title:
Mesoscopic modeling of heterogeneous boundary conditions for
  microchannel flows

abstract:
  We present a mesoscopic model of the fluid-wall interactions for flows in
microchannel geometries. We define a suitable implementation of the boundary
conditions for a discrete version of the Boltzmann equations describing a
wall-bounded single phase fluid. We distinguish different slippage properties
on the surface by introducing a slip function, defining the local degree of
slip for mesoscopic molecules at the boundaries. The slip function plays the
role of a renormalizing factor which incorporates, with some degree of
arbitrariness, the microscopic effects on the mesoscopic description. We
discuss the mesoscopic slip properties in terms of slip length, slip velocity,
pressure drop reduction (drag reduction), and mass flow rate in microchannels
as a function of the degree of slippage and of its spatial distribution and
localization, the latter parameter mimicking the degree of roughness of the
ultra-hydrophobic material in real experiments. We also discuss the increment
of the slip length in the transition regime, i.e. at O(1) Knudsen numbers.
Finally, we compare our results with Molecular Dynamics investigations of the
dependency of the slip length on the mean channel pressure and local slip
properties (Cottin-Bizonne et al. 2004) and with the experimental dependency of
the pressure drop reduction on the percentage of hydrophobic material deposited
on the surface -- Ou et al. (2004).


title: Two Cellular Automata for the 3x+1 Map

abstract:
  Two simple Cellular Automata, which mimic the Collatz-Ulam iterated map (3x+1
map), are introduced. These Cellular Automata allow to test efficiently the
Collatz conjecture for very large numbers.


title: Hydrodynamic Reductions of Dispersionless Harry Dym Hierarchy

abstract:
  We investigate the reductions of dispersionless Harry Dym hierarchy to
systems of finitely many partial differential equations. These equations must
satisfy the compatibility condition and they are diagonalizable and
semi-Hamiltonian. By imposing a further constraint, the compatibility is
reduced to a system of algebraic equations, whose solutions are described.


title:
Compactlike discrete breathers in systems with nonlinear and nonlocal
  dispersive terms

abstract:
  Discrete breathers with purely anharmonic short-range interaction potentials
localize super-exponentially becoming compact-like. We analyze their spatial
localization properties and their dynamical stability. Several branches of
solutions are identified. One of them connects to the well-known Page and
Sievers-Takeno lattice modes, another one connects with the compacton solutions
of Rosenau. The absence of linear dispersion allows for extremely long-lived
time-quasiperiodic localized excitations. Adding long-range anharmonic
interactions leads to an extreme case of competition between length scales
defining the spatial breather localization. We show that short- and long-range
interaction terms competition results in the appearance of several
characteristic cross-over lengths and essentially breaks the concept of
compactness of the corresponding discrete breathers.


title:
Macroscopic evidence of microscopic dynamics in the Fermi-Pasta-Ulam
  oscillator chain from nonlinear time series analysis

abstract:
  The problem of detecting specific features of microscopic dynamics in the
macroscopic behavior of a many-degrees-of-freedom system is investigated by
analyzing the position and momentum time series of a heavy impurity embedded in
a chain of nearest-neighbor anharmonic Fermi-Pasta-Ulam oscillators. Results
obtained in a previous work [M. Romero-Bastida, Phys. Rev. E {\bf69}, 056204
(2004)] suggest that the impurity does not contribute significantly to the
dynamics of the chain and can be considered as a probe for the dynamics of the
system to which the impurity is coupled. The ($r,\tau$) entropy, which measures
the amount of information generated by unit time at different scales $\tau$ of
time and $r$ of the observable, is numerically computed by methods of nonlinear
time-series analysis using the position and momentum signals of the heavy
impurity for various values of the energy density $\epsilon$ (energy per degree
of freedom) of the system and some values of the impurity mass $M$. Results
obtained from these two time series are compared and discussed.


title:
Solving the Korteweg-de Vries Equation by Its Bilinear Form: Wronskian
  Solutions

abstract:
  A broad set of sufficient conditions consisting of systems of linear partial
differential equations is presented which guarantees that the Wronskian
determinant solves the Korteweg-de Vries equation in the bilinear form. A
systematical analysis is made for solving the resultant linear systems of
second-order and third-order partial differential equations, along with
solution formulas for their representative systems. The key technique is to
apply variation of parameters in solving the involved non-homogeneous partial
differential equations. The obtained solution formulas provide us with a
comprehensive approach to construct the existing solutions and many new
solutions including rational solutions, solitons, positons, negatons,
breathers, complexitons and interaction solutions of the Korteweg-de Vries
equation.


title: Nodal domain distribution of rectangular drums

abstract:
  We consider the sequence of nodal counts for eigenfunctions of the
Laplace-Beltrami operator in two dimensional domains. It was conjectured
recently that this sequence stores some information pertaining to the geometry
of the domain, and we show explicitly that this is the case for the family of
rectangular domains with Dirichlet boundary conditions.


title:
A phenomenological approach to normal form modeling: a case study in
  laser induced nematodynamics

abstract:
  An experimental setting for the polarimetric study of optically induced
dynamical behavior in nematic liquid crystal films has allowed to identify most
notably some behavior which was recognized as gluing bifurcations leading to
chaos. This analysis of the data used a comparison with a model for the
transition to chaos via gluing bifurcations in optically excited nematic liquid
crystals previously proposed by G. Demeter and L. Kramer. The model of these
last authors, proposed about twenty years before, does not have the central
symmetry which one would expect for minimal dimensional models for chaos in
nematics in view of the time series. What we show here is that the simplest
truncated normal forms for gluing, with the appropriate symmetry and minimal
dimension, do exhibit time signals that are embarrassingly similar to the ones
found using the above mentioned experimental settings. The gluing bifurcation
scenario itself is only visible in limited parameter ranges and substantial
aspect of the chaos that can be observed is due to other factors. First, out of
the immediate neighborhood of the homoclinic curve, nonlinearity can produce
expansion leading to chaos when combined with the recurrence induced by the
homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme
sensitivity to noise, so that when the noiseless approach contains a rich
behavior, minute noise can transform the complex damping into sustained chaos.
Leonid Shil'nikov taught us that combining global considerations and local
spectral analysis near critical points is crucial to understand the
phenomenology associated to homoclinic bifurcations. Here this helps us
construct a phenomenological approach to modeling experiments in nonlinear
dissipative contexts.


title: Generalized problem of two and four Newtonian centers

abstract:
  We consider integrable spherical analogue of the Darboux potential, which
appear in the problem (and its generalizations) of the planar motion of a
particle in the field of two and four fixed Newtonian centers. The obtained
results can be useful when constructing a theory of motion of satellites in the
field of an oblate spheroid in constant curvature spaces.


title:
Weak turbulence and collapses in the Majda-Mc-Laughlin-Tabak equation:
  Fluxes in wavenumber and in amplitude space

abstract:
  The turbulent energy flow of the onedimensional Majda-McLaughlin-Tabak
equation is studied numerically. The system exhibits weak turbulence for weak
driving forces, while weak turbulence coexists with strongly nonlinear
intermittent collapses when the system is strongly driven. These two types of
dynamics can be distinguished by their energy and particle fluxes. The weakly
turbulent process can be characterized by fluxes in wavenumber space, while
additional fluxes in amplitude space emerge in the intermittent process. The
particle flux is directed from low amplitudes towards high amplitudes, and the
energy flows in the opposite direction.


title: Alternative determinism principle for topological analysis of chaos

abstract:
  The topological analysis of chaos based on a knot-theoretic characterization
of unstable periodic orbits has proved a powerful method, however knot theory
can only be applied to three-dimensional systems. Still, the core principles
upon which this approach is built, determinism and continuity, apply in any
dimension. We propose an alternative framework in which these principles are
enforced on triangulated surfaces rather than curves and show that in dimension
three our approach numerically predicts the correct topological entropies for
periodic orbits of the horseshoe map.


title:
Joint statistics of acceleration and vorticity in fully developed
  turbulence

abstract:
  We report results from a high resolution numerical study of fluid particles
transported by a fully developed turbulent flow. Single particle trajectories
were followed for a time range spanning more than three decades, from less than
a tenth of the Kolmogorov time-scale up to one large-eddy turnover time. We
present results concerning acceleration statistics and the statistics of
trapping by vortex filaments conditioned to the local values of vorticity and
enstrophy. We distinguish two different behaviors between the joint statistics
of vorticity and centripetal acceleration or vorticity and longitudinal
acceleration.


title:
Varying the Population Size of Artificial Foraging Swarms on Time
  Varying Landscapes

abstract:
  Swarm Intelligence (SI) is the property of a system whereby the collective
behaviors of (unsophisticated) entities interacting locally with their
environment cause coherent functional global patterns to emerge. SI provides a
basis with wich it is possible to explore collective (or distributed) problem
solving without centralized control or the provision of a global model. In this
paper we present a Swarm Search Algorithm with varying population of agents.
The swarm is based on a previous model with fixed population which proved its
effectiveness on several computation problems. We will show that the variation
of the population size provides the swarm with mechanisms that improves its
self-adaptability and causes the emergence of a more robust self-organized
behavior, resulting in a higher efficiency on searching peaks and valleys over
dynamic search landscapes represented here - for the purpose of different
experiments - by several three-dimensional mathematical functions that suddenly
change over time. We will also show that the present swarm, for each function,
self-adapts towards an optimal population size, thus self-regulating.


title:
On elliptic solutions of the cubic complex one-dimensional
  Ginzburg-Landau equation

abstract:
  The cubic complex one-dimensional Ginzburg-Landau equation is considered.
Using the Hone's method, based on the use of the Laurent-series solutions and
the residue theorem, we have proved that this equation has neither elliptic
standing wave nor elliptic travelling wave solutions. This result amplifies the
Hone's result, that this equation has no elliptic travelling wave solutions.

categories: nlin.PS cond-mat.other hep-th math-ph math.MP


title: Resolution of a shock in hyperbolic systems modified by weak dispersion

abstract:
  We present a way to deal with dispersion-dominated ``shock-type'' transition
in the absence of completely integrable structure for the systems that one may
characterize as strictly hyperbolic regularized by a small amount of
dispersion. The analysis is performed by assuming that, the dispersive shock
transition between two different constant states can be modelled by an
expansion fan solution of the associated modulation (Whitham) system for the
short-wavelength nonlinear oscillations in the transition region (the so-called
Gurevich -- Pitaevskii problem). We consider as single-wave so bi-directional
systems. The main mathematical assumption is that of hyperbolicity of the
Whitham system for the solutions of our interest. By using general properties
of the Whitham averaging for a certain class of nonlinear dispersive systems
and specific features of the Cauchy data prescription on characteristics we
derive a set of transition conditions for the dispersive shock, actually
bypassing full integration of the modulation equations. Along with model KdV
and mKdV examples, we consider a non-integrable system describing fully
nonlinear ion-acoustic waves in collisionless plasma. In all cases our
transition conditions are in complete agreement with previous analytical and
numerical results.


title: How much information can one store in a non-equilibrium medium?

abstract:
  It has recently been emphasized again that the very existence of stationary
stable localized structures with short range interactions might allow to store
information in non-equilibrium media, opening new perspectives on information
storage. We show how to use generalized topological entropies to measure
aspects of the quantities of storable and non-storable information. This leads
us to introduce a measure of the long term stably storable information. As a
first example to illustrate these concepts, we revisit a mechanism for the
appearance of stationary stable localized structures that is related to the
stabilization of fronts between structured and unstructured states (or between
differently structured states).


title:
Universal Statistics of the Scattering Coefficient of Chaotic Microwave
  Cavities

abstract:
  We consider the statistics of the scattering coefficient S of a chaotic
microwave cavity coupled to a single port. We remove the non-universal effects
of the coupling from the experimental S data using the radiation impedance
obtained directly from the experiments. We thus obtain the normalized, complex
scattering coefficient whose Probability Density Function (PDF) is predicted to
be universal in that it depends only on the loss (quality factor) of the
cavity. We compare experimental PDFs of the normalized scattering coefficients
with those obtained from Random Matrix Theory (RMT), and find excellent
agreement. The results apply to scattering measurements on any wave chaotic
system.


title: Flow distributed oscillation, flow velocity modulation and resonance

abstract:
  We examine the effects of a periodically varying flow velocity on the
standing and travelling wave patterns formed by the flow-distributed
oscillation (FDO) mechanism. In the kinematic (or diffusionless) limit, the
phase fronts undergo a simple, spatiotemporally periodic longitudinal
displacement. On the other hand, when the diffusion is significant, periodic
modulation of the velocity can disrupt the wave pattern, giving rise in the
downstream region to travelling waves whose frequency is a rational multiple of
the velocity perturbation frequency. We observe frequency locking at ratios of
1:1, 2:1 and 3:1, depending on the amplitude and frequency of the velocity
modulation. This phenomenon can be viewed as a novel, rather subtle type of
resonant forcing.


title:
Chaotic Dynamics of a Nonlinear Ring Cavity Driven by an External
  Multi-frequency Signal

abstract:
  Complex dynamics of a ring cavity filled by the medium with cubic phase
non-linearity driven by an external multi-frequency signal is studied. To
describe the dynamics of envelope amplitudes the system of coupled Ikeda maps
was derived. The results of numerical simulations in the case of two-frequency
external signal are presented. Complex dynamics observed in these numerical
simulations is expected to be peculiar for other physical systems that can be
treated as nonlinear resonators driven by an external force.


title:
Eulerian spectral closures for isotropic turbulence using a time-ordered
  fluctuation-dissipation relation

abstract:
  Procedures for time-ordering the covariance function, as given in a previous
paper (K. Kiyani and W.D. McComb Phys. Rev. E 70, 066303 (2004)), are extended
and used to show that the response function associated at second order with the
Kraichnan-Wyld perturbation series can be determined by a local (in wavenumber)
energy balance. These time-ordering procedures also allow the two-time
formulation to be reduced to time-independent form by means of exponential
approximations and it is verified that the response equation does not have an
infra-red divergence at infinite Reynolds number. Lastly, single-time
Markovianised closure equations (stated in the previous paper above) are
derived and shown to be compatible with the Kolmogorov distribution without the
need to introduce an ad hoc constant.


title:
Statistical properties of periodic orbits in 4-disk billiard system:
  pruning-proof property

abstract:
  Periodic orbit theory for classical hyperbolic system is very significant
matter of how we can interpret spectral statistics in terms of semiclassical
theory. Although pruning is significant and generic property for almost all
hyperbolic systems, pruning-proof property for the correlation among the
periodic orbits which gains a resurgence of second term of the random matrix
form factor remains open problem. In the light of the semiclassical form
factor, our attention is paid to statistics for the pairs of periodic orbits.
Also in the context of pruning, we investigated statistical properties of the
"actual" periodic orbits in 4-disk billiard system. This analysis presents some
universality for pair-orbits' statistics. That is, even if the pruning
progresses, there remains the periodic peak structure in the statistics for
periodic orbit pairs. From that property, we claim that if the periodic peak
structure contributes to the correlation, namely the off-diagonal part of the
semiclassical form factor, then the correlation must remain while pruning
progresse.


title: Notes on diffusion in collisionless medium

abstract:
  A collisionless continuous medium in Euclidean space is discussed, i.e. a
continuum of free particles moving inertially, without interacting with each
other. It is shown that the distribution density of such medium is weakly
converging to zero as time increases indefinitely. In the case of Maxwell's
velocity distribution of particles, this density satisfies the well-known
diffusion equation, the diffusion coefficient increasing linearly with time.


title: Billiards, invariant measures, and equilibrium thermodynamics. II

abstract:
  The kinetics of collisionless continuous medium is studied in a bounded
region on a curved manifold. We have assumed that in statistical equilibrium,
the probability distribution density depends only on the total energy. It is
shown that in this case, all the fundamental relations for a multi-dimensional
ideal gas in thermal equilibrium hold true.


title: Perturbative analysis of wave interactions in nonlinear systems

abstract:
  This work proposes a new way for handling obstacles to asymptotic
integrability in perturbed nonlinear PDEs within the method of Normal Forms -
NF - for the case of multi-wave solutions. Instead of including the whole
obstacle in the NF, only its resonant part is included, and the remainder is
assigned to the homological equation. This leaves the NF intergable and its
solutons retain the character of the solutions of the unperturbed equation. We
exploit the freedom in the expansion to construct canonical obstacles which are
confined to te interaction region of the waves. Fo soliton solutions, e.g., in
the KdV equation, the interaction region is a finite domain around the origin;
the canonical obstacles then do not generate secular terms in the homological
equation. When the interaction region is infifnite, or semi-infinite, e.g., in
wave-front solutions of the Burgers equation, the obstacles may contain
resonant terms. The obstacles generate waves of a new type, which cannot be
written as functionals of the solutions of the NF. When an obstacle contributes
a resonant term to the NF, this leads to a non-standard update of th wave
velocity.


title: Wave interactions and the analysis of the perturbed Burgers equation

abstract:
  In multiple-front solutions of the Burgers equation, all the fronts, except
for two, are generated through the inelastic interaction of exponential wave
solutions of the Lax pair associated with the equation. The inelastically
generated fronts are the source of two difficulties encountered in the standard
Normal Form expansion of the approximate solution of the perturbed Burgers
equation, when the zero-order term is a multiple-front solution: (i) The
higher-order terms in the expansion are not bounded; (ii) The Normal Form
(equation obeyed by the zero-order approximation) is not asymptotically
integrable; its solutions lose the simple wave structure of the solutions of
the un-perturbed equation. The freedom inherent in the Normal Form method
allows a simple modification of the expansion procedure, making it possible to
overcome both problems in more than one way. The loss of asymptotic
integrability is shifted from the Normal Form to the higher-order terms (part
of which has to be computed numerically) in the expansion of the solution. The
front-velocity update is different from the one obtained in the standard
analysis.


title:
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread
  and Singularity

abstract:
  We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
  For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
  For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian.


title: Kinetics of Collisionless Continuous Medium

abstract:
  In this article we develop Poincar\'e ideas about a heat balance of ideal gas
considered as a collisionless continuous medium. We obtain the theorems on
diffusion in nondegenerate completely integrable systems. As a corollary we
show that for any initial distribution the gas will be eventually irreversibly
and uniformly distributed over all volume, although every particle during this
process approaches arbitrarily close to the initial position indefinitely many
times. However, such individual returnability is not uniform, which results in
diffusion in a reversible and conservative system. Balancing of pressure and
internal energy of ideal gas is proved, the formulas for limit values of these
quantities are given and the classical law for ideal gas in a heat balance is
deduced. It is shown that the increase of entropy of gas under the adiabatic
extension follows from the law of motion of a collisionless continuous medium.


title:
Spatio temporal characterization of interfacial Faraday waves: A new
  absorption technique

abstract:
  We present measurements of the complete spatio-temporal Fourier spectrum of
Faraday waves. The Faraday waves are generated at the interface of two
immiscible index matched liquids of different density. By use of a new
absorption technique we are able to determine the bifurcation scenario from the
flat surface to the patterned state for each complex spatial and temporal
Fourier component separately. The measured surface spectra at onset are in good
agreement with the predictions from a linear stability analysis. For the
nonlinear state our measurements show in a direct manner how energy is
transferred from lower to higher harmonics and we quantify the nonlinear
coupling coefficients. Furthermore we find that the nonlinear coupling
generates constant components in the spatial Fourier spectrum leading to a
contribution of a non oscillating permanent sinusoidal deformed surface state.
A comparison of a hexagonal and a rectangular pattern reveals that spatial
resonances only can give rise to a spectrum that violates the temporal
resonance conditions given by the weakly nonlinear theory.


title: Galton Board

abstract:
  In this paper, we present results of simulations of a model of the Galton
board for various degrees of elasticity of the ball-to-nail collision.


title:
Validity of the linear marginal stability principle for monotonic fronts
  of the extended Fisher-Kolmogorov equation

abstract:
  The extended Fisher Kolmogorov equation $u_t = u_{xx} - \gamma u_{xxxx} +
f(u)$ with arbitrary positive $f(u)$, satisfying $f(0) = f(1) =0$, has
monotonic traveling fronts for $\gamma < 1/12$. We find a simple lower bound on
the speed of the fronts which allows to assess the validity of linear marginal
stability.


title:
On the principal bifurcation branch of a third order nonlinear long-wave
  equation

abstract:
  We study the principal bifurcation curve of a third order equation which
describes the nonlinear evolution of several systems with a long--wavelength
instability. We show that the main bifurcation branch can be derived from a
variational principle. This allows to obtain a close estimate of the complete
branch. In particular, when the bifurcation is subcritical, the large amplitude
stable branch can be found in a simple manner.


title: On the Integration Theory of Equations of Nonholonomic Mechanics

abstract:
  The paper deals with the problem of integration of equations of motion in
nonholonomic systems. By means of well-known theory of the differential
equations with an invariant measure the new integrable systems are discovered.
Among them there are the generalization of Chaplygin's problem of rolling
nonsymmetric ball in the plane and the Suslov problem of rotation of rigid body
with a fixed point. The structure of dynamics of systems on the invariant
manifold in the integrable problems is shown. Some new ideas in the theory of
integration of the equations in nonholonomic mechanics are suggested. The first
of them consists in using known integrals as the constraints. The second is the
use of resolvable groups of symmetries in nonholonomic systems. The existence
conditions of invariant measure with analytical density for the differential
equations of nonholonomic mechanics is given.


title: Statistics of tumbling of a single polymer molecule in shear flow

abstract:
  We present experimental results on statistics of polymer orientation angles
relatively to shear plane and tumbling times in shear flow with thermal noise.
Strong deviation of probability distribution functions (PDF) of these
parameters from Gaussian was observed and a good accord with theory was found.
The scaling relations of PDF widths for both angles as a function of the
control parameter $Wi$ are verified and compared with numerics. An universal
exponential PDF tail for the tumbling times and its predicted scaling with $Wi$
are also tested experimentally against numerics.


title: Polymers in linear shear flow: a numerical study

abstract:
  We study the dynamics of a single polymer subject to thermal fluctuations in
a linear shear flow. The polymer is modeled as a finitely extendable nonlinear
elastic FENE dumbbell. Both orientation and elongation dynamics are
investigated numerically as a function of the shear strength, by means of a new
efficient integration algorithm. The results are in agreement with recent
experiments.


title:
A dispersionless integrable system associated to Diff$(S^1)$ gauge
  theory

abstract:
  A dispersionless integrable system underlying 2+1-dimensional hyperCR
Einstein--Weyl structures is obtained as a symmetry reduction of the
anti--self--dual Yang--Mills equations with the gauge group Diff$(S^1)$. Two
special classes of solutions are obtained from well known soliton equations by
embedding SU(1,1) in ${Diff}(S^1)$.


title:
A remark on the existence of breather solutions for the Discrete
  Nonlinear Schr\"odinger Equation in infinite lattices: The case of site
  dependent anharmonic parameter

abstract:
  We discuss the existence of breather solutions for a Discrete Nonlinear
Schr\"odinger equation in an infinite $N$-dimensional lattice, involving site
dependent anharmonic parameter. We give a simple proof on the existence of
(nontrivial) breather solutions based on variational approach, assuming that
the sequence of anharmonic parmeters is in an appropriate sequence space
(decays with an appropriate rate). We also give a proof on the non-existence of
(non-trivial) breather solutions, and discuss a possible physical
interpretation of the restrictions, both in the existence and nonexistence
case.


title: Geometric reduction of Hamiltonian systems

abstract:
  Given a foliation S of a manifold M, a distribution Z in M transveral to S
and a Poisson bivector \Pi on M we present a geometric method of reducing this
operator on the foliation S along the distribution Z. It encompasses the
classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden
and T. Ratiu, but our method leads to formulas that allow for an explicit
calculation of the reduced Poisson bracket. Moreover, we analyse the reduction
of Hamiltonian systems corresponding to the bivector \Pi.


title: Multi-peakon solutions of the Degasperis-Procesi equation

abstract:
  We present an inverse scattering approach for computing n-peakon solutions of
the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH)
shallow water equation). The associated non-self-adjoint spectral problem is
shown to be amenable to analysis using the isospectral deformations induced
from the n-peakon solution, and the inverse problem is solved by a method
generalizing the continued fraction solution of the peakon sector of the CH
equation.


title:
Pattern dynamics in a two-dimensional gas discharge system :
  Intermittent collapse and recreation, moving of spots, dynamic change of
  clusters

abstract:
  Reaction-diffusion equation model for a 2-dimensional gas discharge system is
introduced in close relationship with the recent experiment by Nasuno. The
model shows formation of spots, molecule-like organization of a cluster of
spots, moving of spots, and intermittent collapse and recreation of spots, in
agreement with experiments. The pattern dynamics are classified into
distributed, localized, and moving spots, and periodic wave pattern. The phase
diagram displayed against the current and pressure has some agreement with that
observed in the experiment. Dynamic change of spot numbers as well as the
structure of the spot cluster is studied in terms of the collective flow of
charge, and discussed as chaotic itinerancy.


title: Properties making a chaotic system a good Pseudo Random Number Generator

abstract:
  We discuss two properties making a deterministic algorithm suitable to
generate a pseudo random sequence of numbers: high value of Kolmogorov-Sinai
entropy and high-dimensionality. We propose the multi dimensional Anosov
symplectic (cat) map as a Pseudo Random Number Generator. We show what chaotic
features of this map are useful for generating Pseudo Random Numbers and
investigate numerically which of them survive in the discrete version of the
map. Testing and comparisons with other generators are performed.


title: Degasperis-Procesi peakons and the discrete cubic string

abstract:
  We use an inverse scattering approach to study multi-peakon solutions of the
Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm
shallow water equation. The spectral problem associated to the DP equation is
equivalent under a change of variables to what we call the cubic string
problem, which is a third order non-selfadjoint generalization of the
well-known equation describing the vibrational modes of an inhomogeneous string
attached at its ends. We give two proofs that the eigenvalues of the cubic
string are positive and simple; one using scattering properties of DP peakons,
and another using the Gantmacher-Krein theory of oscillatory kernels.
  For the discrete cubic string (analogous to a string consisting of n point
masses) we solve explicitly the inverse spectral problem of reconstructing the
mass distribution from suitable spectral data, and this leads to explicit
formulas for the general n-peakon solution of the DP equation. Central to our
study of the inverse problem is a peculiar type of simultaneous rational
approximation of the two Weyl functions of the cubic string, similar to
classical Pade-Hermite approximation but with lower order of approximation and
an additional symmetry condition instead. The results obtained are intriguing
and nontrivial generalizations of classical facts from the theory of Stieltjes
continued fractions and orthogonal polynomials.


title:
Semiclassical Study of Statistical Measures of Spectral Fluctuations in
  Some Pseudointegrable Billiards

abstract:
  Using semi-classical formalism and asymptotic proliferation law of periodic
orbits, we obtain an analytical expressions for the two-level cluster function,
spectral form factor, level spacing distribution and the number variance for
pseudo-integrable billiards. Our analysis shows that these spectral fluctuation
measures depend on two parameters as well as on the number of levels
considered. These parameters are, (1) projected phase space area occupied by
different classes of bands of periodic orbits and (2) length of the shortest
periodic orbit characteristic of these classes.


title: An analytically solvable model of probabilistic network dynamics

abstract:
  We present a simple model of network dynamics that can be solved analytically
for uniform networks. We obtain the dynamics of response of the system to
perturbations. The analytical solution is an excellent approximation for random
networks. A comparison with the scale-free network, though qualitatively
similar, shows the effect of distinct topology.


title: Boundary effects on localized structures in spatially extended systems

abstract:
  We present a general method of analyzing the influence of finite size and
boundary effects on the dynamics of localized solutions of non-linear spatially
extended systems. The dynamics of localized structures in infinite systems
involve solvability conditions that require projection onto a Goldstone mode.
Our method works by extending the solvability conditions to finite sized
systems, by incorporating the finite sized modifications of the Goldstone mode
and associated nonzero eigenvalue. We apply this method to the special case of
non-equilibrium domain walls under the influence of Dirichlet boundary
conditions in a parametrically forced complex Ginzburg Landau equation, where
we examine exotic nonuniform domain wall motion due to the influence of
boundary conditions.


title:
How Far Can We Go Through Social System?: Algorithmic Information Social
  Theories

abstract:
  The paper elaborates an endeavor on applying the algorithmic
information-theoretic computational complexity to meta-social-sciences. It is
motivated by the effort on seeking the impact of the well-known incompleteness
theorem to the scientific methodology approaching social phenomena. The paper
uses the binary string as the model of social phenomena to gain understanding
on some problems faced in the philosophy of social sciences or some traps in
sociological theories. The paper ends on showing the great opportunity in
recent social researches and some boundaries that limit them.


title:
An example of physical system with hyperbolic attractor of Smale -
  Williams type

abstract:
  A simple and transparent example of a non-autonomous flow system, with
hyperbolic strange attractor is suggested. The system is constructed on a basis
of two coupled van der Pol oscillators, the characteristic frequencies differ
twice, and the parameters controlling generation in both oscillators undergo a
slow periodic counter-phase variation in time. In terms of stroboscopic
Poincar\'{e} section, the respective four-dimensional mapping has a hyperbolic
strange attractor of Smale - Williams type. Qualitative reasoning and
quantitative data of numerical computations are presented and discussed, e.g.
Lyapunov exponents and their parameter dependencies. A special test for
hyperbolicity based on statistical analysis of distributions of angles between
stable and unstable subspaces of a chaotic trajectory has been performed.
Perspectives of further comparative studies of hyperbolic and non-hyperbolic
chaotic dynamics in physical aspect are outlined.


title:
Integrability of one degree of freedom symplectic maps with polar
  singularities

abstract:
  In this paper, we treat symplectic difference equations with one degree of
freedom. For such cases, we resolve the relation between that the dynamics on
the two dimensional phase space is reduced to on one dimensional level sets by
a conserved quantity and that the dynamics is integrable, under some
assumptions. The process which we introduce is related to interval exchange
transformations.


title: Classical Loschmidt echo in chaotic many-body systems

abstract:
  General theoretic approach to classical Loschmidt echoes in chaotic systems
with many degrees of freedom is developed. For perturbations which affect
essentially all degrees of freedom we find a doubly exponential decay with the
rate determined by the largest Lyapunov exponent. The scaling of the decay rate
on the perturbation strength depends on whether the initial phase-space density
is continuous or not.


title:
The permutation entropy rate equals the metric entropy rate for ergodic
  information sources and ergodic dynamical systems

abstract:
  Permutation entropy quantifies the diversity of possible orderings of the
values a random or deterministic system can take, as Shannon entropy quantifies
the diversity of values. We show that the metric and permutation entropy
rates--measures of new disorder per new observed value--are equal for ergodic
finite-alphabet information sources (discrete-time stationary stochastic
processes). With this result, we then prove that the same holds for
deterministic dynamical systems defined by ergodic maps on $n$% -dimensional
intervals. This result generalizes a previous one for piecewise monotone
interval maps on the real line (Bandt, Keller and Pompe, "Entropy of interval
maps via permutations",\textit{Nonlinearity} \textbf{15}, 1595-602, (2002)), at
the expense of requiring ergodicity and using a definition of permutation
entropy rate differing in the order of two limits. The case of non-ergodic
finite-alphabet sources is also studied and an inequality developed. Finally,
the equality of permutation and metric entropy rates is extended to ergodic
non-discrete information sources when entropy is replaced by differential
entropy in the usual way.


title: Fluctuations of temperature gradients in turbulent thermal convection

abstract:
  Broad theoretical arguments are proposed to show, formally, that the
magnitude G of the temperature gradients in turbulent thermal convection at
high Rayleigh numbers obeys the same advection-diffusion equation that governs
the temperature fluctuation T, except that the velocity field in the new
equation is substantially smoothed. This smoothed field leads to a -1 scaling
of the spectrum of G in the same range of scales for which the spectral
exponent of T lies between -7/5 and -5/3. This result is confirmed by
measurements in a confined container with cryogenic helium gas as the working
fluid for Rayleigh number Ra=1.5x10^{11}. Also confirmed is the logarithmic
form of the autocorrelation function of G. The anomalous scaling of
dissipation-like quantities of T and G are identical in the inertial range,
showing that the analogy between the two fields is quite deep.


title: Logarithmic scaling in the near-dissipation range of turbulence

abstract:
  A logarithmic scaling for structure functions, in the form $S_p \sim [\ln
(r/\eta)]^{\zeta_p}$, where $\eta$ is the Kolmogorov dissipation scale and
$\zeta_p$ are the scaling exponents, is suggested for the statistical
description of the near-dissipation range for which classical power-law scaling
does not apply. From experimental data at moderate Reynolds numbers, it is
shown that the logarithmic scaling, deduced from general considerations for the
near-dissipation range, covers almost the entire range of scales (about two
decades) of structure functions, for both velocity and passive scalar fields.
This new scaling requires two empirical constants, just as the classical
scaling does, and can be considered the basis for extended self-similarity.


title:
Modulational instability of two pairs of counter-propagating waves and
  energy exchange in two-component media

abstract:
  The dynamics of two pairs of counter-propagating waves in two-component media
is considered within the framework of two generally nonintegrable coupled
Sine-Gordon equations. We consider the dynamics of weakly nonlinear wave
packets, and using an asymptotic multiple-scales expansion we obtain a suite of
evolution equations to describe energy exchange between the two components of
the system. Depending on the wave packet length-scale vis-a-vis the wave
amplitude scale, these evolution equations are either four non-dispersive and
nonlinearly coupled envelope equations, or four non-locally coupled nonlinear
Schroedinger equations. We also consider a set of fully coupled nonlinear
Schroedinger equations, even though this system contains small dispersive terms
which are strictly beyond the leading order of the asymptotic multiple-scales
expansion method. Using both the theoretical predictions following from these
asymptotic models and numerical simulations of the original unapproximated
equations, we investigate the stability of plane-wave solutions, and show that
they may be modulationally unstable. These instabilities can then lead to the
formation of localized structures, and to a modification of the energy exchange
between the components. When the system is close to being integrable, the
time-evolution is distinguished by a remarkable almost periodic sequence of
energy exchange scenarios, with spatial patterns alternating between
approximately uniform wavetrains and localized structures.


title: Gray soliton solution in the extended nonlinear Schrodinger equation

abstract:
  In the framework of extended nonlinear Schrodinger equation (ENSE) the
classification of self-similar solutions by the relation between the amplitude
and phase is performed. New solutions of ENSE - ``gray soliton'' and ``gray
soliton chain'' are presented. The properties of these solutions and the
possibility for using theirs in physical applications are discussed.


title: Dynamics and statics of vortices on a plane and a sphere - I

abstract:
  In the present paper a description of a problem of point vortices on a plane
and a sphere in the "internal" variables is discussed. The hamiltonian
equations of motion of vortices on a plane are built on the Lie-Poisson
algebras, and in the case of vortices on a sphere on the quadratic Jacobi
algebras. The last ones are obtained by deformation of the corresponding linear
algebras. Some partial solutions of the systems of three and four vortices are
considered. Stationary and static vortex configurations are found.


title:
Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda
  orbits and isomonodromic tau functions

abstract:
  We consider the class of biorthogonal polynomials that are used to solve the
inverse spectral problem associated to elementary co-adjoint orbits of the
Borel group of upper triangular matrices; these orbits are the phase space of
generalized integrable lattices of Toda type. Such polynomials naturally
interpolate between the theory of orthogonal polynomials on the line and
orthogonal polynomials on the unit circle and tie together the theory of Toda,
relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish
corresponding Christoffel-Darboux formulae . For all these classes of
polynomials a 2x2 system of Differential-Difference-Deformation equations is
analyzed in the most general setting of pseudo measures with arbitrary rational
logarithmic derivative. They provide particular classes of isomonodromic
deformations of rational connections on the Riemann sphere. The corresponding
isomonodromic tau function is explicitly related to the shifted Toeplitz
determinants of the moments of the pseudo-measure. In particular the results
imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure)
with arbitrary rational logarithmic derivative is an isomonodromic tau
function.


title: Resemblances and differences in mechanisms of noise-induced resonance

abstract:
  Systems showing stochastic resonance (SR) or coherent resonance (CR) share
some features, in particular the nearby periodic character of the signal. We
show that in spite of this resemblance the different underlying dynamics can be
detected in experimental data by studying the histogram of inter-spikes times
and some statistical properties like two-times correlation functions. We
discuss the possible relevance for climate modeling.


title: Periodic-Orbit Theory of Universality in Quantum Chaos

abstract:
  We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory,
that full classical chaos is paralleled by quantum energy spectra with
universal spectral statistics, in agreement with random-matrix theory. For
dynamics from all three Wigner-Dyson symmetry classes, we calculate the
small-time spectral form factor $K(\tau)$ as power series in the time $\tau$.
Each term $\tau^n$ of that series is provided by specific families of pairs of
periodic orbits. The contributing pairs are classified in terms of close
self-encounters in phase space. The frequency of occurrence of self-encounters
is calculated by invoking ergodicity. Combinatorial rules for building pairs
involve non-trivial properties of permutations. We show our series to be
equivalent to perturbative implementations of the non-linear sigma models for
the Wigner-Dyson ensembles of random matrices and for disordered systems; our
families of orbit pairs are one-to-one with Feynman diagrams known from the
sigma model.


title:
The Evolution of Cellar Automaton based on Dilemmma Games with Selfish
  Strategy

abstract:
  We have proposed two new evolutionary rules on spatio-iterated games that is
not mimic evolution of strategies, and mainly discussed the Prisoner's Dilemma
game \cite{toyota2} by the two evoutionary rules \cite{toyota3}. In this paper
we focus the first rule, that is, the selfish evolutionary rule for various
dilemma games. In contrast to the Prisoner's Dilemma, there are gererally rich
pase structures in the dilemma games. First we analytically clear the structure
to present phase diagrams in the various dilemma games. Forthermore we simulate
the time evolution of the soatio-games in the some representatives of the
parameters according to the phase diagrams.
  Including some mutations, detail investigations are made by a computer
simulation for five kinds of initial configurations. As results we find some
dualities and game invariant properties. They show a sort of bifurcation as a
mutation parameter are varied. In the path from one period to two one some
common features are observed in most of games and some chaotic behaviors appear
in the middle of the transition. Lastly we estimate the total hamiltonian,
which is defined by the sum of the total payoff of all agents in the system,
and show that the chaotic period is best from the perspective of the payoff. We
also made some primitive discussions on them.


title:
Dynamics of three vortices on a plane and a sphere - II. General compact
  case

abstract:
  Integrable problem of three vorteces on a plane and sphere are considered.
The classification of Poisson structures is carried out. We accomplish the
bifurcational analysis using the variables introduced in previous part of the
work.


title: Discrete Toda field equations

abstract:
  Discrete analogs of the finite and affine Toda field equations are found
corresponding to the Lie algebras of series $C_N$ and $\tilde{C_N}$. Their Lax
pairs are represented.


title: On the motion of a heavy rigid body in an ideal fluid with circulation

abstract:
  Chaplygin's equations describing the planar motion of a rigid body in an
unbounded volume of an ideal fluid involved in a circular flow around the body
are considered. Hamiltonian structures, new integrable cases, and partial
solutions are revealed, and their stability is examined. The problems of
non-integrability of the equations of motion because of a chaotic behavior of
the system are discussed.


title:
Dynamics of three vortices on a plane and a sphere -- III. Noncompact
  case. Problems of collaps and scattering

abstract:
  In this article we considered the integrable problems of three vortices on a
plane and sphere for noncompact case. We investigated explicitly the problems
of a collapse and a scattering of vortices and obtained the conditions of its
realization. We completed the bifurcation analysis and investigated for
collinear and Thomson's configurations the dependence of stability in linear
approximation and frequency of rotation in relative coordinates from value of a
full moment. We indicated the geometric interpretation for characteristic
situations. We constructed a phase portrait and geometric projection for the
integrable configuration of four vortices on a plane.


title: Correlation Function Bootstrapping in Quantum Chaotic Systems

abstract:
  We discuss a general and efficient approach for "bootstrapping" short-time
correlation data in chaotic or complex quantum systems to obtain information
about long-time dynamics and stationary properties, such as the local density
of states. When the short-time data is sufficient to identify an individual
quantum system, we obtain a systematic approximation for the spectrum and wave
functions. Otherwise, we obtain statistical properties, including wave function
intensity distributions, for an ensemble of all quantum systems sharing the
given short-time correlations. The results are valid for open or closed
systems, and are stable under perturbation of the short-time input data.
Numerical examples include quantum maps and two-dimensional anharmonic
oscillators.


title:
Nonlinear Schr\"odinger Equation with a White-Noise Potential:
  Phase-space Approach to Spread and Singularity

abstract:
  We propose a phase-space formulation for the nonlinear Schr\"odinger equation
with a white-noise potential in order to shed light on two issues: the rate of
spread and the singularity formation in the average sense. Our main tools are
the energy law and the variance identity. The method is completely elementary.
  For the problem of wave spread, we show that the ensemble-averaged dispersion
in the critical or defocusing case follows the cubic-in-time law while in the
supercritical and subcritical focusing cases the cubic law becomes an upper and
lower bounds respectively.
  We have also found that in the critical and supercritical focusing cases the
presence of a white-noise random potential results in different conditions for
singularity-with-positive-probability from the homogeneous case but does not
prevent singularity formation. We show that in the supercritical focusing case
the ensemble-averaged self-interaction energy and the momentum variance can
exceed any fixed level in a finite time with positive probability.


title: Effective dynamics in Hamiltonian systems with mixed phase space

abstract:
  An adequate characterization of the dynamics of Hamiltonian systems at
physically relevant scales has been largely lacking. Here we investigate this
fundamental problem and we show that the finite-scale Hamiltonian dynamics is
governed by effective dynamical invariants, which are significantly different
from the dynamical invariants that describe the asymptotic Hamiltonian
dynamics. The effective invariants depend both on the scale of resolution and
the region of the phase space under consideration, and they are naturally
interpreted within a framework in which the nonhyperbolic dynamics of the
Hamiltonian system is modeled as a chain of hyperbolic systems.


title: Kovalevskaya Exponents and Poisson Structures

abstract:
  We consider generalizations of pairing relations for Kovalevskaya exponents
in quasihomogeneous systems with quasihomogeneous tensor invariants. The case
of presence of a Poisson structure in the system is investigated in more
detail. We give some examples which illustrate general theorems.


title:
Breather trapping and breather transmission in a DNA model with an
  interface

abstract:
  We study the dynamics of moving discrete breathers in an interfaced piecewise
DNA molecule.
  This is a DNA chain in which all the base pairs are identical and there
exists an interface such that the base pairs dipole moments at each side are
oriented in opposite directions.
  The Hamiltonian of the Peyrard--Bishop model is augmented with a term that
includes the dipole--dipole coupling between base pairs. Numerical simulations
show the existence of two dynamical regimes. If the translational kinetic
energy of a moving breather launched towards the interface is below a critical
value, it is trapped in a region around the interface collecting vibrational
energy. For an energy larger than the critical value, the breather is
transmitted and continues travelling along the double strand with lower
velocity. Reflection phenomena never occur.
  The same study has been carried out when a single dipole is oriented in
opposite direction to the other ones.
  When moving breathers collide with the single inverted dipole, the same
effects appear. These results emphasize the importance of this simple type of
local inhomogeneity as it creates a mechanism for the trapping of energy.
  Finally, the simulations show that, under favorable conditions, several
launched moving breathers can be trapped successively at the interface region
producing an accumulation of vibrational energy. Moreover, an additional
colliding moving breather can produce a saturation of energy and a moving
breather with all the accumulated energy is transmitted to the chain.


title: Convection-induced nonlinear-symmetry-breaking in wave mixing

abstract:
  We show that the combined action of diffraction and convection (walk-off) in
wave mixing processes leads to a nonlinear-symmetry-breaking in the generated
traveling waves. The dynamics near to threshold is reduced to a Ginzburg-Landau
model, showing an original dependence of the nonlinear self-coupling term on
the convection. Analytical expressions of the intensity and the velocity of
traveling waves emphasize the utmost importance of convection in this
phenomenon. These predictions are in excellent agreement with the numerical
solutions of the full dynamical model.


title:
Convective instability induced by nonlocality in nonlinear diffusive
  systems

abstract:
  We consider a large class of nonlinear diffusive systems with nonlocal
coupling. By using a non-perturbative analytical approach we are able to
determine the convective and absolute instabilities of all the uniform states
of these systems. We find a huge window of convective instability that should
provide a great opportunity to study experimentally and theoretically noise
sustained patterns.


title:
Coupled mode theory for photonic band-gap inhibition of spatial
  instabilities

abstract:
  We study the inhibition of pattern formation in nonlinear optical systems
using intracavity photonic crystals. We consider mean field models for single
and doubly degenerate optical parametric oscillators. Analytical expressions
for the new (higher) modulational thresholds and the size of the ''band-gap``
as function of the system and photonic crystal parameters are obtained via a
coupled-mode theory. Then, by means of a nonlinear analysis, we derive
amplitude equations for the unstable modes and find the stationary solutions
above threshold. The form of the unstable mode is different in the lower and
upper part of the bandgap. In each part there is bistability between two
spatially shifted patterns. In large systems stable wall defects between the
two solutions are formed and we provide analytical expressions for their shape.
The analytical results are favorably compared with results obtained from the
full system equations. Inhibition of pattern formation can be used to spatially
control signal generation in the transverse plane.


title: Lie algebras in vortex dynamics and celestial mechanics - IV

abstract:
  The work of A.V. Borisov, A.E. Pavlov, Dynamics and Statics of Vortices on a
Plane and a Sphere - I (Reg. & Ch. Dynamics, 1998, Vol. 3, No 1, p.28-39)
introduces a naive description of dynamics of point vortices on a plane in
terms of variables of distances and areas which generate Lie-Poisson structure.
Using this approach a qualitative description of dynamics of point vortices on
a plane and a sphere is obtained in the works Dynamics of Three Vortices on a
Plane and a Sphere - II. General compact case by A.V. Borisov, V.G. Lebedev
(Reg. & Ch. Dynamics, 1998, Vol. 3, No 2, p.99-114), Dynamics of three vortices
on a plane and a sphere - III. Noncompact case. Problem of collaps and
scattering by A.V. Borisov, V.G. Lebedev (Reg. & Ch. Dynamics, 1998, Vol. 3, No
4, p.76-90). In this paper we consider more formal constructions of the general
problem of n vortices on a plane and a sphere. The developed methods of
algebraization are also applied to the classical problem of the reduction in
the three-body problem.


title: Periodic orbits and semiclassical form factor in barrier billiards

abstract:
  Using heuristic arguments based on the trace formulas, we analytically
calculate the semiclassical two-point correlation form factor for a family of
rectangular billiards with a barrier of height irrational with respect to the
side of the billiard and located at any rational position p/q from the side. To
do this, we first obtain the asymptotic density of lengths for each family of
periodic orbits by a Siegel-Veech formula. The result K(0)=1/2+1/q obtained for
these pseudo-integrable, non-Veech billiards is different but not far from the
value of 1/2 expected for semi-Poisson statistics and from values of K(0)
obtained previously in the case of Veech billiards.


title: Stability of Thomson's Configurations of Vortices on a Sphere

abstract:
  In this work stability of polygonal configurations on a plane and sphere is
investigated. The conditions of linear stability are obtained. A nonlinear
analysis of the problem is made with the help of Birkhoff normalization. Some
problems are also formulated.


title: Finite geometries and diffractive orbits in isospectral billiards

abstract:
  Several examples of pairs of isospectral planar domains have been produced in
the two-dimensional Euclidean space by various methods. We show that all these
examples rely on the symmetry between points and blocks in finite projective
spaces; from the properties of these spaces, one can derive a relation between
Green functions as well as a relation between diffractive orbits in isospectral
billiards.


title: Multi-particle dispersion in fully developed turbulence

abstract:
  The statistical geometry of dispersing Lagrangian clusters of four particles
(tetrahedra) is studied by means of high-resolution direct numerical
simulations of three-dimensional homogeneous isotropic turbulence. We give the
first evidence of a self-similar regime of shape dynamics characterized by
almost two-dimensional, strongly elongated geometries. The analysis of
four-point velocity-difference statistics and orientation shows that
inertial-range eddies typically generate a straining field with a strong
extensional component aligned with the elongation direction and weak
extensional/compressional components in the orthogonal plane.


title: On Deviations from Gaussian Statistics for Surface Gravity Waves

abstract:
  Here we discuss some issues concerning the statistical properties of ocean
surface waves. We show that, using the approach of weak turbulence theory,
deviations from Gaussian statistics can be naturally included. In particular we
discuss the role of bound and free modes for the determination of the
statistical properties of the surface elevation. General formulas for skewness
and kurtosis as a function of the spectral wave action density are here
derived.


title: Doubling the Intensity of an ERL Based Light Source

abstract:
  A light source based on an Energy Recovered Linac (ERL) [1] consist of a
superconducting linac and a transfer line that includes wigglers and undulators
to produce the synchrotron light. The transfer line brings the electrons
bunches back to the beginning of the linac so that their energy can be
recovered when they traverse the linac a second time, lambda/2 out of phase.
There is another interesting condition when the length of the transfer line is
(n+/- 1/4) lambda. In this case, the electrons drift through on the zero RF
crossing, and make a further pass around the transfer line, effectively
doubling the circulating current in the wigglers and undulators. On the third
pass through the linac, they will be decelerated and their energy recovered.
The longitudinal focusing at the zero crossing is a problem, but it can be
canceled if the drifting beam sees a positive energy gradient for the first
half of the linac and a negative gradient for the second half (or vice versa).
This paper presents a proposal to use a double chicane at the center of the
linac to provide this focusing inversion fo rthe drifting beam while leaving
the accelerating and decelerating beams on crest.


title: Synchronization in driven versus autonomous coupled chaotic maps

abstract:
  The phenomenon of synchronization occurring in a locally coupled map lattice
subject to an external drive is compared to the synchronization process in an
autonomous coupled map system with similar local couplings plus a global
interaction. It is shown that chaotic synchronized states in both systems are
equivalent, but the collective states arising after the chaotic synchronized
state becomes unstable can be different in these two systems. It is found that
the external drive induces chaotic synchronization as well as synchronization
of unstable periodic orbits of the local dynamics in the driven lattice. On the
other hand, the addition of a global interaction in the autonomous system
allows for chaotic synchronization that is not possible in a large coupled map
system possessing only local couplings.


title: Kovalevskaya Top and Generalizations of Integrable Systems

abstract:
  Generalizations of the Kovalevskaya, Chaplygin, Goryachev-Chaplygin and
Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a
method of introduction of separating variables and action-angle variables is
described. Another integration method for the Kovalevskaya top on the bundle is
found. This method uses a coordinate transformation that reduces the
Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A
generalization of a recent Gaffet system to the bundle of Poisson brackets is
obtained at the end of the paper.


title:
Localization of response functions of spiral waves in the
  FitzHugh-Nagumo system

abstract:
  Dynamics of spiral waves in perturbed, e. g. slightly inhomogeneous or
subject to a small periodic external force, two-dimensional autowave media can
be described asymptotically in terms of Aristotelean dynamics, so that the
velocities of the spiral wave drift in space and time are proportional to the
forces caused by the perturbation. The forces are defined as a convolution of
the perturbation with the spiral's Response Functions, which are eigenfunctions
of the adjoint linearised problem. In this paper we find numerically the
Response Functions of a spiral wave solution in the classic excitable
FitzHugh-Nagumo model, and show that they are effectively localised in the
vicinity of the spiral core.


title: Incoherent multi-gap optical solitons in nonlinear photonic lattices

abstract:
  We demonstrate numerically that partially incoherent light can be trapped in
the spectral band gaps of a photonic lattice, creating partially incoherent
multi-component spatial optical solitons in a self-defocusing nonlinear
periodic medium. We find numerically such incoherent multi-gap optical solitons
and discuss how to generate them in experiment by interfering incoherent light
beams at the input of a nonlinear periodic medium.


title: Nonequilibrium Energy Profiles for a Class of 1-D Models

abstract:
  As a paradigm for heat conduction in 1 dimension, we propose a class of
models represented by chains of identical cells, each one of which containing
an energy storage device called a "tank". Energy exchange among tanks is
mediated by tracer particles, which are injected at characteristic temperatures
and rates from heat baths at the two ends of the chain. For stochastic and
Hamiltonian models of this type, we develop a theory that allows one to derive
rigorously -- under physically natural assumptions -- macroscopic equations for
quantities related to heat transport, including mean energy profiles and tracer
densities. Concrete examples are treated for illustration, and the validity of
the Fourier Law in the present context is discussed.

categories: nlin.CD cond-mat.stat-mech math-ph math.MP


title: On Justification of Gibbs Distribution

abstract:
  The paper develop a new approach to the justification of Gibbs canonical
distribution for Hamiltonian systems with finite number of degrees of freedom.
It uses the condition of nonintegrability of the ensemble of weak interacting
Hamiltonian systems.


title:
Canonical Gibbs distribution and thermodynamics of mechanical systems
  with a finite number of degrees of freedom

abstract:
  Traditional derivation of Gibbs canonical distribution and the justification
of thermodynamics are based on the assumption concerning an isoenergetic
ergodicity of a system of $n$ weakly interacting identical subsystems and
passage to the limit $n\to\infty$. In the presented work we develop another
approach to these problems assuming that $n$ is fixed and $n\ge2$. The ergodic
hypothesis (which frequently is not valid due to known results of the
KAM-theory) is substituted by a weaker assumption that the perturbed system
does not have additional first integrals independent of the energy integral.
The proof of nonintegrability of perturbed Hamiltonian systems is based on the
Poincare method. Moreover, we use the natural Gibbs assumption concerning a
thermodynamic equilibrium of bsystems at vanishing interaction. The general
results are applied to the system of the weakly connected pendula. The
averaging with respect to the Gibbs measure allows to pass from usual dynamics
of mechanical systems to the classical thermodynamic model.


title: Hexagons become second if symmetry is broken

abstract:
  Pattern formation on the free surface of a magnetic fluid subjected to a
magnetic field is investigated experimentally. By tilting the magnetic field
the symmetry can be broken in a controllable manner. When increasing the
amplitude of the tilted field, the flat surface gives way to liquid ridges. A
further increase results in a hysteretic transition to a pattern of stretched
hexagons. The instabilities are detected by means of a linear array of magnetic
hall sensors and compared with theoretical predictions.


title: Two-dimensional solitons on the surface of magnetic fluids

abstract:
  We report an observation of a stable soliton-like structure on the surface of
a ferrofluid, generated by a local perturbation in the hysteretic regime of the
Rosensweig instability. Unlike other pattern-forming systems with localized 2D
structures, magnetic fluids are characterized by energy conservation; hence
their mechanism of soliton stabilization is different from the previously
discussed gain/loss balance mechanism. The radioscopic measurements of the
soliton's surface profile suggest that locking on the underlying periodic
structure is instrumental in its stabilization.


title: Parameter fluctuations in coupled chaotic systems

abstract:
  We study the effect of parameter fluctuations on synchronization of a coupled
chaotic system. The fluctuations to the parameter can be random or it can be a
periodic modulation. For random fluctuations we introduce a new quantity, the
\emph{rate of fluctuations}, apart from the statistical features of the
fluctuations. Fluctuation rate in our study refers to the number of random
modifications to parameters occurring in unit time. With a periodic modulation,
the fluctuation rates can be the frequency of the modulating term. It is found
that with high fluctuation rates the synchronization is stable irrespective of
the statistical or mathematical features of fluctuation. It was also found that
the low fluctuation rates can destroy synchronization even with a small
amplitude. We analytically explain the observed phenomenon using the dynamical
equations and numerically verified with a coupled system of Rossler attractor
as an example. We also numerically quantify the relation between
synchronization error and fluctuation rates.


title: Optimizing the bulk modulus of cellular networks

abstract:
  We present an alternative derivation of upper-bounds for the bulk modulus of
both two-dimensional and three-dimensional cellular materials. For
two-dimensional materials, we recover exactly the expression of the
Hashin-Shtrikman (HS) upper-bound in the low-density limit, while for
three-dimensional materials we even improve the HS bound. Furthermore, we
establish necessary and sufficient conditions on the cellular structure for
maximizing the bulk modulus, for a given solid volume fraction. These
conditions are found to be exactly those under which the electrical (or
thermal) conductivity of the material reaches its maximal value as well. These
results provide a set of straightforward criteria allowing to address the
design of optimized cellular materials, and shed light on recent studies of
structures with both maximal bulk modulus and maximal conductivity. Finally, we
discuss the compatibility of the criteria presented here with the geometrical
constraints caused by minimization of surface energy in a real foam.

categories: nlin.AO cond-mat.dis-nn cond-mat.soft cond-mat.stat-mech physics.class-ph


title: Anomalous synchronization threshold in coupled logistic maps

abstract:
  We consider regular lattices of coupled chaotic maps. Depending on lattice
size, there may exist a window in parameter space where complete
synchronization is eventually attained after a transient regime. Close outside
this window, an intermittent transition to synchronization occurs. While
asymptotic transversal Lyapunov exponents allow to determine the
synchronization threshold, the distribution of finite-time Lyapunov exponents,
in the vicinity of the critical frontier, is expected to provide relevant
information on phenomena such as intermittency. In this work we scrutinize the
distribution of finite-time exponents when the local dynamics is ruled by the
logistic map $x \mapsto 4x(1-x)$. We obtain a theoretical estimate for the
distribution of finite-time exponents, that is markedly non-Gaussian. The
existence of correlations, that spoil the central limit approximation, is shown
to modify the typical intermittent bursting behavior. The present scenario
could apply to a wider class of systems with different local dynamics and
coupling schemes.


title: Nonlinearly driven transverse synchronization in coupled chaotic systems

abstract:
  Synchronization transitions are investigated in coupled chaotic maps.
Depending on the relative weight of linear versus nonlinear instability
mechanisms associated to the single map two different scenarios for the
transition may occur. When only two maps are considered we always find that the
critical coupling $\epsilon_l$ for chaotic synchronization can be predicted
within a linear analysis by the vanishing of the transverse Lyapunov exponent
$\lambda_T$. However, major differences between transitions driven by linear or
nonlinear mechanisms are revealed by the dynamics of the transient toward the
synchronized state. As a representative example of extended systems a one
dimensional lattice of chaotic maps with power-law coupling is considered. In
this high dimensional model finite amplitude instabilities may have a dramatic
effect on the transition. For strong nonlinearities an exponential divergence
of the synchronization times with the chain length can be observed above
$\epsilon_l$, notwithstanding the transverse dynamics is stable against
infinitesimal perturbations at any instant. Therefore, the transition takes
place at a coupling $\epsilon_{nl}$ definitely larger than $\epsilon_l$ and its
origin is intrinsically nonlinear. The linearly driven transitions are
continuous and can be described in terms of mean field results for
non-equilibrium phase transitions with long range interactions. While the
transitions dominated by nonlinear mechanisms appear to be discontinuous.


title: Architecture of optimal transport networks

abstract:
  We analyze the structure of networks minimizing the global resistance to flow
(or dissipated energy) with respect to two different constraints: fixed total
channel volume and fixed total channel surface area. First, we determine the
shape of channels in such optimal networks and show that they must be straight
with uniform cross-sectional areas. Then, we establish a relation between the
cross-sectional areas of adjoining channels at each junction. Indeed, this
relation is a generalization of Murray's law, originally established in the
context of local optimization. Moreover, we establish a relation between angles
and cross-sectional areas of adjoining channels at each junction, which can be
represented as a vectorial force balance equation, where the force weight
depends on the channel cross-sectional area. A scaling law between the minimal
resistance value and the total volume or surface area value is also derived
from the analysis. Furthermore, we show that no more than three or four
channels meet in one junction of optimal bi-dimensional networks, depending on
the flow profile (e.g.: Poiseuille-like or plug-like) and the considered
constraint (fixed volume or surface area). In particular, we show that sources
are directly connected to wells, without intermediate junctions, for minimal
resistance networks preserving the total channel volume in case of plug flow
regime. Finally, all these results are illustrated with a simple example, and
compared with the structure of natural networks.


title: Some aspects of the synchronization in coupled maps

abstract:
  Through numerical simulations we analyze the synchronization time and the
Lyapunov dimension of a coupled map lattice consisting of a chain of chaotic
logistic maps exhibiting power law interactions. From the observed behaviors we
find a lower bound for the size $N$ of the lattice, independent of the range
and strength of the interaction, which imposes a practical lower bound in
numerical simulations for the system to be considered in the thermodynamic
limit. We also observe the existence of a strong correlation between the
averaged synchronization time and the Lyapunov dimension. This is an
interesting result because it allows an analytical estimation of the
synchronization time, which otherwise requires numerical simulations.


title: Counting function for a sphere of anisotropic quartz

abstract:
  We calculate the leading Weyl term of the counting function for a
mono-crystalline quartz sphere. In contrast to other studies of counting
functions, the anisotropy of quartz is a crucial element in our investigation.
Hence, we do not obtain a simple analytical form, but we carry out a numerical
evaluation. To this end we employ the Radon transform representation of the
Green's function. We compare our result to a previously measured unique data
set of several tens of thousands of resonances.


title: Optimizing transport in a homogeneous network

abstract:
  Many situations in physics, biology, and engineering consist of the transport
of some physical quantity through a network of narrow channels. The ability of
a network to transport such a quantity in every direction can be described by
the average conductivity associated with. When the flow through each channel is
conserved and derives from a potential function, we show that there exist an
upper bound of the average conductivity, and explicitly give the expression for
this upper bound as a function of the channel permeability and channel length
distributions. Moreover, we express the necessary and sufficient conditions on
the network structure to maximize the average conductivity. These conditions
are found to be independent of the connectivity of the vertices.

categories: nlin.AO cond-mat.dis-nn cond-mat.soft cond-mat.stat-mech


title: Superintegrable systems on sphere

abstract:
  We consider various generalizations of the Kepler problem to
three-dimensional sphere $S^3$, a compact space of constant curvature. These
generalizations include, among other things, addition of a spherical analog of
the magnetic monopole (the Poincar\'e--Appell system) and addition of a more
complicated field, which itself is a generalization of the MICZ-system. The
mentioned systems are integrable -- in fact, superintegrable. The latter is due
to the vector integral, which is analogous to the Laplace--Runge--Lenz vector.
We offer a classification of the motions and consider a trajectory isomorphism
between planar and spatial motions. The presented results can be easily
extended to Lobachevsky space $L^3$.


title: Chaotic Turing Patterns displaying the beauty of chaos

abstract:
  The problem of Turing pattern formation has attracted much attention in
nonlinear science as well as physics, chemistry and biology. So far all Turing
patterns have been observed in stationary and oscillatory media only. In this
letter we find for the first time that ordered Turing patterns exist in chaotic
extended systems. And chaotic Turing patterns are strikingly rich and
surprisingly beautiful with their space structures. These findings are in sharp
contrast with the intuition of pseudo-randomness of chaos. The richness and
beauty of the chaotic Turing patterns are attributed to a large variety of
symmetry properties realized by various types of self-organizations of partial
chaos synchronizations. Some statistical measurements are performed to confirm
and heuristically understand these findings.


title: Vortex solitons in dispersive nonlinear Kerr type media

abstract:
  We have investigated the nonlinear amplitude vector equation governing the
evolution of optical pulses in optical and UV region. We are normalizing this
equation for the cases of different and equal transverse and longitudinal size
of optical pulses, of week and strong dispersion. This gives us the possibility
to reduce the amplitude equation to different nonlinear evolution equations in
the partial cases. For some of these nonlinear equations exact vortex solutions
are found. Conditions for experimental observations of these vortices are
determined.


title:
Classical and quantum nonlinear localized excitations in discrete
  systems

abstract:
  Discrete breathers, or intrinsic localized modes, are spatially localized,
time--periodic, nonlinear excitations that can exist and propagate in systems
of coupled dynamical units. Recently, some experiments show the sighting of a
form of discrete breather that exist at the atomic scale in a magnetic solid.
Other observations of breathers refer to systems such as Josephson--junction
arrays, photonic crystals and optical-switching waveguide arrays. All these
observations underscore their importance in physical phenomena at all scales.
The authors review some of their latest theoretical contributions in the field
of classical and quantum breathers, with possible applications to these widely
different physical systems and to many other such as DNA, proteins, quantum
dots, quantum computing, etc.


title: Spatiotemporal intermittency and scaling laws in coupled map lattices

abstract:
  We discuss the spatiotemporal intermittency (STI) seen in coupled map
lattices (CML-s). We identify the types of intermittency seen in such systems
in the context of several specific CML-s. The Chat\'e-Manneville CML is
introduced and the on-going debate on the connection of the spatiotemporal
intermittency seen in this model with the problem of directed percolation is
summarised. We also discuss the STI seen in the sine circle map model and its
connection with the directed percolation problem, as well as the inhomogenous
logistic map lattice which shows the novel phenomenon of spatial intermittency
and other types of behaviour not seen in the other models. The connection of
the bifurcation behaviour in this model with STI is touched upon. We conclude
with a discussion of open problems.


title: Optimal disorder for segregation in annealed small worlds

abstract:
  We study a model for microscopic segregation in a homogeneous system of
particles moving on a one-dimensional lattice. Particles tend to separate from
each other, and evolution ceases when at least one empty site is found between
any two particles. Motion is a mixture of diffusion to nearest-neighbour sites
and long-range jumps, known as annealed small-world propagation. The long-range
jump probability plays the role of the small-world disorder. We show that there
is an optimal value of this probability, for which the segregation process is
fastest. Moreover, above a critical probability, the time needed to reach a
fully segregated state diverges for asymptotically large systems. These special
values of the long-range jump probability depend crucially on the particle
density. Our system is a novel example of the rare dynamical processes with
critical behaviour at a finite value of the small-world disorder.


title: Extended phase diagram of the Lorenz model

abstract:
  The parameter dependence of the various attractive solutions of the three
variable nonlinear Lorenz model equations for thermal convection in
Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been
investigated as a function of r, the normalized Rayleigh number, at fixed
Prandtl number \sigma. The present work extends the analysis to the entire
(r,\sigma) parameter plane. An onion like periodic pattern is found which is
due to the alternating stability of symmetric and non-symmetric periodic
orbits. This periodic pattern is explained by considering non-trivial limits of
large r and \sigma. In addition to the limit which was previously analyzed by
Sparrow, we identify two more distinct asymptotic regimes in which either
\sigma/r or \sigma^2/r is constant. In both limits the dynamics is
approximately described by Airy functions whence the periodicity in parameter
space can be calculated analytically. Furthermore, some observations about
sequences of bifurcations and coexistence of attractors, periodic as well as
chaotic, are reported.


title: Order in Binary Sequences and the Routes to Chaos

abstract:
  The natural order in the space of binary sequences permits to recover the
$U$-sequence. Also the scaling laws of the period-doubling cascade and the
intermittency route to chaos defined in that ordered set are explained. These
arise as intrinsic properties of this ordered set, and independent from any
consideration about dynamical systems.

categories: nlin.CD cond-mat.stat-mech math.DS math.NT


title: Finite-order meromorphic solutions and the discrete Painleve equations

abstract:
  Let w(z) be a finite-order meromorphic solution of the second-order
difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is
rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference
linear or Riccati equation or else equation (1) can be transformed to one of a
list of canonical difference equations. This list consists of all known
difference Painleve equation of the form (1), together with their autonomous
versions. This suggests that the existence of finite-order meromorphic
solutions is a good detector of integrable difference equations.


title: Diophantine Integrability

abstract:
  The heights of iterates of the discrete Painleve equations over number fields
appear to grow no faster than polynomials while the heights of generic
solutions of non-integrable discrete equations grow exponentially. This gives
rise to a simple and effective numerical test for the integrability of discrete
equations. Numerical evidence and theoretical results are presented.
Connections with other tests for integrability and Vojta's dictionary are
discussed.


title:
Periodic and Chaotic Events in a Discrete Model of Logistic Type for the
  Competitive Interaction of Two Species

abstract:
  Two symmetrically coupled logistic equations are proposed to mimic the
competitive interaction between two species. The phenomena of coexistence,
oscillations and chaos are present in this cubic discrete system. This work,
together with two other similar ones recently published by the authors,
completes a triptych dedicated to the two-species relationships present in
Nature, namely the symbiosis, the predator-prey and the competition. These
models can be used as basic ingredients to build up more complex interactions
in the ecological networks.


title: Spectral stability and time evolution of N-solitons in KdV hierarchy

abstract:
  This paper concerns spectral stability and time evolution of $N$-solitons in
the KdV hierarchy with mixed commuting time flows. Spectral stability problem
is analyzed by using a pair of self-adjoint operators with finite numbers of
negative eigenvalues. We show that the absence of unstable eigenvalues in the
stability problem is related to the absence of negative eigenvalues of these
operators in the constrained function spaces. Time evolution of $N$-solitons is
uniquely characterized from the inverse scattering transform technique.


title:
General Approach to the Quantum Kicked Particle in a Magnetic Field:
  Quantum-Antiresonance Transition

abstract:
  The quantum kicked particle in a magnetic field is studied in a weak-chaos
regime under realistic conditions, i.e., for {\em general} values of the
conserved coordinate $x_{{\rm c}}$ of the cyclotron orbit center. The system
exhibits spectral structures [``Hofstadter butterflies'' (HBs)] and quantum
diffusion depending sensitively on $x_{{\rm c}}$. Most significant changes take
place when $x_{{\rm c}}$ approaches the value at which quantum antiresonance
(exactly periodic recurrences) can occur: the HB essentially ``doubles'' and
the quantum-diffusion coefficient $D(x_{{\rm c}})$ is strongly reduced. An
explanation of these phenomena, including an approximate formula for $D(x_{{\rm
c}})$ in a class of wave packets, is given on the basis of an effective
Hamiltonian which is derived as a power expansion in a small parameter. The
global quantum diffusion of a two-dimensional wave packet for all $x_{{\rm c}}$
is briefly considered.

categories: nlin.CD cond-mat.mes-hall math-ph math.MP physics.atom-ph quant-ph


title: Bifurcation scenario to Nikolaevskii turbulence in small systems

abstract:
  We show that the chaos in Kuramoto-Sivashinsky equation occurs through
period-doubling cascade (Feigenbaum scenario), in contrast, the chaos in
Nikolaevskii equation occurs through torus-doubling bifurcation
(Ruelle-Takens-Newhouse scenario).


title:
Lyapunov spectra of billiards with cylindrical scatterers: comparison
  with many-particle systems

abstract:
  The dynamics of a system consisting of many spherical hard particles can be
described as a single point particle moving in a high-dimensional space with
fixed hypercylindrical scatterers with specific orientations and positions. In
this paper, the similarities in the Lyapunov exponents are investigated between
systems of many particles and high-dimensional billiards with cylindrical
scatterers which have isotropically distributed orientations and homogeneously
distributed positions. The dynamics of the isotropic billiard are calculated
using a Monte-Carlo simulation, and a reorthogonalization process is used to
find the Lyapunov exponents. The results are compared to numerical results for
systems of many hard particles as well as the analytical results for the
high-dimensional Lorentz gas. The smallest three-quarters of the positive
exponents behave more like the exponents of hard-disk systems than the
exponents of the Lorentz gas. This similarity shows that the hard-disk systems
may be approximated by a spatially homogeneous and isotropic system of
scatterers for a calculation of the smaller Lyapunov exponents, apart from the
exponent associated with localization. The method of the partial stretching
factor is used to calculate these exponents analytically, with results that
compare well with simulation results of hard disks and hard spheres.


title: Control of stochasticity in magnetic field lines

abstract:
  We present a method of control which is able to create barriers to magnetic
field line diffusion by a small modification of the magnetic perturbation. This
method of control is based on a localized control of chaos in Hamiltonian
systems. The aim is to modify the perturbation locally by a small control term
which creates invariant tori acting as barriers to diffusion for Hamiltonian
systems with two degrees of freedom. The location of the invariant torus is
enforced in the vicinity of the chosen target. Given the importance of
confinement in magnetic fusion devices, the method is applied to two examples
with a loss of magnetic confinement. In the case of locked tearing modes, an
invariant torus can be restored that aims at showing the current quench and
therefore the generation of runaway electrons. In the second case, the method
is applied to the control of stochastic boundaries allowing one to define a
transport barrier within the stochastic boundary and therefore to monitor the
volume of closed field lines.


title: Dynamic scaling in stick-slip friction

abstract:
  We introduce a generalized homogeneous function to describe the joint
probability density for magnitude and duration of events in self-organized
critical systems (SOC). It follows that the cumulative distributions of
magnitude and of duration are power-laws with exponents $\alpha$ and $\tau$
respectively. A power-law relates duration and magnitude (exponent $\gamma$) on
the average. The exponents satisfy the dynamic scaling relation
$\alpha=\gamma\tau$. The exponents classify SOC systems into universality
classes that do not depend on microscopic details provided that both $\alpha<1$
and $\tau<1$. We also present new experimental results on the stick-slip motion
of a sandpaper slowly pulled across a carpet that are consistent with our
criteria for SOC systems. Our experiments, as well as experiments by others,
satisfy our dynamic scaling relation. We discuss the relevance of our results
to earthquake statistics.


title: Energy diffusion in strongly driven quantum chaotic systems

abstract:
  The energy evolution of a quantum chaotic system under the perturbation that
harmonically depends on time is studied for the case of large perturbation, in
which the rate of transition calculated from the Fermi golden rule exceeds the
frequency of perturbation. It is shown that the energy evolution retains its
diffusive character, with the diffusion coefficient that is asymptotically
proportional to the magnitude of perturbation and to the square root of the
density of states. The results are supported by numerical calculation. They
imply the absence of the quantum-classical correspondence for the energy
diffusion and the energy absorption in the classical limit $\hbar \to 0$.


title: q-Breathers and the Fermi-Pasta-Ulam Problem

abstract:
  The Fermi-Pasta-Ulam (FPU) paradox consists of the nonequipartition of energy
among normal modes of a weakly anharmonic atomic chain model. In the harmonic
limit each normal mode corresponds to a periodic orbit in phase space and is
characterized by its wave number $q$. We continue normal modes from the
harmonic limit into the FPU parameter regime and obtain persistence of these
periodic orbits, termed here $q$-Breathers (QB). They are characterized by time
periodicity, exponential localization in the $q$-space of normal modes and
linear stability up to a size-dependent threshold amplitude. Trajectories
computed in the original FPU setting are perturbations around these exact QB
solutions. The QB concept is applicable to other nonlinear lattices as well.


title: Enlarged spectral problems and nonintegrability

abstract:
  The method of obtaining new integrable coupled equations through enlarging
spectral problems of known integrable equations, which was recently proposed by
W.-X. Ma, can produce nonintegrable systems as well. This phenomenon is
demonstrated and explained by the example of the enlarged spectral problem of
the Korteweg - de Vries equation.


title: Leray and LANS-$\alpha$ modeling of turbulent mixing

abstract:
  Mathematical regularisation of the nonlinear terms in the Navier-Stokes
equations provides a systematic approach to deriving subgrid closures for
numerical simulations of turbulent flow. By construction, these subgrid
closures imply existence and uniqueness of strong solutions to the
corresponding modelled system of equations. We will consider the large eddy
interpretation of two such mathematical regularisation principles, i.e., Leray
and LANS$-\alpha$ regularisation. The Leray principle introduces a {\bfi
smoothed transport velocity} as part of the regularised convective
nonlinearity. The LANS$-\alpha$ principle extends the Leray formulation in a
natural way in which a {\bfi filtered Kelvin circulation theorem},
incorporating the smoothed transport velocity, is explicitly satisfied. These
regularisation principles give rise to implied subgrid closures which will be
applied in large eddy simulation of turbulent mixing. Comparison with filtered
direct numerical simulation data, and with predictions obtained from popular
dynamic eddy-viscosity modelling, shows that these mathematical regularisation
models are considerably more accurate, at a lower computational cost.


title:
On localized solutions of discrete nonlinear Schrodinger equation. An
  exact result

abstract:
  Local and global existence of localized solutions of a discrete nonlinear
Schrodinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In
particular, it is shown that an initially localized excitation persists
localized during infinite time. Moreover, if initial localization is stronger
than |n|^{-d} with any power d, it maintains itself as such during infinite
time. The results are generalized to various types of inter-side and saturable
nonlinearities, to lattices with long range interactions, as well as DNLS with
dissipation.

categories: nlin.PS cond-mat.dis-nn math-ph math.AP math.MP


title: Persistent Chaos in High Dimensions

abstract:
  An extensive statistical survey of universal approximators shows that as the
dimension of a typical dissipative dynamical system is increased, the number of
positive Lyapunov exponents increases monotonically and the number of parameter
windows with periodic behavior decreases. A subset of parameter space remains
in which topological change induced by small parameter variation is very
common. It turns out, however, that if the system's dimension is sufficiently
high, this inevitable, and expected, topological change is never catastrophic,
in the sense chaotic behavior is preserved. One concludes that deterministic
chaos is persistent in high dimensions.


title: Integrable Systems and Discrete Geometry

abstract:
  This is an expository article for Elsevier's Encyclopedia of Mathematical
Physics on the subject in the title. Comments/corrections welcome.


title: Fidelity amplitude of the scattering matrix in microwave cavities

abstract:
  The concept of fidelity decay is discussed from the point of view of the
scattering matrix, and the scattering fidelity is introduced as the parametric
cross-correlation of a given S-matrix element, taken in the time domain,
normalized by the corresponding autocorrelation function. We show that for
chaotic systems, this quantity represents the usual fidelity amplitude, if
appropriate ensemble and/or energy averages are taken. We present a microwave
experiment where the scattering fidelity is measured for an ensemble of chaotic
systems. The results are in excellent agreement with random matrix theory for
the standard fidelity amplitude. The only parameter, namely the perturbation
strength could be determined independently from level dynamics of the system,
thus providing a parameter free agreement between theory and experiment.


title: Supersymmetric Non-local Gas Equation

abstract:
  In this paper we study systematically the question of supersymmetrization of
the non-local gas equation. We obtain both the N=1 and the N=2 supersymmetric
generalizations of the system which are integrable. We show that both the
systems are bi-Hamiltonian. While the N=1 supersymmetrization allows the
hierarchy of equations to be extended to negative orders (local equations), we
argue that this is not the case for the N=2 supersymmetrization. In the bosonic
limit, however, the N=2 system of equations lead to a new coupled integrable
system of equations.