this area can require accurate
calculation of heteroclinic orbits, which are important in a wide range of
dynamical system problems. For very strongly chaotic systems initial deviations
from a true orbit are magnified by a large exponential rate making direct
computational methods fail quickly. Here, a method is developed that avoids
direct calculation of the orbit by making use of the well-known stability
property of the invariant unstable and stable manifolds. Under an
area-preserving map, this property assures that any initial deviation from the
stable (unstable) manifold collapses onto themselves under inverse (forward)
iterations of the map. Using a set of judiciously chosen auxiliary points on
the manifolds, long orbit segments can be calculated using the stable and
unstable manifold intersections of the heteroclinic (homoclinic) tangle.
Detailed calculations using the example of the kicked rotor are provided along
with verification of the relation between action differences. The loop
structure of the heteroclinic tangle is necessarily quite different from that
of the turnstile for a homoclinic tangle, its analogous partner. | nlin_CD |
Test of the Fluctuation Relation in lagrangian turbulence on a free
surface: The statistics of lagrangian velocity divergence are studied for an assembly
of particles in compressible turbulence on a free surface. Under an appropriate
definition of entropy, the two-dimensional lagrangian velocity divergence of a
particle trajectory represents the local entropy rate, a random variable. The
statistics of this rate are shown to be in agreement with the fluctuation
relation (FR) over a limited range. The probability distribution functions
(PDFs) obtained in this analysis exhibit features different from those observed
in previous experimental tests. | nlin_CD |
Permutation Complexity and Coupling Measures in Hidden Markov Models: In [Haruna, T. and Nakajima, K., 2011. Physica D 240, 1370-1377], the authors
introduced the duality between values (words) and orderings (permutations) as a
basis to discuss the relationship between information theoretic measures for
finite-alphabet stationary stochastic processes and their permutation
analogues. It has been used to give a simple proof of the equality between the
entropy rate and the permutation entropy rate for any finite-alphabet
stationary stochastic process and show some results on the excess entropy and
the transfer entropy for finite-alphabet stationary ergodic Markov processes.
In this paper, we extend our previous results to hidden Markov models and show
the equalities between various information theoretic complexity and coupling
measures and their permutation analogues. In particular, we show the following
two results within the realm of hidden Markov models with ergodic internal
processes: the two permutation analogues of the transfer entropy, the symbolic
transfer entropy and the transfer entropy on rank vectors, are both equivalent
to the transfer entropy if they are considered as the rates, and the directed
information theory can be captured by the permutation entropy approach. | nlin_CD |
Synchronized states in dissipatively coupled harmonic oscillator
networks: The question under which conditions oscillators with slightly different
frequencies synchronize appears in various settings. We show that
synchronization can be achieved even for harmonic oscillators that are
bilinearly coupled via a purely dissipative interaction. By appropriately tuned
gain/loss stable dynamics may be achieved where for the cases studied in this
work all oscillators are synchronized. These findings are interpreted using the
complex eigenvalues and eigenvectors of the non-Hermitian matrix describing the
dynamics of the system. | nlin_CD |
Strong field double ionization: What is under the "knee"?: Both uncorrelated ("sequential") and correlated ("nonsequential") processes
contribute to the double ionization of the helium atom in strong laser pulses.
The double ionization probability has a characteristic "knee" shape as a
function of the intensity of the pulse. We investigate the phase-space dynamics
of this system, specifically by finding the dynamical structures that regulate
the ionization processes. The emerging picture complements the recollision
scenario by clarifying the distinct roles played by the recolliding and core
electrons. Our analysis leads to verifiable predictions of the intensities
where qualitiative changes in ionization occur, leading to the hallmark "knee"
shape. | nlin_CD |
Spectral statistics of chaotic many-body systems: We derive a trace formula that expresses the level density of chaotic
many-body systems as a smooth term plus a sum over contributions associated to
solutions of the nonlinear Schr\"odinger (or Gross-Pitaevski) equation. Our
formula applies to bosonic systems with discretised positions, such as the
Bose-Hubbard model, in the semiclassical limit as well as in the limit where
the number of particles is taken to infinity. We use the trace formula to
investigate the spectral statistics of these systems, by studying interference
between solutions of the nonlinear Schr\"odinger equation. We show that in the
limits taken the statistics of fully chaotic many-particle systems becomes
universal and agrees with predictions from the Wigner-Dyson ensembles of random
matrix theory. The conditions for Wigner-Dyson statistics involve a gap in the
spectrum of the Frobenius-Perron operator, leaving the possibility of different
statistics for systems with weaker chaotic properties. | nlin_CD |
Dynamics in hybrid complex systems of switches and oscillators: While considerable progress has been made in the analysis of large systems
containing a single type of coupled dynamical component (e.g., coupled
oscillators or coupled switches), systems containing diverse components (e.g.,
both oscillators and switches) have received much less attention. We analyze
large, hybrid systems of interconnected Kuramoto oscillators and Hopfield
switches with positive feedback. In this system, oscillator synchronization
promotes switches to turn on. In turn, when switches turn on they enhance the
synchrony of the oscillators to which they are coupled. Depending on the choice
of parameters, we find theoretically coexisting stable solutions with either
(i) incoherent oscillators and all switches permanently off, (ii) synchronized
oscillators and all switches permanently on, or (iii) synchronized oscillators
and switches that periodically alternate between the on and off states.
Numerical experiments confirm these predictions. We discuss how transitions
between these steady state solutions can be onset deterministically through
dynamic bifurcations or spontaneously due to finite-size fluctuations. | nlin_CD |
A Graphical User Interface to Simulate Classical Billiard Systems: Classical billiards constitute an important class of dynamical systems. They
have not only been in used in mathematical disciplines such as ergodic theory,
but their properties demonstrate fundamental physical phenomena that can be
observed in laboratory settings. This document provides instructions for a
Matlab module that simulates classical billiard systems. It is intended to be
used as both a research and teaching tool. At present, the program efficiently
simulates tables that are constructed entirely from line segments and
elliptical arcs. It functions less reliably for tables with more complex
boundary components. The program and documentation can be downloaded from
\textit{http://www.math.gatech.edu/$\sim$mason/papers/}. | nlin_CD |
Dynamics of the Harper map: Localized states, Cantor spectra and Strange
nonchaotic attractors: The Harper (or ``almost Mathieu'') equation plays an important role in
studies of localization. Through a simple transformation, this equation can be
converted into an iterative two dimensional skew--product mapping of the
cylinder to itself. Localized states of the Harper system correspond to fractal
attractors with nonpositive maximal Lyapunov exponent in the dynamics of the
associated Harper map. We study this map and these strange nonchaotic
attractors (SNAs) in detail in this paper. The spectral gaps of the Harper
system have a unique labeling through a topological invariant of orbits of the
Harper map. This labeling associates an integer index with each gap, and the
scaling properties of the width of the gaps as a function of potential
strength, $\epsilon$ depends on the index. SNAs occur in a large region in
parameter space: these regions have a tongue--like shape and end on a Cantor
set on the line $\epsilon = 1$ where the states are critically localized, and
the spectrum is singular continuous. The SNAs of the Harper map are described
in terms of their fractal properties, and the scaling behaviour of their
power--spectra. These are created by unusual bifurcations and differ in many
respects from SNAs that have hitherto been studied. The technique of studying a
quantum eigenvalue problem in terms of the dynamics of an associated mapping
can be applied to a number of related problems in 1~dimension. We discuss
generalizations of the Harper potential as well as other quasiperiodic
potentials in this context. | nlin_CD |
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