title:
Convergence properties of the 8, 10 and 12 mode representations of
  quasipatterns

abstract:
  Spatial Fourier transforms of quasipatterns observed in Faraday wave
experiments suggest that the patterns are well represented by the sum of 8, 10
or 12 Fourier modes with wavevectors equally spaced around a circle. This
representation has been used many times as the starting point for standard
perturbative methods of computing the weakly nonlinear dependence of the
pattern amplitude on parameters. We show that nonlinear interactions of n such
Fourier modes generate new modes with wavevectors that approach the original
circle no faster than a constant times n^{-2}, and that there are combinations
of modes that do achieve this limit. As in KAM theory, small divisors cause
difficulties in the perturbation theory, and the convergence of the standard
method is questionable in spite of the bound on the small divisors. We compute
steady quasipattern solutions of the cubic Swift--Hohenberg equation up to 33rd
order to illustrate the issues in some detail, and argue that the standard
method does not converge sufficiently rapidly to be regarded as a reliable way
of calculating properties of quasipatterns.


title: Matrix integrals as Borel sums of Schur function expansions

abstract:
  The partition function for unitary two matrix models is known to be a double
KP tau-function, as well as providing solutions to the two dimensional Toda
hierarchy. It is shown how it may also be viewed as a Borel sum regularization
of divergent sums over products of Schur functions in the two sequences of
associated KP flow variables.


title: Soliton molecules in trapped vector Nonlinear Schrodinger systems

abstract:
  We study a new class of vector solitons in trapped Nonlinear Schrodinger
systems modelling the dynamics of coupled light beams in GRIN Kerr media and
atomic mixtures in Bose-Einstein condensates. These solitons exist for
different spatial dimensions, their existence is studied by means of a
systematic mathematical technique and the analysis is made for inhomogeneous
media.


title: Selection of Catalysts through Cellular Reproduction

abstract:
  It has been pointed out that if an autocatalytic cycle produces primitive
membrane chemicals, it can compose a self-maintaining proto cell. Moreover, it
is known that a proto cell can divide itself spontaneously as it grows. An
unsolved problem is how such a catalytic system can evolve in the pre-cellular
environment. Here we examine, as the first step, the evolution of catalysts
that have different activity in generating membrane chemicals using a
Lattice-Gas-like model. We demonstrated that a self-replicating proto-cell
emerges from random initial configuration. We also showed that cells with
higher activity of membrane production evolve through cellular selection.


title: Self-Organized Criticality and Thermodynamic formalism

abstract:
  We introduce a dissipative version of the Zhang's model of Self-Organized
Criticality, where a parameter allows to tune the local energy dissipation. We
analyze the main dynamical features of the model and relate in particular the
Lyapunov spectrum with the transport properties in the stationary regime. We
develop a thermodynamic formalism where we define formal Gibbs measure,
partition function and pressure characterizing the avalanche distributions. We
discuss the infinite size limit in this setting. We show in particular that a
Lee-Yang phenomenon occurs in this model, for the only conservative case. This
suggests new connexions to classical critical phenomena.


title:
Lyapunov spectra and nonequilibrium ensembles equivalence in 2D fluid
  mechanics

abstract:
  We perform numerical experiments to study the Lyapunov spectra of dynamical
systems associated with the Navier--Stokes (NS) equation in two spatial
dimensions truncated over the Fourier basis. Recently new equations, called GNS
equations, have been introduced and conjectured to be equivalent to the NS
equations at large Reynolds numbers. The Lyapunov spectra of the NS and of the
corresponding GNS systems overlap, adding evidence in favor of the conjectured
equivalence already studied and partially extended in previous papers. We make
use of the Lyapunov spectra to study a fluctuation relation which had been
proposed to extend the ``fluctuation theorem'' to strongly dissipative systems.
Preliminary results towards the formulation of a local version of the
fluctuation formula are also presented.


title: Loop Groups and Discrete KdV Equations

abstract:
  A study is presented of fully discretized lattice equations associated with
the KdV hierarchy. Loop group methods give a systematic way of constructing
discretizations of the equations in the hierarchy. The lattice KdV system of
Nijhoff et al. arises from the lowest order discretization of the trivial,
lowest order equation in the hierarchy, b_t=b_x. Two new discretizations are
also given, the lowest order discretization of the first nontrivial equation in
the hierarchy, and a "second order" discretization of b_t=b_x. The former,
which is given the name "full lattice KdV" has the (potential) KdV equation as
a standard continuum limit. For each discretization a Backlund transformation
is given and soliton content analyzed. The full lattice KdV system has, like
KdV itself, solitons of all speeds, whereas both other discretizations studied
have a limited range of speeds, being discretizations of an equation with
solutions only of a fixed speed.


title: HexaKdV

abstract:
  An analog of the lattice KdV equation of Nijhoff et al. is constructed on a
hexagonal lattice. The resulting system of difference equations exhibits
soliton solutions with interesting local structure: there is a nontrivial phase
shift on moving between adjacent lattice sites, with the magnitude of the shift
tending to zero in the continuum limit.


title: The Onset of Dissipation in the Kinematic Dynamo

abstract:
  The kinematic regime of the magnetic dynamo neglects the backreaction of the
magnetic field on the flow. For small magnetic diffusivity, in the early stage
of evolution, there is an ideal phase where dissipative effects can also be
neglected. We estimate the magnitude of the energy dissipation term (Ohmic
heating), taking into account differential constraints on chaotic flows. We
find that the period of ideal evolution is roughly doubled over an estimate
without constraints. The helicity generation terms are exponentially smaller
than the energy dissipation, so that large quantities of energy are dissipated
before any helicity can be created. Helicity flow is exponentially larger than
net helicity generation. The constraints also lead to the existence of a
singular initial condition for the magnetic field for which sizable amounts of
helicity can potentially be created.


title: Lorenz integrable system moves \`a la Poinsot

abstract:
  A transformation is derived which takes Lorenz integrable system into the
well-known Euler equations of a free-torque rigid body with a fixed point, i.e.
the famous motion \`a la Poinsot. The proof is based on Lie group analysis
applied to two third order ordinary differential equations admitting the same
two-dimensional Lie symmetry algebra. Lie's classification of two-dimensional
symmetry algebra in the plane is used. If the same transformation is applied to
Lorenz system with any value of parameters, then one obtains Euler equations of
a rigid body with a fixed point subjected to a torsion depending on time and
angular velocity. The numerical solution of this system yields a
three-dimensional picture which looks like a "tornado" whose cross-section has
a butterfly-shape. Thus, Lorenz's {\em butterfly} has been transformed into a
{\em tornado}.


title: Intrinsic chaos and external noise in population dynamics

abstract:
  We address the problem of the relative importance of the intrinsic chaos and
the external noise in determining the complexity of population dynamics. We use
a recently proposed method for studying the complexity of nonlinear random
dynamical systems. The new measure of complexity is defined in terms of the
average number of bits per time-unit necessary to specify the sequence
generated by the system. This measure coincides with the rate of divergence of
nearby trajectories under two different realizations of the noise. In
particular, we show that the complexity of a nonlinear time-series model
constructed from sheep populations comes completely from the environmental
variations. However, in other situations, intrinsic chaos can be the crucial
factor. This method can be applied to many other systems in biology and
physics.


title:
Exact soliton solutions of the one-dimensional complex Swift-Hohenberg
  equation

abstract:
  Using Painlev\'e analysis, the Hirota multi-linear method and a direct ansatz
technique, we study analytic solutions of the (1+1)-dimensional complex cubic
and quintic Swift-Hohenberg equations. We consider both standard and
generalized versions of these equations. We have found that a number of exact
solutions exist to each of these equations, provided that the coefficients are
constrained by certain relations. The set of solutions include particular types
of solitary wave solutions, hole (dark soliton) solutions and periodic
solutions in terms of elliptic Jacobi functions and the Weierstrass $\wp$
function. Although these solutions represent only a small subset of the large
variety of possible solutions admitted by the complex cubic and quintic
Swift-Hohenberg equations, those presented here are the first examples of exact
analytic solutions found thus far.

categories: nlin.PS cond-mat.stat-mech math-ph math.MP nlin.SI physics.optics


title: Entry and exit sets in the dynamics of area preserving Henon map

abstract:
  In this paper we study dynamical properties of the area preserving Henon map,
as a discrete version of open Hamiltonian systems, that can exhibit chaotic
scattering. Exploiting its geometric properties we locate the exit and entry
sets, i.e. regions through which any forward, respectively backward, unbounded
orbit escapes to infinity. In order to get the boundaries of these sets we
prove that the right branch of the unstable manifold of the hyperbolic fixed
point is the graph of a function, which is the uniform limit of a sequence of
functions whose graphs are arcs of the symmetry lines of the Henon map, as a
reversible map.


title: Darboux transformation for classical acoustic spectral problem

abstract:
  We study discrete isospectral symmetries for the classical acoustic spectral
problem in spatial dimensions one and two, by developing a Darboux (Moutard)
transformation formalism for this problem. The procedure follows the steps,
similar to those for the Schr\"{o}dinger operator. However, there is no
one-to-one correspondence between the two problems. The technique developed
enables one to construct new families of integrable potentials for the acoustic
problem, in addition to those already known.
  The acoustic problem produces a non-linear Harry Dym PDE. Using the
technique, we reproduce a pair of simple soliton solutions of this equation.
These solutions are further used to construct a new positon solution for this
PDE. Furthermore, using the dressing chain approach, we build a modified Harry
Dym equation together with its LA-pair.
  As an application, we construct some singular and non-singular integrable
potentials (dielectric permitivity) for the Maxwell equations in a 2D
inhomogeneous medium.


title: Thermosolutal and binary fluid convection as a 2 x 2 matrix problem

abstract:
  We describe an interpretation of convection in binary fluid mixtures as a
superposition of thermal and solutal problems, with coupling due to advection
and proportional to the separation parameter S. Many properties of binary fluid
convection are then consequences of generic properties of 2 x 2 matrices. The
eigenvalues of 2 x 2 matrices varying continuously with a parameter r undergo
either avoided crossing or complex coalescence, depending on the sign of the
coupling (product of off-diagonal terms). We first consider the matrix
governing the stability of the conductive state. When the thermal and solutal
gradients act in concert (S>0, avoided crossing), the growth rates of
perturbations remain real and of either thermal or solutal type. In contrast,
when the thermal and solutal gradients are of opposite signs (S<0, complex
coalescence), the growth rates become complex and are of mixed type.
Surprisingly, the kinetic energy of nonlinear steady states is governed by an
eigenvalue problem very similar to that governing the growth rates. There is a
quantitative analogy between the growth rates of the linear stability problem
for infinite Prandtl number and the amplitudes of steady states of the minimal
five-variable Veronis model for arbitrary Prandtl number. For positive S,
avoided crossing leads to a distinction between low-amplitude solutal and
high-amplitude thermal regimes. For negative S, the transition between real and
complex eigenvalues leads to the creation of branches of finite amplitude, i.e.
to saddle-node bifurcations. The codimension-two point at which the saddle-node
bifurcations disappear, separating subcritical from supercritical pitchfork
bifurcations, is exactly analogous to the Bogdanov codimension-two point at
which the Hopf bifurcations disappear in the linear problem.


title: Air parcel random walk and droplet spectra broadening in clouds

abstract:
  We study the effect of turbulent flow on the droplet growth in a cloud during
the condensational phase. Using the air parcel model, we describe analytically
how the distribution of droplets over sizes evolves at the different stages of
parcel movement. We show that turbulent random walk superimposed on an
accelerated ascent of the parcel make the relative width of droplet
distribution to grow initially as $t^{1/2}$ and then decay as $t^{-3/2}$.


title: A hodograph transformation which applies to the heavenly equation

abstract:
  A hodograph transformation for a wide family of multidimensional nonlinear
partial differential equations is presented. It is used to derive solutions of
the heavenly equation (dispersionless Toda equation) as well as a family of
explicit ultra-hyperbolic selfdual vacuum spaces admiting only one Killing
vector which is not selfdual, we also give the corresponding explicit
Einstein--Weyl structures.


title: On the Whitham Hierarchies: Reductions and Hodograph Solutions

abstract:
  A general scheme for analyzing reductions of Whitham hierarchies is
presented. It is based on a method for determining the $S$-function by means of
a system of first order partial differential equations. Compatibility systems
of differential equations characterizing both reductions and hodograph
solutions of Whitham hierarchies are obtained. The method is illustrated by
exhibiting solutions of integrable models such as the dispersionless Toda
equation (heavenly equation) and the generalized Benney system.


title:
Optical soliton solutions of the quintic complex Swift-Hohenberg
  equation

abstract:
  Using a direct ansatz approach, we have found a number of zero-velocity
analytic solutons of the complex quintic Swift-Hohenberg equation. These find
application in assorted optical problems.


title:
A quantitative approximation scheme for the traveling wave solutions in
  the Hodgkin-Huxley model

abstract:
  We introduce an approximation scheme for the Hodgkin-Huxley model of nerve
conductance which allows to calculate both the speed of the traveling pulses
and their shape in quantitative agreement with the solutions of the model. We
demonstrate that the reduced problem for the front of the traveling pulse
admits a unique solution. We obtain an explicit analytical expression for the
speed of the pulses which is valid with good accuracy in a wide range of the
parameters.


title: Symmetric invariant manifolds in the Fermi-Pasta-Ulam lattice

abstract:
  The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$
particles admits a large group of discrete symmetries. The fixed point sets of
these symmetries naturally form invariant symplectic manifolds that are
investigated in this short note. For each $k$ dividing $n$ we find $k$ degree
of freedom invariant manifolds. They represent short wavelength solutions
composed of $k$ Fourier-modes and can be interpreted as embedded lattices with
periodic boundary conditions and only $k$ particles. Inside these invariant
manifolds other invariant structures and exact solutions are found which
represent for instance periodic and quasi-periodic solutions and standing and
traveling waves. Some of these results have been found previously by other
authors via a study of mode coupling coefficients and recently also by
investigating `bushes of normal modes'. The method of this paper is similar to
the latter method and much more systematic than the former. We arrive at
previously unknown results without any difficult computations. It is shown
moreover that similar invariant manifolds exist also in the Klein-Gordon
lattice and in the thermodynamic and continuum limits.


title: Reversible Polynomial Automorphisms of the Plane: the Involutory Case

abstract:
  Planar polynomial automorphisms are polynomial maps of the plane whose
inverse is also a polynomial map. A map is reversible if it is conjugate to its
inverse. Here we obtain a normal form for automorphisms that are reversible by
an involution that is also in the group of polynomial automorphisms. This form
is a composition of a sequence of generalized Henon maps together with two
simple involutions. We show that the coefficients in the normal form are unique
up to finitely many choices.


title:
Saddle points in the chaotic analytic function and Ginibre
  characteristic polynomial

abstract:
  Comparison is made between the distribution of saddle points in the chaotic
analytic function and in the characteristic polynomials of the Ginibre
ensemble. Realising the logarithmic derivative of these infinite polynomials as
the electric field of a distribution of coulombic charges at the zeros, a
simple mean-field electrostatic argument shows that the density of saddles
minus zeros falls off as $\pi^{-1}|z|^{-4}$ from the origin. This behaviour is
expected to be general for finite or infinite polynomials with zeros uniformly
randomly distributed in the complex plane, and which repel quadratically.


title: Integrable Systems and Factorization Problems

abstract:
  The present lectures were prepared for the Faro International Summer School
on Factorization and Integrable Systems in September 2000. They were intended
for participants with the background in Analysis and Operator Theory but
without special knowledge of Geometry and Lie Groups. In order to make the main
ideas reasonably clear, I tried to use only matrix algebras such as
$\frak{gl}(n)$ and its natural subalgebras; Lie groups used are either GL(n)
and its subgroups, or loop groups consisting of matrix-valued functions on the
circle (possibly admitting an extension to parts of the Riemann sphere). I hope
this makes the environment sufficiently easy to live in for an analyst. The
main goal is to explain how the factorization problems (typically, the matrix
Riemann problem) generate the entire small world of Integrable Systems along
with the geometry of the phase space, Hamiltonian structure, Lax
representations, integrals of motion and explicit solutions. The key tool will
be the \emph{% classical r-matrix} (an object whose other guise is the
well-known Hilbert transform). I do not give technical details, unless they may
be exposed in a few lines; on the other hand, all motivations are given in full
scale whenever possible.


title: A new Eulerian-Lagrangian length-scale in turbulent flows

abstract:
  We introduce a time-dependent Eulerian-Lagrangian length-scale and an inverse
locality hypothesis which explain scalings of second order one-particle
Lagrangian structure functions observed in Kinematic Simulations (KS) of
homogeneous isotropic turbulence. Our KS results are consistent with the
physical picture that particle trajectories are more/less autocorrelated if
they are smoother/rougher as a result of encountering less/more straining
stagnation points, thus leading to enhanced/reduced turbulent diffusion.


title: Singularities of Euler flow? Not out of the blue!

abstract:
  Does three-dimensional incompressible Euler flow with smooth initial
conditions develop a singularity with infinite vorticity after a finite time?
This blowup problem is still open. After briefly reviewing what is known and
pointing out some of the difficulties, we propose to tackle this issue for the
class of flows having analytic initial data for which hypothetical real
singularities are preceded by singularities at complex locations. We present
some results concerning the nature of complex space singularities in two
dimensions and propose a new strategy for the numerical investigation of
blowup.(A version of the paper with higher-quality figures is available at
http://www.obs-nice.fr/etc7/complex.pdf)


title:
Spectral theory for the failure of linear control in a nonlinear
  stochastic system

abstract:
  We consider the failure of localized control in a nonlinear spatially
extended system caused by extremely small amounts of noise. It is shown that
this failure occurs as a result of a nonlinear instability. Nonlinear
instabilities can occur in systems described by linearly stable but strongly
nonnormal evolution operators. In spatially extended systems the nonnormality
manifests itself in two different but complementary ways: transient
amplification and spectral focusing of disturbances. We show that temporal and
spatial aspects of the nonnormality and the type of nonlinearity are all
crucially important to understanding and describing the mechanism of nonlinear
instability. Presented results are expected to apply equally to other physical
systems where strong nonnormality is due to the presence of mean flow rather
than the action of control.


title: Soft billiards with corners

abstract:
  We develop a framework for dealing with smooth approximations to billiards
with corners in the two-dimensional setting. Let a polygonal trajectory in a
billiard start and end up at the same billiard's corner point. We prove that
smooth Hamiltonian flows which limit to this billiard have a nearby periodic
orbit if and only if the polygon angles at the corner are ''acceptable''. The
criterion for a corner polygon to be acceptable depends on the smooth potential
behavior at the corners, which is expressed in terms of a {scattering
function}. We define such an asymptotic scattering function and prove the
existence of it, explain how it can be calculated and predict some of its
properties. In particular, we show that it is non-monotone for some potentials
in some phase space regions. We prove that when the smooth system has a
limiting periodic orbit it is hyperbolic provided the scattering function is
not extremal there. We then prove that if the scattering function is extremal,
the smooth system has elliptic periodic orbits limiting to the corner polygon,
and, furthermore, that the return map near these periodic orbits is conjugate
to a small perturbation of the Henon map and therefore has elliptic islands. We
find from the scaling that the island size is typically algebraic in the
smoothing parameter and exponentially small in the number of reflections of the
polygon orbit.


title: On testing integrability

abstract:
  We demonstrate, using the symbolic method together with p-adic and resultant
methods,the existence of systems with exactly one or two generalized
symmetries. Since the existence of one or two symmetries is often taken as a
sure sign (or as the definition) of integrability, that is, the existence of
symmetries on infinitely many orders,this shows that such practice is devoid of
any mathematical foundation. Extensive computations show that systems with one
symmetry are rather common, and with two symmetries are fairly rare, at least
within the class we have been considering in this paper.


title: New Solvable Matrix Integrals

abstract:
  We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals
(Gross-Witten integral and integrals over complex matrices) using the notion of
tau function of matrix argument. In this case one can reduce the matrix
integral to the integral over eigenvalues, which in turn is certain tau
function. We also consider a generalization of the Kontsevich integral.


title: Forced Burgers Equation in an Unbounded Domain

abstract:
  The inviscid Burgers equation with random and spatially smooth forcing is
considered in the limit when the size of the system tends to infinity. For the
one-dimensional problem, it is shown both theoretically and numerically that
many of the features of the space-periodic case carry over to infinite domains
as intermediate time asymptotics. In particular, for large time $T$ we
introduce the concept of $T$-global shocks replacing the notion of main shock
which was considered earlier in the periodic case (1997, E et al., Phys. Rev.
Lett. 78, 1904). In the case of spatially extended systems these objects are no
anymore global. They can be defined only for a given time scale and their
spatial density behaves as $\rho(T) \sim T^{-2/3}$ for large $T$. The
probability density function $p(A)$ of the age $A$ of shocks behaves
asymptotically as $A^{-5/3}$. We also suggest a simple statistical model for
the dynamics and interaction of shocks and discuss an analogy with the problem
of distribution of instability islands for a simple first-order stochastic
differential equation.


title:
Deformations of plane algebraic curves and integrable systems of
  hydrodynamic type

abstract:
  We describe our recent work on deformations of hyperelliptic curves by means
of integrable hierarchy of hydrodynamic type (nlin.SI/0205012). We also discuss
a further extension to the case of non-hyperelliptic curves.


title: Picture of the low-dimensional structure in chaotic dripping faucets

abstract:
  Chaotic dynamics of the dripping faucet was investigated both experimentally
and theoretically. We measured continuous change in drop position and velocity
using a high-speed camera. Continuous trajectories of a low-dimensional chaotic
attractor were reconstructed from these data, which was not previously obtained
but predicted in our fluid dynamic simulation. From the simulation, we further
obtained an approximate potential function with only two variables, the drop
mass and its position of the center of mass. The potential landscape helps one
to understand intuitively how the dripping dynamics can exhibit low-dimensional
chaos.


title: Sand stirred by chaotic advection

abstract:
  We study the spatial structure of a granular material, N particles subject to
inelastic mutual collisions, when it is stirred by a bidimensional smooth
chaotic flow. A simple dynamical model is introduced where four different time
scales are explicitly considered: i) the Stokes time, accounting for the
inertia of the particles, ii) the mean collision time among the grains, iii)
the typical time scale of the flow, and iv) the inverse of the Lyapunov
exponent of the chaotic flow, which gives a typical time for the separation of
two initially close parcels of fluid. Depending on the relative values of these
different times a complex scenario appears for the long-time steady spatial
distribution of particles, where clusters of particles may or not appear.


title:
Stable two-dimensional solitary pulses in linearly coupled dissipative
  Kadomtsev-Petviashvili equations

abstract:
  A two-dimensional (2D) generalization of the stabilized Kuramoto -
Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili
(KP) equation including dissipation of the generic (Newell -- Whitehead --
Segel, NWS) type and gain. The system directly applies to the description of
gravity-capillary waves on the surface of a liquid layer flowing down an
inclined plane, with a surfactant diffusing along the layer's surface.
Actually, the model is quite general, offering a simple way to stabilize
nonlinear waves in media combining the weakly-2D dispersion of the KP type with
gain and NWS dissipation. Parallel to this, another model is introduced, whose
dissipative terms are isotropic, rather than of the NWS type. Both models
include an additional linear equation of the advection-diffusion type, linearly
coupled to the main KP-NWS equation. The extra equation provides for stability
of the zero background in the system, opening a way to the existence of stable
localized pulses. The consideration is focused on the case when the dispersive
part of the system of the KP-I type, admitting the existence of 2D localized
pulses. Treating the dissipation and gain as small perturbations and making use
of the balance equation for the field momentum, we find that the equilibrium
between the gain and losses may select two 2D solitons, from their continuous
family existing in the conservative counterpart of the model (the latter family
is found in an exact analytical form). The selected soliton with the larger
amplitude is expected to be stable. Direct simulations completely corroborate
the analytical predictions.


title: Self-organized and driven phase synchronization in coupled maps

abstract:
  We study the phase synchronization and cluster formation in coupled maps on
different networks. We identify two different mechanisms of cluster formation;
(a) {\it Self-organized} phase synchronization which leads to clusters with
dominant intra-cluster couplings and (b) {\it driven} phase synchronization
which leads to clusters with dominant inter-cluster couplings. In the novel
driven synchronization the nodes of one cluster are driven by those of the
others. We also discuss the dynamical origin of these two mechanisms for small
networks with two and three nodes.


title: Application of Weak Turbulence Theory to FPU Model

abstract:
  The foundations of weak turbulence theory is explored through its application
to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear
dispersive system. A direct application of the standard kinetic equations would
miss interesting dynamics of the energy transfer process starting from a
large-scale excitation. This failure is traced to an enforcement of the exact
resonance condition, whereas mathematically the resonance should be broadened
due to the energy transfer happening on large but finite time scales. By
allowing for the broadened resonance, a modified three-wave kinetic equation is
derived for the FPU model. This kinetic equation produces some correct scaling
predictions about the statistical dynamics of the FPU model, but does not model
accurately the detailed evolution of the energy spectrum. The reason for the
failure seems not to be one of the previously clarified reasons for breakdown
in the weak turbulence theory.


title: Stages of Energy Transfer in the FPU Model

abstract:
  The Alpha version of the Fermi-Pasta-Ulam problem is revisited through direct
numerical simulations and an application of weak turbulence theory. The energy
spectrum, initialized with a large scale excitation, is traced through a series
of distinct qualitative phases en route to eventual equipartition. Weak
turbulence theory is applied in an attempt to provide and effective
quantitative description of the evolution of the energy spectrum. Some scaling
predictions are well-confirmed byt the numerical simulations.


title: Folded solitary waves and Foldons in 2+1 dimensions

abstract:
  A general type of localized excitations, folded solitary waves and foldons,
are defined and studied both analytically and graphically. The folded solitary
waves and foldons may be "folded" in quite complicated ways and possess quite
rich structures and abundant interaction properties. The folded phenomenon is
quite universal in the real natural world. The folded solitary waves and
foldons are obtained from a quite universal formula and the universal formula
is valid for some quite universal (2+1)-dimensional physical models. The
"universal" formula is also extended to a more general form with many more
independent arbitrary functions.


title: Derivative moments in turbulent shear flows

abstract:
  We propose a generalized perspective on the behavior of high-order derivative
moments in turbulent shear flows by taking account of the roles of small-scale
intermittency and mean shear, in addition to the Reynolds number. Two
asymptotic regimes are discussed with respect to shear effects. By these means,
some existing disagreements on the Reynolds number dependence of derivative
moments can be explained. That odd-order moments of transverse velocity
derivatives tend not vanish as expected from elementary scaling considerations
does not necessarily imply that small-scale anisotropy persists at all Reynolds
numbers.


title:
Variable separation approach for a differential-difference system:
  special Toda equation

abstract:
  A multi-linear variable separation approach is developed to solve a
differential-difference Toda equation. The semi-discrete form of the continuous
universal formula is found for a suitable potential of the
differential-difference Toda system. Abundant semi-discrete localized coherent
structures of the potential can be found by appropriately selecting the
arbitrary functions of the semi-discrete form of the universal formula.


title:
Darboux Transformation and Variable Separation Approach: the
  Nizhnik-Novikov-Veselov equation

abstract:
  Darboux transformation is developed to systematically find variable
separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a
seed solution with some arbitrary functions, the once Darboux transformation
yields the variable separable solutions which can be obtained from the
truncated Painlev\'e analysis and the twice Darboux transformation leads to
some new variable separable solutions which are the generalization of the known
results obtained by using a guess ansatz to solve the generalized trilinear
equation.


title: The impressive complexity in the Nautilus pompilius shell

abstract:
  The complexity of the Nautilus pompilius shell is analyzed in terms of its
fractal dimension and its equiangular spiral form. Our findings assert that the
shell is fractal from its birth and that its growth is dictated by a
self-similar criterion (we obtain the fractal dimension of the shell as a
function of time). The variables that have been used for the analysis show an
exponential dependence on the number of chambers/age of the cephalopod, a
property inherited from its form.


title: Fractal dimension and self-similarity in Asparagus plumosus

abstract:
  We measure the fractal dimension of an African plant that is widely
cultivated as ornamental, the Asparagus plumosus. This plant presents
self-similarity, remarkable in at least two different scalings. In the
following, we present the results obtained by analyzing this plant via the box
counting method for three different scalings. We show in a quantitatively way
that this species is a fractal.


title: The Fractal Dimension of an Oil Spray

abstract:
  We study the fractal dimension of the contour of the liquid-gas interface in
a spray. Our images include both, the linking region and the break-up region
and are obtained with a high-resolution shadowgraph technique; this means that
the images can then be subject to an intensity filtering, equivalent to a
threshold analysis, that enables the establishment of the fractal range.


title: Self-averaging of Wigner transforms in random media

abstract:
  We establish the self-averaging properties of the Wigner transform of a
mixture of states in the regime when the correlation length of the random
medium is much longer than the wave length but much shorter than the
propagation distance. The main ingredients in the proof are the error estimates
for the semiclassical approximation of the Wigner transform by the solution of
the Liouville equations, and the limit theorem for two-particle motion along
the characteristics of the Liouville equations. The results are applied to a
mathematical model of the time-reversal experiments for the acoustic waves, and
self-averaging properties of the re-transmitted wave are proved.


title:
New variable separation approach: application to nonlinear diffusion
  equations

abstract:
  The concept of the derivative-dependent functional separable solution, as a
generalization to the functional separable solution, is proposed. As an
application, it is used to discuss the generalized nonlinear diffusion
equations based on the generalized conditional symmetry approach. As a
consequence, a complete list of canonical forms for such equations which admit
the derivative-dependent functional separable solutions is obtained and some
exact solutions to the resulting equations are described.


title: Survival in the replicator model: Dynamics and statistics

abstract:
  We study the multi-species replicator model with linear fitness and random
fitness matrices of various classes. By means of numerical resolution of the
replicator equations, we determine the survival probability of a species in
terms of its average interaction with the rest of the system. The role of the
interaction pattern of the ecosystem in defining survival and extinction
probabilities is emphasized.


title:
Route to chaotic synchronisation in coupled map lattices: Rigorous
  results

abstract:
  Two-dimensional mappings obtained by coupling two piecewise increasing
expanding maps are considered. Their dynamics is described when the coupling
parameter increases in the expanding domain. By introducing a coding and by
analysing an admissibility condition, approximations of the corresponding
symbolic systems are obtained. These approximations imply that the topological
entropy is located between two decreasing step functions of the coupling
parameter. The analysis firstly applies to mappings with piecewise affine local
maps which allow explicit expressions and, in a second step, is extended by
continuity to mappings with piecewise smooth local maps.


title:
Quasi-integrability in a class of systems generalizing the problem of
  two fixed centers

abstract:
  The problem of two fixed centers is a classical integrable problem, stated
and integrated by Euler in 1760. The integrability is due to the unexpected
first integral $G$. Some straightforward generalizations of the problem still
have the generalization of $G$ as a first integral, but do not possess the
energy integral. We present some numerical integrations suggesting that in the
domain of bounded orbits the behavior of these {\it a priori} non hamiltonian
systems is very similar to the behavior of usual quasi-integrable systems.


title:
Relaxation of finite perturbations: Beyond the Fluctuation-Response
  relation

abstract:
  We study the response of dynamical systems to finite amplitude perturbation.
A generalized Fluctuation-Response relation is derived, which links the average
relaxation toward equilibrium to the invariant measure of the system and points
out the relevance of the amplitude of the initial perturbation. Numerical
computations on systems with many characteristic times show the relevance of
the above relation in realistic cases.


title: Solitons in Bose-Einstein condensates trapped in a double-well potential

abstract:
  We investigate, analytically and numerically, families of bright solitons in
a system of two linearly coupled nonlinear Schrodinger/Gross-Pitaevskii
equations, describing two Bose-Einstein condensates trapped in an asymmetric
double-well potential, in particular, when the scattering lengths in the
condensates have arbitrary magnitudes and opposite signs. The solitons are
found to exist everywhere where they are permitted by the dispersion law. Using
the Vakhitov-Kolokolov criterion and numerical methods, we show that, except
for small regions in the parameter space, the solitons are stable to small
perturbations. Some of them feature self-trapping of almost all the atoms in
the condensate with no atomic interaction or weak repulsion coupled to the
self-attractive condensate. An unusual bifurcation is found, when the soliton
bifurcates from the zero solution without a visible jump in the shape, but with
a jump in the number of trapped atoms. By means of numerical simulations, it is
found that, depending on values of the parameters and the initial perturbation,
unstable solitons either give rise to breathers or completely break down into
incoherent waves ("radiation"). A version of the model with the self-attraction
in both components, which applies to the description of dual-core fibers in
nonlinear optics, is considered too, and new results are obtained for this much
studied system.


title: Lattice Dynamics and Melting of a Nonequilibrium Pattern

abstract:
  We present a new description of nonequilibrium square patterns as a
harmonically coupled crystal lattice. In a vertically oscillating granular
layer, different transverse normal modes of the granular square-lattice pattern
are observed for different driving frequencies ($f_d$) and accelerations. The
amplitude of a mode can be further excited by either frequency modulation of
$f_d$ or reduction of friction between the grains and the plate. When the mode
amplitude becomes large, the lattice melts (disorders), in accord with the
Lindemann criterion for melting in two-dimensions.


title:
Number and Amplitude of Limit Cycles emerging from {\it Topologically
  Equivalent} Perturbed Centers

abstract:
  We consider three examples of weekly perturbed centers which do not have {\it
geometrical equivalence}: a linear center, a degenerate center and a
non-hamiltonian center. In each case the number and amplitude of the limit
cycles emerging from the period annulus are calculated following the same
strategy: we reduce of all of them to locally equivalent perturbed integrable
systems of the form: $dH(x,y)+\epsilon(f(x,y)dy-g(x,y)dx)=0$, with
$H(x,y)={1/2}(x^2+y^2)$. This reduction allows us to find the Melnikov
function, $M(h)=\int_{H=h}fdy-gdx$, associated to each particular problem. We
obtain the information on the bifurcation curves of the limit cycles by solving
explicitly the equation $M(h)=0$ in each case.


title:
Boundary effects in the stepwise structure of the Lyapunov spectra for
  quasi-one-dimensional systems

abstract:
  Boundary effects in the stepwise structure of the Lyapunov spectra and the
corresponding wavelike structure of the Lyapunov vectors are discussed
numerically in quasi-one-dimensional systems consisting of many hard-disks.
Four kinds of boundary conditions constructed by combinations of periodic
boundary conditions and hard-wall boundary conditions are considered, and lead
to different stepwise structures of the Lyapunov spectra in each case. We show
that a spatial wavelike structure with a time-oscillation appears in the
spatial part of the Lyapunov vectors divided by momenta in some steps of the
Lyapunov spectra, while a rather stationary wavelike structure appears in the
purely spatial part of the Lyapunov vectors corresponding to the other steps.
Using these two kinds of wavelike structure we categorize the sequence and the
kinds of steps of the Lyapunov spectra in the four different boundary condition
cases.


title: Diversity Evolution

abstract:
  Bedau has developed a general set of evolutionary statistics that quantify
the adaptive component of evolutionary processes. On the basis of these
measures, he has proposed a set of 4 classes of evolutionary system. All
artificial life sytems so far looked at fall into the first 3 classes, whereas
the biosphere, and possibly the human economy belongs to the 4th class. The
challenge to the artificial life community is to identify exactly what is
difference between these natural evolutionary systems, and existing artificial
life systems. At ALife VII, I presented a study using an artificial
evolutionary ecology called \EcoLab. Bedau's statistics captured the
qualitative behaviour of the model. \EcoLab{} exhibited behaviour from the
first 3 classes, but not class 4, which is characterised by unbounded growth in
diversity. \EcoLab{} exhibits a critical surface given by an inverse
relationship between connectivity and diversity, above which the model cannot
tarry long. Thus in order to get unbounded diversity increase, there needs to
be a corresponding connectivity reducing (or food web pruning) process. This
paper reexamines this question in light of two possible processes that reduce
ecosystem connectivity: a tendency for specialisation and increase in
biogeographic zones through continental drift.


title: Open-Ended Artificial Evolution

abstract:
  Of all the issues discussed at {\em Alife VII: Looking Forward,
  Looking Backward}, the issue of whether it was possible to create an
artificial life system that exhibits {\em open-ended evolution} of novelty is
by far the biggest. Of the 14 open problems settled on as a result of debate at
the conference, some 6 are directly, or indirectly related to this issue. Most
people equate open-ended evolution with complexity growth, although a priori
these seem to be different things. In this paper I report on experiments to
measure the complexity of Tierran organisms, and show the results for a {\em
size-neutral} run of Tierra. In this run, no increase in organismal complexity
was observed, although organism size did increase through the run. This result
is discussed, offering some signposts on path to solving the issue of open
ended evolution.


title: Recursion operator for stationary Nizhnik--Veselov--Novikov equation

abstract:
  We present a new general construction of recursion operator from zero
curvature representation. Using it, we find a recursion operator for the
stationary Nizhnik--Veselov--Novikov equation and present a few low order
symmetries generated with the help of this operator.


title: Spectral Statistics for the Dirac Operator on Graphs

abstract:
  We determine conditions for the quantisation of graphs using the Dirac
operator for both two and four component spinors. According to the
Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry
the energy level statistics are expected, in the semiclassical limit, to
correspond to those of random matrices from the Gaussian symplectic ensemble.
This is confirmed by numerical investigation. The scattering matrix used to
formulate the quantisation condition is found to be independent of the type of
spinor. We derive an exact trace formula for the spectrum and use this to
investigate the form factor in the diagonal approximation.


title: A Uniform Approximation for the Fidelity in Chaotic Systems

abstract:
  In quantum/wave systems with chaotic classical analogs, wavefunctions evolve
in highly complex, yet deterministic ways. A slight perturbation of the system,
though, will cause the evolution to diverge from its original behavior
increasingly with time. This divergence can be measured by the fidelity, which
is defined as the squared overlap of the two time evolved states. For chaotic
systems, two main decay regimes of either Gaussian or exponential behavior have
been identified depending on the strength of the perturbation. For perturbation
strengths intermediate between the two regimes, the fidelity displays both
forms of decay. By applying a complementary combination of random matrix and
semiclassical theory, a uniform approximation can be derived that covers the
full range of perturbation strengths. The time dependence is entirely fixed by
the density of states and the so-called transition parameter, which can be
related to the phase space volume of the system and the classical action
diffusion constant, respectively. The accuracy of the approximations are
illustrated with the standard map.


title:
Integration of nonlinear Partial Differential Equations by using matrix
  algebraic systems

abstract:
  The paper develops the method for construction of families of particular
solutions to some classes of nonlinear Partial Differential Equations (PDE).
Method is based on the specific link between algebraic matrix equations and
PDE. Admittable solutions involve arbitrary functions of either single or
several variables.


title: The ILW hierarchy

abstract:
  In this paper, we present a new hierarchy which includes the intermediate
long wave (ILW) equation at the lowest order. This hierarchy is thought to be a
novel reduction of the 1st modified KP type hierarchy. The framework of our
investigation is Sato theory.


title: Universal Power-law Decay in Hamiltonian Systems?

abstract:
  The understanding of the asymptotic decay of correlations and of the
distribution of Poincar\'e recurrence times $P(t)$ has been a major challenge
in the field of Hamiltonian chaos for more than two decades. In a recent
Letter, Chirikov and Shepelyansky claimed the universal decay $P(t) \sim
t^{-3}$ for Hamiltonian systems. Their reasoning is based on renormalization
arguments and numerical findings for the sticking of chaotic trajectories near
a critical golden torus in the standard map. We performed extensive numerics
and find clear deviations from the predicted asymptotic exponent of the decay
of $P(t)$. We thereby demonstrate that even in the supposedly simple case, when
a critical golden torus is present, the fundamental question of asymptotic
statistics in Hamiltonian systems remains unsolved.


title:
Evidence for directed percolation universality at the onset of
  spatiotemporal intermittency in coupled circle maps

abstract:
  We consider a lattice of coupled circle maps, a model arising naturally in
descriptions of solid state phenomena such as Josephson junction arrays. We
find that the onset of spatiotemporal intermittency (STI) in this system is
analogous to directed percolation (DP), with the transition being to an unique
absorbing state for low nonlinearities, and to weakly chaotic absorbing states
for high nonlinearities. We find that the complete set of static exponents and
spreading exponents at all critical points match those of DP very convincingly.
Further, hyperscaling relations are fulfilled, leading to independent controls
and consistency checks of the values of all the critical exponents. These
results lend strong support to the conjecture that the onset of STI in
deterministic models belongs to the DP universality class.


title:
Nonlinear dynamics in the Cournot duopoly game with heterogeneous
  players

abstract:
  We analyze a nonlinear discrete-time Cournot duopoly game, where players have
heterogeneous expectations. Two types of players are considered: boundedly
rational and naive expectations. In this study we show that the dynamics of the
duopoly game with players whose beliefs are heterogeneous, may become
complicated. The model gives more complex chaotic and unpredictable
trajectories as a consequence of increasing the speed of adjustment of
boundedly rational player. The equilibrium points and local stability of the
duopoly game are investigated. As some parameters of the model are varied, the
stability of the Nash equilibrium point is lost and the complex (periodic or
chaotic) behavior occurs. Numerical simulations is presented to show that
players with heterogeneous beliefs make the duopoly game behaves chaotically.
Also we get the fractal dimension of the chaotic attractor of our map which is
equivalent to the dimension of Henon map.


title: Chaos and flights in the atom-photon interaction in cavity QED

abstract:
  We study dynamics of the atom-photon interaction in cavity quantum
electrodynamics (QED), considering a cold two-level atom in a single-mode
high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three
coupled degrees of freedom: translational, internal atomic, and the field. The
system proves to have different types of motion including L\'{e}vy flights and
chaotic walkings of an atom in a cavity. It is shown that the translational
motion, related to the atom recoils, is governed by an equation of a parametric
nonlinear pendulum with a frequency modulated by the Rabi oscillations. This
type of dynamics is chaotic with some width of the stochastic layer that is
estimated analytically. The width is fairly small for realistic values of the
control parameters, the normalized detuning $\delta$ and atomic recoil
frequency $\alpha$. It is demonstrated how the atom-photon dynamics with a
given value of $\alpha$ depends on the values of $\delta$ and initial
conditions. Two types of L\'{e}vy flights, one corresponding to the ballistic
motion of the atom and another one corresponding to small oscillations in a
potential well, are found. These flights influence statistical properties of
the atom-photon interaction such as distribution of Poincar\'{e} recurrences
and moments of the atom position $x$. The simulation shows different regimes of
motion, from slightly abnormal diffusion with $<x^2>\sim\tau^{1.13}$ at $\delta
=1.2$ to a superdiffusion with $<x^2> \sim \tau^{2.2}$ at $\delta=1.92$ that
corresponds to a superballistic motion of the atom with an acceleration. The
obtained results can be used to find new ways to manipulate atoms, to cool and
trap them by adjusting the detuning $\delta$.


title: Commutator-errors in large-eddy simulation

abstract:
  A new formulation is derived for the commutator-errors in large-eddy
simulation of incompressible flow. These commutator-errors arise from the
application of non-uniform filters to the Navier-Stokes equations. As a
consequence, the filtered velocity field is no longer solenoidal. The order of
magnitude of the commutator-errors is compared with the divergence of the
turbulent stresses. This shows that one can not reduce the size of the
commutator-errors independently of the turbulent stress terms by some judicious
construction of the filter operator. Similarity modeling for the
commutator-errors is presented, including an extension of Bardina's approach
and the application of Leray regularization. The performance of the
commutator-error parameterization is illustrated with the one-dimensional
Burgers equation. For large filter-width variations the Leray approach is shown
to capture the filtered flow with better accuracy than is possible with
Bardina's approach.


title:
Instability of two interacting, quasi-monochromatic waves in shallow
  water

abstract:
  We study the nonlinear interactions of waves with a doubled-peaked power
spectrum in shallow water. The starting point is the prototypical equation for
nonlinear uni-directional waves in shallow water, i.e. the Korteweg de Vries
equation. Using a multiple-scale technique two defocusing coupled Nonlinear
Schr\"odinger equations are derived. We show analytically that plane wave
solutions of such a system can be unstable to small perturbations. This
surprising result suggests the existence of a new energy exchange mechanism
which could influence the behaviour of ocean waves in shallow water.


title:
Generalized r-matrix structure and algebro-geometric solution for
  integrable systems

abstract:
  The purpose of this paper is to construct a generalized r-matrix structure of
finite dimensional systems and an approach to obtain the algebro-geometric
solutions of integrable nonlinear evolution equations (NLEEs). Our starting
point is a generalized Lax matrix instead of usual Lax pair. The generalized
r-matrix structure and Hamiltonian functions are presented on the basis of
fundamental Poisson bracket. It can be clearly seen that various nonlinear
constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV,
c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure.
All these nonlinear systems have {\it r}-matrices, and are completely
integrable in Liouville's sense. Furthermore, our generalized structure is
developed to become an approach to obtain the algebro-geometric solutions of
integrable NLEEs. Finally, the two typical examples are considered to
illustrate this approach: the infinite or periodic Toda lattice equation and
the AKNS equation with the condition of decay at infinity or periodic boundary.


title:
Riccati Solutions of Discrete Painlev\'e Equations with Weyl Group
  Symmetry of Type $E_8^{(1)}$

abstract:
  We present a special solutions of the discrete Painlev\'e equations
associated with $A_0^{(1)}$, $A_0^{(1)*}$ and $A_0^{(1)**}$-surface. These
solutions can be expressed by solutions of linear difference equations. Here
the $A_0^{(1)}$-surface discrete Painlev\'e equation is the most generic
difference equation, as all discrete Painlev\'e equations can be obtained by
its degeneration limit. These special solutions exist when the parameters of
the discrete Painlev\'e equation satisfy a particular constraint. We consider
that these special functions belong to the hypergeometric family although they
seems to go beyond the known discrete and $q$-discrete hypergeometric
functions. We also discuss the degeneration scheme of these solutions.


title:
Simple Model for the Dynamics of Correlations in the Evolution of
  Economic Entities Under Varying Economic Conditions

abstract:
  From some observations on economic behaviors, in particular changing economic
conditions with time and space, we develop a very simple model for the
evolution of economic entities within a geographical type of framework. We
raise a few questions and attempt to investigate whether some of them can be
tackled by our model. Several cases of interest are reported. It is found that
the model even in its simple forms can lead to a large variety of situations,
including: delocalization and cycles, but also pre-chaotic behavior.

categories: nlin.AO cond-mat.stat-mech cs.CE physics.soc-ph


title: In phase and anti-phase syncronization of coupled homoclinic oscillators

abstract:
  We numerically investigate the dynamics of a closed chain of unidirectionally
coupled oscillators in a regime of homoclinic chaos. The emerging
synchronization regimes show analogies with the experimental behavior of a
single chaotic laser subjected to a delayed feedback


title:
Hierarchy of chaotic maps with an invariant measure and their
  compositions

abstract:
  We give a hierarchy of many-parameter families of maps of the interval [0,1]
with an invariant measure and using the measure, we calculate Kolmogorov--Sinai
entropy of these maps analytically. In contrary to the usual one-dimensional
maps these maps do not possess period doubling or period-n-tupling cascade
bifurcation to chaos, but they have single fixed point attractor at certain
region of parameters space, where they bifurcate directly to chaos without
having period-n-tupling scenario exactly at certain values of the parameters.


title:
The matrix Kadomtsev--Petviashvili equation as a source of integrable
  nonlinear equations

abstract:
  A new integrable class of Davey--Stewartson type systems of nonlinear partial
differential equations (NPDEs) in 2+1 dimensions is derived from the matrix
Kadomtsev--Petviashvili equation by means of an asymptotically exact nonlinear
reduction method based on Fourier expansion and spatio-temporal rescaling. The
integrability by the inverse scattering method is explicitly demonstrated, by
applying the reduction technique also to the Lax pair of the starting matrix
equation and thereby obtaining the Lax pair for the new class of systems of
equations. The characteristics of the reduction method suggest that the new
systems are likely to be of applicative relevance. A reduction to a system of
two interacting complex fields is briefly described.


title: Finite Extension of Polymers in Turbulent Flow

abstract:
  The statistics of polymers advected by a turbulent flow are investigated. To
limit the polymer lengths above to coil-stretch transition, a FENE-P type
relaxation law is used. The turbulence is modeled by a random strain,
delta-correlated in time and with Gaussian statistics. The distribution of
polymer lengths for both the coiled and stretched states are derived, from
which are obtained analytical expressions for the moments of the distribution.
The polymer stress on the fluid decreases linearly with the inverse of the
Deborah number. The degradation (breaking) of the polymers is also discussed,
showing that even for large Deborah number some polymers can remain unbroken.


title: B_{n}^{(1)} and A_{2n}^{(2)}reflection K-matrices

abstract:
  We investigate the regular solutions of the boundary Yang-Baxter equation for
the vertex models associated with the $B_{n}^{(1)}$ and $A_{2n}^{(2)}$ affine
Lie algebras. In both class of models we find two general solutions with $n+1$
free parameters. In addition, we have find $2n-1$ diagonal solutions for
$B_{n}^{(1)}$ models and $2n+1$ diagonal solutions for $% A_{2n}^{(2)}$ models.
It turns out that for each $B_{n}^{(1)}$ model there exist a diagonal K-matrix
with one free parameter. Moreover, a three free parameter general solution
exists for the $B_{1}^{(1)}$ model which is the vector representation for the
Zamolodchikov-Fateev model.


title: Path integration over closed loops and Gutzwiller's trace formula

abstract:
  In 1967 M.C. Gutzwiller succeeded to derive the semiclassical expression of
the quantum energy density of systems exhibiting a chaotic Hamiltonian dynamics
in the classical limit. The result is known as the Gutzwiller trace formula.
  The scope of this review is to present in a self-contained way recent
developments in functional determinant theory allowing to revisit the
Gutzwiller trace formula in the spirit of field theory.
  The field theoretic setup permits to work explicitly at every step of the
derivation of the trace formula with invariant quantities of classical periodic
orbits. R. Forman's theory of functional determinants of linear, non singular
elliptic operators yields the expression of quantum quadratic fluctuations
around classical periodic orbits directly in terms of the monodromy matrix of
the periodic orbits.
  The phase factor associated to quadratic fluctuations, the Maslov phase, is
shown to be specified by the Morse index for closed extremals, also known as
Conley and Zehnder index.


title:
System-Size Effects on the Collective Dynamics of Cell Populations with
  Global Coupling

abstract:
  Phase-transitionlike behavior is found to occur in globally coupled systems
of finite number of elements, and its theoretical explanation is provided. The
system studied is a population of globally pulse-coupled integrate-and-fire
cells subject to small additive noise. As the population size is changed, the
system shows a phase-transitionlike behavior. That is, there exits a
well-defined critical system size above which the system stays in a monostable
state with high-frequency activity while below which a new phase characterized
by alternation of high- and low frequency activities appears. The mean field
motion obeys a stochastic process with state-dependent noise, and the above
phenomenon can be interpreted as a noise-induced transition characteristic to
such processes. Coexistence of high- and low frequency activities observed in
finite size systems is reported by N. Cohen, Y. Soen and E. Braun[Physica A249,
600 (1998)] in the experiments of cultivated heart cells. The present report
gives the first qualitative interpretation of their experimental results.


title: The origin of diffusion: the case of non chaotic systems

abstract:
  We investigate the origin of diffusion in non-chaotic systems. As an example,
we consider 1-$d$ map models whose slope is everywhere 1 (therefore the
Lyapunov exponent is zero) but with random quenched discontinuities and
quasi-periodic forcing. The models are constructed as non-chaotic
approximations of chaotic maps showing deterministic diffusion, and represent
one-dimensional versions of a Lorentz gas with polygonal obstacles (e.g., the
Ehrenfest wind tree model). In particular, a simple construction shows that
these maps define non-chaotic billiards in space-time. The models exhibit, in a
wide range of the parameters, the same diffusive behavior of the corresponding
chaotic versions. We present evidence of two sufficient ingredients for
diffusive behavior in one-dimensional, non-chaotic systems: i) a finite-size,
algebraic instability mechanism, and ii) a mechanism that suppresses periodic
orbits.


title:
Long-Time Asymptotics of Solutions to the Cauchy Problem for the
  Defocusing Non-Linear Schr\"{o}dinger Equation with Finite-Density Initial
  Data. II. Dark Solitons on Continua

abstract:
  For Lax-pair isospectral deformations whose associated spectrum, for given
initial data, consists of the disjoint union of a finitely denumerable discrete
spectrum (solitons) and a continuous spectrum (continuum), the matrix
Riemann-Hilbert problem approach is used to derive the leading-order
asymptotics as $| t | \to \infty$ $(x/t \sim \mathcal{O} (1))$ of solutions $(u
= u(x,t))$ to the Cauchy problem for the defocusing non-linear Schr\"{o}dinger
equation (D${}_{f}$NLSE), $\mi \partial_{t}u + \partial_{x}^{2}u - 2(| u |^{2}
- 1) u = 0$, with finite-density initial data $u(x,0) =_{x \to \pm \infty} \exp
(\tfrac{\mi (1 \mp 1) \theta}{2})(1 + o(1))$, $\theta \in [0,2\pi)$. The
D${}_{f}$NLSE dark soliton position shifts in the presence of the continuum are
also obtained.


title:
Dynamical tunneling in molecules: role of the classical resonances and
  chaos

abstract:
  In this letter we study dynamical tunneling in highly excited symmetric
molecules. The role of classical phase space structures like resonances and
chaos on the tunneling splittings are illustrated using the water molecule as
an example. It is argued that the enhancements in the splittings due to
resonances (near-integrable phase space) and due to chaos (mixed phase space)
are best understood away from the fluctuations associated with avoided
crossings. In particular we provide an essential difference between the two
mechanisms in terms of high order perturbation theory. The analysis, apart from
testing the validity of a perturbative approach, suggests such systems as prime
candidates for studying dynamical tunneling.


title: Large-System Phase-Space Dimensionality Loss in Stationary Heat Flows

abstract:
  Thermostated tethered harmonic lattices provide good illustrations of the
phase-space dimensionality loss which occurs in the strange-attractor
distributions characterizing stationary nonequilibrium flows. We use
time-reversible nonequilibrium molecular dynamics, with two Nose-Hoover
thermostats, one hot and one cold, to study a family of square heat-conducting
systems. We find a phase-space dimensionality loss which can exceed the
dimensionality associated with the two driving Nose-Hoover thermostats by as
much as a factor of four. We also estimate the dimensionality loss in the
purely Hamiltonian part of phase space. By measuring the projection of the
total dimensionality loss there we show that nearly all of the loss occurs in
the Hamiltonian part. Thus this loss, which characterizes the extreme rarity of
nonequilibrium states, persists in the large-system thermodynamic limit.


title:
Smaller alignment index (SALI): Determining the ordered or chaotic
  nature of orbits in conservative dynamical systems

abstract:
  We apply the smaller alignment index (SALI) method for distinguishing between
ordered and chaotic motion in some simple conservative dynamical systems. In
particular we compute the SALI for ordered and chaotic orbits in a 2D and a 4D
symplectic map, as well as a two--degree of freedom Hamiltonian system due to
H\'{e}non & Heiles. In all cases, the SALI determines correctly the nature of
the tested orbit, faster than the method of the computation of the maximal
Lyapunov characteristic number. The computation of the SALI for a sample of
initial conditions allows us to clearly distinguish between regions in phase
space where ordered or chaotic motion occurs.


title: Lyapunov instability and collective tangent space dynamics of fluids

abstract:
  The phase space trajectories of many body systems charateristic of simple
fluids are highly unstable. We quantify this instability by a set of Lyapunov
exponents, which are the rates of exponential divergence, or convergence, of
initial (infinitesimal) perturbations along carefully selected directions in
phase space. It is demonstrated that the perturbation associated with the
maximum Lyapunov exponent is localized in space. This localization persists in
the large-particle limit, regardless of the interaction potential. The
perturbations belonging to the smallest positive exponents, however, are
sensitive to the potential. For hard particles they form well-defined
long-wavelength modes. The modes could not be observed for systems interacting
with a soft potential due to surprisingly large fluctuations of the local
time-dependent exponents.


title: Applicability of the Fisher Equation to Bacterial Population Dynamics

abstract:
  The applicability of the Fisher equation, which combines diffusion with
logistic nonlinearity, to population dynamics of bacterial colonies is studied
with the help of explicit analytic solutions for the spatial distribution of a
stationary bacterial population under a static mask. The mask protects the
bacteria from ultraviolet light. The solution, which is in terms of Jacobian
elliptic functions, is used to provide a practical prescription to extract
Fisher equation parameters from observations and to decide on the validity of
the Fisher equation.


title: Stochastic synchronization in globally coupled phase oscillators

abstract:
  Cooperative effects of periodic force and noise in globally Cooperative
effects of periodic force and noise in globally coupled systems are studied
using a nonlinear diffusion equation for the number density. The amplitude of
the order parameter oscillation is enhanced in an intermediate range of noise
strength for a globally coupled bistable system, and the order parameter
oscillation is entrained to the external periodic force in an intermediate
range of noise strength. These enhancement phenomena of the response of the
order parameter in the deterministic equations are interpreted as stochastic
resonance and stochastic synchronization in globally coupled systems.


title: Stick-Slip Motion and Phase Transition in a Block-Spring System

abstract:
  We study numerically stick slip motions in a model of blocks and springs
being pulled slowly. The sliding friction is assumed to change dynamically with
a state variable. The transition from steady sliding to stick-slip is
subcritical in a single block and spring system. However, we find that the
transition is continuous in a long chain of blocks and springs. The size
distribution of stick-slip motions exhibits a power law at the critical point.


title:
Initial-boundary value problems for linear PDEs: the analyticity
  approach

abstract:
  To study initial-boundary value problems for linear PDEs we have recently
proposed two alternative approaches in Fourier space: the "analyticity appoach"
and the "elimination by restriction approach". In this paper we present the
analyticity appoach and we illustrate its power in studying the well-posedness
of initial - boundary value problems for second and third order evolutionary
PDEs, and in constructing their solution. We also show the connection between
the analyticity approach and the elimination by restriction approach in the
study of the Dirichelet and Neumann problems for the Schroedinger equation in
the n-dimensional quadrant.


title: Reply to E.G.D. Cohen, L. Rondoni, Physica A 306 (2002) 117

abstract:
  In their paper Cohen and Rondoni severely question the physical relevance of
studies of transport by means of multibaker maps and other classes of dynamics
considering the motion of independen particles - like e.g. the Lorentz gas. We
argue that this is to a large extent due to inappropriate interpretation of the
models. In particular, the arguments concerning the lack of local thermodynamic
equilibrium are inconsistent for the models previously worked out by ourselves.


title:
Diffusive Lorentz gases and multibaker maps are compatible with
  irreversible thermodynamics

abstract:
  We show that simple diffusive systems, such as the Lorentz gas and multibaker
maps are perfectly compatible with the laws of irreversible thermodynamics,
despite the fact that the moving particles, or their equivalents, in these
models do not interact with each other, and that the dynamics takes place in
low-dimensional phase spaces. The interaction of moving particles with
scatterers provides the dynamical mechanism responsible for an approach to
equilibrium, under appropriate conditions. This analysis provides a refutation
of the criticisms expressed recently by Cohen and Rondoni [Physica A 306 (2002)
117-128].


title: Echoes in classical dynamical systems

abstract:
  Echoes arise when external manipulations to a system induce a reversal of its
time evolution that leads to a more or less perfect recovery of the initial
state. We discuss the accuracy with which a cloud of trajectories returns to
the initial state in classical dynamical systems that are exposed to additive
noise and small differences in the equations of motion for forward and backward
evolution. The cases of integrable and chaotic motion and small or large noise
are studied in some detail and many different dynamical laws are identified.
Experimental tests in 2-d flows that show chaotic advection are proposed.


title: Anomalous Diffusion in Infinite Horizon Billiards

abstract:
  We consider the long time dependence for the moments of displacement < |r|^q
> of infinite horizon billiards, given a bounded initial distribution of
particles. For a variety of billiard models we find <|r|^q> ~ t^g(q) (up to
factors of log t). The time exponent, g(q), is piecewise linear and equal to
q/2 for q<2 and q-1 for q>2. We discuss the lack of dependence of this result
on the initial distribution of particles and resolve apparent discrepancies
between this time dependence and a prior result. The lack of dependence on
initial distribution follows from a remarkable scaling result that we obtain
for the time evolution of the distribution function of the angle of a
particle's velocity vector.


title: Long Time Algebraic Relaxation in Chaotic Billiards

abstract:
  The long time algebraic relaxation process in spatially periodic billiards
with infinite horizon is shown to display a self-similar time asymptotic form.
This form is identical for a class of such billiards, but can be different in
an important special case.


title:
Drag Reduction by Polymers in Turbulent Channel Flows: Energy
  Redistribution Between Invariant Empirical Modes

abstract:
  We address the phenomenon of drag reduction by dilute polymeric additive to
turbulent flows, using Direct Numerical Simulations (DNS) of the FENE-P model
of viscoelastic flows. It had been amply demonstrated that these model
equations reproduce the phenomenon, but the results of DNS were not analyzed so
far with the goal of interpreting the phenomenon. In order to construct a
useful framework for the understanding of drag reduction we initiate in this
paper an investigation of the most important modes that are sustained in the
viscoelastic and Newtonian turbulent flows respectively. The modes are obtained
empirically using the Karhunen-Loeve decomposition, allowing us to compare the
most energetic modes in the viscoelastic and Newtonian flows. The main finding
of the present study is that the spatial profile of the most energetic modes is
hardly changed between the two flows. What changes is the energy associated
with these modes, and their relative ordering in the decreasing order from the
most energetic to the least. Modes that are highly excited in one flow can be
strongly suppressed in the other, and vice versa. This dramatic energy
redistribution is an important clue to the mechanism of drag reduction as is
proposed in this paper. In particular there is an enhancement of the energy
containing modes in the viscoelastic flow compared to the Newtonian one; drag
reduction is seen in the energy containing modes rather than the dissipative
modes as proposed in some previous theories.


title: Pitch perception: A dynamical-systems perspective

abstract:
  Two and a half millennia ago Pythagoras initiated the scientific study of the
pitch of sounds; yet our understanding of the mechanisms of pitch perception
remains incomplete. Physical models of pitch perception try to explain from
elementary principles why certain physical characteristics of the stimulus lead
to particular pitch sensations. There are two broad categories of
pitch-perception models: place or spectral models consider that pitch is mainly
related to the Fourier spectrum of the stimulus, whereas for periodicity or
temporal models its characteristics in the time domain are more important.
Current models from either class are usually computationally intensive,
implementing a series of steps more or less supported by auditory physiology.
However, the brain has to analyse and react in real time to an enormous amount
of information from the ear and other senses. How is all this information
efficiently represented and processed in the nervous system? A proposal of
nonlinear and complex systems research is that dynamical attractors may form
the basis of neural information processing. Because the auditory system is a
complex and highly nonlinear dynamical system it is natural to suppose that
dynamical attractors may carry perceptual and functional meaning. Here we show
that this idea, scarcely developed in current pitch models, can be successfully
applied to pitch perception.


title:
Pattern Formation and Dynamics in Rayleigh-B\'{e}nard Convection:
  Numerical Simulations of Experimentally Realistic Geometries

abstract:
  Rayleigh-B\'{e}nard convection is studied and quantitative comparisons are
made, where possible, between theory and experiment by performing numerical
simulations of the Boussinesq equations for a variety of experimentally
realistic situations. Rectangular and cylindrical geometries of varying aspect
ratios for experimental boundary conditions, including fins and spatial ramps
in plate separation, are examined with particular attention paid to the role of
the mean flow. A small cylindrical convection layer bounded laterally either by
a rigid wall, fin, or a ramp is investigated and our results suggest that the
mean flow plays an important role in the observed wavenumber. Analytical
results are developed quantifying the mean flow sources, generated by amplitude
gradients, and its effect on the pattern wavenumber for a large-aspect-ratio
cylinder with a ramped boundary. Numerical results are found to agree well with
these analytical predictions. We gain further insight into the role of mean
flow in pattern dynamics by employing a novel method of quenching the mean flow
numerically. Simulations of a spiral defect chaos state where the mean flow is
suddenly quenched is found to remove the time dependence, increase the
wavenumber and make the pattern more angular in nature.


title: Fractal asymptotics

abstract:
  Recent advances in the periodic orbit theory of stochastically perturbed
systems have permitted a calculation of the escape rate of a noisy chaotic map
to order 64 in the noise strength. Comparison with the usual asymptotic
expansions obtained from integrals and with a previous calculation of the
electrostatic potential of exactly selfsimilar fractal charge distributions,
suggests a remarkably accurate form for the late terms in the expansion, with
parameters determined independently from the fractal repeller and the critical
point of the map. Two methods give a precise meaning to the asymptotic
expansion, Borel summation and Shafer approximants. These can then be compared
with the escape rate as computed by alternative methods.


title:
Renormalization and destruction of $1/\gamma^2$ tori in the standard
  nontwist map

abstract:
  Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D
{\bf 91}, 1 (1996) and {\bf 100}, 311 (1997) on the standard nontwist map, the
breakup of an invariant torus with winding number equal to the inverse golden
mean squared is studied. Improved numerical techniques provide the greater
accuracy that is needed for this case. The new results are interpreted within
the renormalization group framework by constructing a renormalization operator
on the space of commuting map pairs, and by studying the fixed points of the so
constructed operator.


title: Effect of particle inertia on the turbulence in a suspension

abstract:
  We propose a one-fluid analytical model for a turbulently flowing dilute
suspension, based on modified Navier-Stokes equation with a $k$-dependent
effective density of suspension, $\rho_ {eff}(k)$, and an additional damping
term $\propto \gamma_ p(k)$, representing the fluid-particle friction
(described by Stokes law). The statistical description of turbulence within the
model is simplified by a modification of the usual closure procedure based on
the Richardson-Kolmogorov picture of turbulence with a differential
approximation for the energy transfer term. The resulting ordinary differential
equation for the energy budget is solved analytically for various important
limiting cases and numerically in the general case. In the inertial interval of
scales we describe analytically two competing effects: the energy suppression
due to the fluid particle friction and the energy enhancement during the
cascade process due to decrease of the effective density of the small scale
motions. An additional suppression or enhancement of the energy density may
occur in the viscous subrange, caused by the variation of the extent of the
inertial interval due to the combined effect of the fluid-particle friction and
the decrease of the kinematic viscosity of the suspensions. The analytical
description of the complicated interplay of these effects supported by
numerical calculations is presented. Our findings allow one to rationalize the
qualitative picture of the isotropic homogeneous turbulence of dilute
suspensions as observed in direct numerical simulations.


title: Graph Theory and the Evolution of Autocatalytic Networks

abstract:
  We give a self-contained introduction to the theory of directed graphs,
leading up to the relationship between the Perron-Frobenius eigenvectors of a
graph and its autocatalytic sets. Then we discuss a particular dynamical system
on a fixed but arbitrary graph, that describes the population dynamics of
species whose interactions are determined by the graph. The attractors of this
dynamical system are described as a function of graph topology. Finally we
consider a dynamical system in which the graph of interactions of the species
coevolves with the populations of the species. We show that this system
exhibits complex dynamics including self-organization of the network by
autocatalytic sets, growth of complexity and structure, and collapse of the
network followed by recoveries. We argue that a graph theoretic classification
of perturbations of the network is helpful in predicting the future impact of a
perturbation over short and medium time scales.


title: Euroattractor: a brief introduction to Iterated Function Systems

abstract:
  In this work we propose a definition of an Euroattractor: an attracting
invariant measure of a certain iterated functions system (IFS). An IFS is
defined by specifying a set of functions, defined in subsets of R^N or in a
classical phase space, which act randomly on the initial point, so it may be
considered as a generalization of the notion of classical dynamical system. If
the functions are sufficiently contracting, there exists an invariant measure
of the system, often concentrated on a fractal set. We investigate invariant
measures of a certain class of generalized or weakly contracting IFS's.


title:
Spectral Analysis of Darboux Transformations for the Focusing NLS
  Hierarchy

abstract:
  We study Darboux transformations associated with the focusing nonlinear
Schr\"odinger equation (NLS_-) and their effect on spectral properties of the
underlying Lax operator. The latter is a formally J-self-adjoint (but
non-self-adjoint) one-dimensional Dirac-type operator with potential q. As one
of our principal results we prove that under the most general hypothesis of
local integrability on q, the maximally defined operator D(q) generated by this
Dirac operator is actually J-self-adjoint. We also establish the existence of
Weyl--Titchmarsh-type solutions associated with D(q). The Darboux
transformations considered in this paper guarantee that the resulting
potentials q are locally nonsingular. Moreover, we prove that the construction
of N-soliton NLS_- potentials q^{(N)} with respect to a general NLS_-
background potential q, associated with Dirac operators D(q^{(N)}) and D(q),
amounts to the insertion of N complex conjugate pairs of eigenvalues into the
spectrum of D(q), leaving the rest of the spectrum (especially, the essential
spectrum) invariant. These results are obtained by establishing the existence
of bounded transformation operators which intertwine the background Dirac
operator D(q) and the Dirac operator D(q^{(N)}) obtained after N Darboux
transformations.


title: Spatial Resonator Solitons

abstract:
  Spatial solitons can exist in various kinds of nonlinear optical resonators
with and without amplification. In the past years different types of these
localized structures such as vortices, bright, dark solitons and phase solitons
have been experimentally shown to exist. Many links appear to exist to fields
different from optics, such as fluids, phase transitions or particle physics.
These spatial resonator solitons are bistable and due to their mobility suggest
schemes of information processing not possible with the fixed bistable elements
forming the basic ingredient of traditional electronic processing. The recent
demonstration of existence and manipulation of spatial solitons in emiconductor
microresonators represents a step in the direction of such optical parallel
processing applications. We review pattern formation and solitons in a general
context, show some proof of principle soliton experiments on slow systems, and
describe in more detail the experiments on semiconductor resonator solitons
which are aimed at applications.


title:
Emergence of a complex and stable network in a model ecosystem with
  extinction and mutation

abstract:
  We propose a minimal model of the dynamics of diversity -- replicator
equations with extinction, invasion and mutation. We numerically study the
behavior of this simple model and show that it displays completely different
behavior from the conventional replicator equation and the generalized
Lotka-Volterra equation. We reach several significant conclusions as follows:
(1) a complex ecosystem can emerge when mutants with respect to
species-specific interaction are introduced; (2) such an ecosystem possesses
strong resistance to invasion; (3) a typical fixation process of mutants is
realized through the rapid growth of a group of mutualistic mutants with higher
fitness than majority species; (4) a hierarchical taxonomic structure (like
family-genus-species) emerges; and (5) the relative abundance of species
exhibits a typical pattern widely observed in nature. Several implications of
these results are discussed in connection with the relationship of the present
model to the generalized Lotka-Volterra equation.


title: Stable regions and singular trajectories in chaotic soft wall billiards

abstract:
  We present numerical and experimental results for the development of islands
of stability in atom-optics billiards with soft walls. As the walls are soften,
stable regions appear near singular periodic trajectories in converging
(focusing) and dispersing billiards, and are surrounded by areas of
"stickiness" in phase-space. The size of these islands depends on the softness
of the potential in a very sensitive way.


title: The structure of flame filaments in chaotic flows

abstract:
  The structure of flame filaments resulting from chaotic mixing within a
combustion reaction is considered. The transverse profile of the filaments is
investigated numerically and analytically based on a one-dimensional model that
represents the effect of stirring as a convergent flow. The dependence of the
steady solutions on the Damkohler number and Lewis number is treated in detail.
It is found that, below a critical Damkohler number Da(crit), the flame is
quenched by the flow. The quenching transition appears as a result of a
saddle-node bifurcation where the stable steady filament solution collides with
an unstable one. The shape of the steady solutions for the concentration and
temperature profiles changes with the Lewis number and the value of Da(crit)
increases monotonically with the Lewis number. Properties of the solutions are
studied analytically in the limit of large Damkohler number and for small and
large Lewis number.


title: Labyrinthine Turing Pattern Formation in the Cerebral Cortex

abstract:
  I propose that the labyrinthine patterns of the cortices of mammalian brains
may be formed by a Turing instability of interacting axonal guidance species
acting together with the mechanical strain imposed by the interconnecting
axons.


title:
Hamiltonian mappings and circle packing phase spaces: numerical
  investigations

abstract:
  In a previous paper we introduced examples of Hamiltonian mappings with phase
space structures resembling circle packings. We now concentrate on one
particular mapping and present numerical evidence which supports the conjecture
that the set of circular resonance islands is dense in phase space.


title:
Statistics of defect motion in spatiotemporal chaos in inclined layer
  convection

abstract:
  We report experiments on defect-tracking in the state of undulation chaos
observed in thermal convection of an inclined fluid layer. We characterize the
ensemble of defect trajectories according to their velocities, relative
positions, diffusion, and gain and loss rates. In particular, the defects
exhibit incidents of rapid transverse motion which result in power law
distributions for a number of quantitative measures. We examine connections
between this behavior and L\'evy flights and anomalous diffusion. In addition,
we describe time-reversal and system size invariance for defect creation and
annihilation rates.


title: Catalytic Reaction Sets, Decay, and the Preservation of Information

abstract:
  We study the ability to maintain information in a population of reacting
polymers under the influence of decay, i.e., spontaneous breakdown of large
polymers. At a certain decay rate, it becomes impossible to maintain a
significant concentration of large polymers, while it is still possible to
maintain sets of smaller polymers that can maintain the same amount of
information. We use a genetic algorithm to evolve reaction sets to generate
specific polymer distributions under the influence of decay. In these evolved
reaction sets, the beginnings of hypercycle-type structures can be observed,
which are believed to have been an important step toward the evolution of the
first living cells.


title:
Intermittency in Turbulence: Multiplicative random process in space and
  time

abstract:
  We present a simple stochastic algorithm for generating multiplicative
processes with multiscaling both in space and in time. With this algorithm we
are able to reproduce a synthetic signal with the same space and time
correlation as the one coming from shell models for turbulence and the one
coming from a turbulent velocity field in a quasi-Lagrangian reference frame.


title: Form Factor of a Quantum Graph in a Weak Magnetic Field

abstract:
  Using periodic orbit theory, we evaluate the form factor of a quantum graph
to which a very weak magnetic field is applied. The first correction to the
diagonal approximation describing the transition between the universality
classes is shown to be in agreement with Pandey and Mehta's formula of
parametric random matrix theory.


title: Asymptotic function for multi-growth surfaces using power-law noise

abstract:
  Numerical simulations are used to investigate the multiaffine exponent
$\alpha_q$ and multi-growth exponent $\beta_q$ of ballistic deposition growth
for noise obeying a power-law distribution. The simulated values of $\beta_q$
are compared with the asymptotic function $\beta_q = \frac{1}{q}$ that is
approximated from the power-law behavior of the distribution of height
differences over time. They are in good agreement for large $q$. The simulated
$\alpha_q$ is found in the range $\frac{1}{q} \leq \alpha_q \leq
\frac{2}{q+1}$. This implies that large rare events tend to break the KPZ
universality scaling-law at higher order $q$.


title:
Classical R-matrix theory of dispersionless systems: I. (1+1)-dimension
  theory

abstract:
  A systematic way of construction of (1+1)-dimensional dispersionless
integrable Hamiltonian systems is presented. The method is based on the
classical R-matrix on Poisson algebras of formal Laurent series. Results are
illustrated with the known and new (1+1)-dimensional dispersionless systems.


title: Four-wave mixing instabilities in tapered and photonic crystal fibers

abstract:
  We present an analytical study of four-wave mixing instabilities in tapered
fibers and photonic crystal fibers. Our approach avoids the use of Taylor
expansion for the linear susceptibility and the slowly-varying envelope
approximation. This allows us to describe the generation of sidebands strongly
detuned from the pump wave with simultaneous account for the entire dispersion
characteristic of a fiber, which is found to be important for describing
properly the key role of the parametric instabilities in the supercontinuum
generation in these kind of fibers.


title: Evolution and anti-evolution in a minimal stock market model

abstract:
  We present a novel microscopic stock market model consisting of a large
number of random agents modeling traders in a market. Each agent is
characterized by a set of parameters that serve to make iterated predictions of
two successive returns. The future price is determined according to the offer
and the demand of all agents. The system evolves by redistributing the capital
among the agents in each trading cycle. Without noise the dynamics of this
system is nearly regular and thereby fails to reproduce the stochastic return
fluctuations observed in real markets. However, when in each cycle a small
amount of noise is introduced we find the typical features of real financial
time series like fat-tails of the return distribution and large temporal
correlations in the volatility without significant correlations in the price
returns. Introducing the noise by an evolutionary process leads to different
scalings of the return distributions that depend on the definition of fitness.
Because our realistic model has only very few parameters, and the results
appear to be robust with respect to the noise level and the number of agents we
expect that our framework may serve as new paradigm for modeling self generated
return fluctuations in markets.

categories: nlin.AO cond-mat.stat-mech cs.MA q-fin.TR


title: Diffusive transport and self-consistent dynamics in coupled maps

abstract:
  The study of diffusion in Hamiltonian systems has been a problem of interest
for a number of years.
  In this paper we explore the influence of self-consistency on the diffusion
properties of systems described by coupled symplectic maps. Self-consistency,
i.e. the back-influence of the transported quantity on the velocity field of
the driving flow, despite of its critical importance, is usually overlooked in
the description of realistic systems, for example in plasma physics. We propose
a class of self-consistent models consisting of an ensemble of maps globally
coupled through a mean field. Depending on the kind of coupling, two different
general types of self-consistent maps are considered: maps coupled to the field
only through the phase, and fully coupled maps, i.e. through the phase and the
amplitude of the external field. The analogies and differences of the diffusion
properties of these two kinds of maps are discussed in detail.


title:
Fractality of deterministic diffusion in the nonhyperbolic climbing sine
  map

abstract:
  The nonlinear climbing sine map is a nonhyperbolic dynamical system
exhibiting both normal and anomalous diffusion under variation of a control
parameter. We show that on a suitable coarse scale this map generates an
oscillating parameter-dependent diffusion coefficient, similarly to hyperbolic
maps, whose asymptotic functional form can be understood in terms of simple
random walk approximations. On finer scales we find fractal hierarchies of
normal and anomalous diffusive regions as functions of the control parameter.
By using a Green-Kubo formula for diffusion the origin of these different
regions is systematically traced back to strong dynamical correlations.
Starting from the equations of motion of the map these correlations are
formulated in terms of fractal generalized Takagi functions obeying generalized
de Rham-type functional recursion relations. We finally analyze the measure of
the normal and anomalous diffusive regions in the parameter space showing that
in both cases it is positive, and that for normal diffusion it increases by
increasing the parameter value.


title: A Spin Glass Model of Human Logic Systems

abstract:
  In this paper, we discuss different models for human logic systems and
describe a game with nature. G\"odel`s incompleteness theorem is taken into
account to construct a model of logical networks based on axioms obtained by
symmetry breaking. These classical logic networks are then coupled using rules
that depend on whether two networks contain axioms or anti-axioms. The social
lattice of axiom based logic networks is then placed with the environment
network in a game including entropy as a cost factor. The classical logical
networks are then replaced with ``preference axioms'' to the role of fuzzy
logic.


title:
Noise Transformation in Nonlinear System with Intensity Dependent Phase
  Rotation

abstract:
  The statistical behavior of a nonlinear system described by a mapping with
phase rotation is studied. We use the Kolmogorov-Chapman equations for the
multi-time probability distribution functions for investigation of dynamics
under the external noise perturbations. We find a stationary solution in the
long-time limit as a power series around a state with complete phase
randomization ("phase mixing"). The Ornstein-Uhlenbeck and Kubo-Andersen models
of noise statistics are considered; the conditions of convergence of the power
expansions are established.