title: Spatial Patterns in Chemically and Biologically Reacting Flows

abstract:
  We present here a number of processes, inspired by concepts in Nonlinear
Dynamics such as chaotic advection and excitability, that can be useful to
understand generic behaviors in chemical or biological systems in fluid flows.
Emphasis is put on the description of observed plankton patchiness in the sea.
The linearly decaying tracer, and excitable kinetics in a chaotic flow are
mainly the models described. Finally, some warnings are given about the
difficulties in modeling discrete individuals (such as planktonic organisms) in
terms of continuous concentration fields.


title: Painlev\'e expressions for LOE, LSE and interpolating ensembles

abstract:
  We consider an ensemble which interpolates the Laguerre orthogonal ensemble
and the Laguerre symplectic ensemble. This interpolating ensemble was
introduced earlier by the author and Rains in connection with a last passage
percolation model with a symmetry condition. In this paper, we obtain a
Painelev\'e V expression for the distribution of the rightmost particle of the
interpolating ensemble. Special cases of this result yield the Painlev\'e V
expressions for the largest eigenvalues of Laguerre orthogonal ensemble and
Laguerre symplectic ensemble of finite size.


title: Portfolio Allocation to Corporate Bonds with Correlated Defaults

abstract:
  This article deals with the problem of optimal allocation of capital to
corporate bonds in fixed income portfolios when there is the possibility of
correlated defaults. Under fairly general assumptions for the distribution of
the total net assets of a set of firms we show that retaining the first few
moments of the portfolio default loss distribution gives an extremely good
approximation to the full solution of the asset allocation problem. We provide
detailed results on the convergence of the moment expansion. We also provide
explicit results for the inverse problem, i.e. for a given allocation to the
set of risky bonds, what is the average risk premium required to make the
portfolio optimal. Numerous numerical illustrations exhibit the results for
realistic portfolios and utility functions.


title:
Singular sector of the Burgers-Hopf hierarchy and deformations of
  hyperelliptic curves

abstract:
  We discuss the structure of shock singularities of the Burgers-Hopf
hierarchy. It is shown that the set of singular solutions defines a
stratification of the affine space of the flow parameters in the hierarchy. The
stratification is associated with the Birkhoff decomposition of the
Grassmannian given by the set of linear spaces spanned by the hierarchy. We
then construct integrable hierarchy on each stratum and demonstrate that it
describes a deformation of a hyperelliptic curve parametrizing the stratum. The
hierarchy is called the hidden Burgers-Hopf hierarchy, and we found the Riemann
invarint form and the hodograph solution.


title:
Blow-ups of the Toda lattices and their intersections with the Bruhat
  cells

abstract:
  We study the topology of the set of singular points (blow-ups) in the
solution of the nonperiodic Toda lattice defined on real split semisimple Lie
algebra $\mathfrak g$. The set of blow-ups is called the Painlev\'e divisor.
The isospectral manifold of the Toda lattice is compactified through the
companion embedding which maps themanifold to the flag manifold associated with
the underlying Lie algebra $\mathfrak g$. The Painlev\'e divisor is then given
by the intersections of the compactified manifold with the Bruhat cells in the
flag manifold. In this paper, we give explicit description of the topology of
the Painlev\'e divisor for the cases of all the rank two Lie algebra, $A_2,B_2,
C_2, G_2$, and $A_3$ type. The results are obtained by using the Mumford system
and the limit matrices introduced originally for the periodic Toda lattice. We
also give a Lie theoretic description of the Painlev\'e divisor of codimension
one case, and propose several conjecturesfor the general case.


title: Form factor for a family of quantum graphs: An expansion to third order

abstract:
  For certain types of quantum graphs we show that the random-matrix form
factor can be recovered to at least third order in the scaled time $\tau$ from
periodic-orbit theory. We consider the contributions from pairs of periodic
orbits represented by diagrams with up to two self-intersections connected by
up to four arcs and explain why all other diagrams are expected to give
higher-order corrections only.
  For a large family of graphs with ergodic classical dynamics the diagrams
that exist in the absence of time-reversal symmetry sum to zero. The mechanism
for this cancellation is rather general which suggests that it may also apply
at higher-orders in the expansion. This expectation is in full agreement with
the fact that in this case the linear-$\tau$ contribution, the diagonal
approximation, already reproduces the random-matrix form factor for $\tau<1$.
For systems with time-reversal symmetry there are more diagrams which
contribute at third order. We sum these contributions for quantum graphs with
uniformly hyperbolic dynamics, obtaining $+2\tau^{3}$, in agreement with
random-matrix theory. As in the previous calculation of the leading-order
correction to the diagonal approximation we find that the third order
contribution can be attributed to exceptional orbits representing the
intersection of diagram classes.


title:
Two-dimensional solitary pulses in driven diffractive-diffusive complex
  Ginzburg-Landau equations

abstract:
  Two models of driven optical cavities, based on two-dimensional
Ginzburg-Landau equations, are introduced. The models include loss, the Kerr
nonlinearity, diffraction in one transverse direction, and a combination of
diffusion and dispersion in the other one (which is, actually, a temporal
direction). Each model is driven either parametrically or directly by an
external field. By means of direct simulations, stable completely localized
pulses are found (in the directly driven model, they are built on top of a
nonzero flat background). These solitary pulses correspond to spatio-temporal
solitons in the optical cavities. Basic results are presented in a compact form
as stability regions for the solitons in a full three-dimensional parameter
space of either model. The stability region is bounded by two surfaces; beyond
the left one, any two-dimensional (2D) pulse decays to zero, while quasi-1D
pulses, representing spatial solitons in the optical cavity, are found beyond
the right boundary. The spatial solitons are found to be stable both inside the
stability region of the 2D pulses (hence, bistability takes place in this
region) and beyond the right boundary of this region (although they are not
stable everywhere). Unlike the spatial solitons, their quasi-1D counterparts in
the form of purely temporal solitons are always subject to modulational
instability, which splits them into an array of 2D pulses, that further
coalesce into two final pulses. A uniform nonzero state in the parametrically
driven model is also modulationally unstable, which leads to formation of many
2D pulses that subsequently merge into few ones.


title: Superfast front propagation in reactive systems with anomalous diffusion

abstract:
  We study a reaction diffusion system where we consider a non-gaussian process
instead of a standard diffusion. If the process increments follow a probability
distribution with tails approaching to zero faster than a power law, the usual
qualitative behaviours of the standard reaction diffusion system, i.e.,
exponential tails for the reacting field and a constant front speed, are
recovered. On the contrary if the process has power law tails, also the
reacting field shows power law tail and the front speed increases exponentially
with time. The comparison with other reaction-transport systems which exhibit
anomalous diffusion shows that, not only the presence of anomalous diffusion,
but also the detailed mechanism, is relevant for the front propagation.


title: Mating Instabilities Lead to Sympatric Speciation

abstract:
  One of the most challenging issues of evolutionary biology concerns
speciation, the emergence of new species from an initial one. The huge amount
of species found in nature demands a simple and robust mechanism. Yet, no
consensus has been reached concerning a reasonable disruptive selection
mechanism that prevents mixing genes among the emerging species, especially
when they live in sympatry. Usually it is assumed that females select males
according to their displaying traits, but males perform no selection on female
traits. However, recent experimental evidence accumulates towards the existence
of male choice. Here we propose a robust mechanism for sympatric speciation,
based on the assumption that sexual selection operates in two directions:
selection of males by females and of females by males. Complex mating
instabilities emerge, creating differential fitness depending on the
individuals displaying traits and preferences. When a secondary sexual trait is
introduced in a population, due to mutations, the activation of previously
neutral genes or due to a different perception of already existent displaying
traits, sympatric speciation may result (together with a species recognition
system) from a competitive exclusion principle. We suggest that potential
candidates to test our theory could be yeasts.


title:
The Onset of Synchronization in Systems of Globally Coupled Chaotic and
  Periodic Oscillators

abstract:
  A general stability analysis is presented for the determination of the
transition from incoherent to coherent behavior in an ensemble of globally
coupled, heterogeneous, continuous-time dynamical systems. The formalism allows
for the simultaneous presence of ensemble members exhibiting chaotic and
periodic behavior, and, in a special case, yields the Kuramoto model for
globally coupled periodic oscillators described by a phase. Numerical
experiments using different types of ensembles of Lorenz equations with a
distribution of parameters are presented.


title:
On a q-difference Painlev\'e III equation: I. Derivation, symmetry and
  Riccati type solutions

abstract:
  A q-difference analogue of the Painlev\'e III equation is considered. Its
derivations, affine Weyl group symmetry, and two kinds of special function type
solutions are discussed.


title:
Effect of asynchronicity on the universal behaviour of coupled map
  lattices

abstract:
  We investigate the spatiotemporal dynamics of coupled circle map lattices,
evolving under synchronous (parallel) updating on one hand and asynchronous
(random) updating rules on the other. Synchronous evolution of extended
spatiotemporal systems, such as coupled circle map lattices, commonly yields
multiple co-existing attractors, giving rise to phenomena strongly dependent on
the initial lattice. By marked contrast numerical evidence here strongly
indicates that asynchronous evolution eliminates most of the attractor states
arising from special sets of initial conditions in synchronous systems, and
tends to yield more global attractors. Thus the phenomenology arising from
asynchronous evolution is more generic and robust in that it is obtained from
many different classes of initial states. Further we show that in parameter
regions where both asynchronous and synchronous evolution yield spatio-temporal
intermittency, asynchronicity leads to better scaling behaviour.


title: Complexity in some Physical Systems

abstract:
  The LMC-complexity introduced by Lopez-Ruiz, Mancini and Calbet [Phys. Lett.
A 209, 321-326 (1995)] is calculated for different physical situations: one
instance of classical statistical mechanics, normal and exponential
distributions, and a simplified laser model. We stand out the specific value of
the population inversion for which the laser presents maximun complexity.


title: Tendency to Maximum Complexity in a Non-Equilibrium Isolated System

abstract:
  The time evolution equations of a simplified isolated ideal gas, the
"tetrahe- dral" gas, are derived. The dynamical behavior of the LMC complexity
[R. Lopez-Ruiz, H. L. Mancini, and X. Calbet, Phys. Lett. A 209, 321 (1995)] is
studied in this system. In general, it is shown that the complexity remains
within the bounds of minimum and maximum complexity. We find that there are
certain restrictions when the isolated "tetrahedral" gas evolves towards
equilibrium. In addition to the well-known increase in entropy, the quantity
called disequilibrium decreases monotonically with time. Furthermore, the
trajectories of the system in phase space approach the maximum complexity.


title: A New Integrable Equation with Peakon Solutions

abstract:
  We consider a new partial differential equation, of a similar form to the
Camassa-Holm shallow water wave equation, which was recently obtained by
Degasperis and Procesi using the method of asymptotic integrability. We prove
the exact integrability of the new equation by constructing its Lax pair, and
we explain its connection with a negative flow in the Kaup-Kupershmidt
hierarchy via a reciprocal transformation. The infinite sequence of conserved
quantities is derived together with a proposed bi-Hamiltonian structure. The
equation admits exact solutions in the form of a superposition of
multi-peakons, and we describe the integrable finite-dimensional peakon
dynamics and compare it with the analogous results for Camassa-Holm peakons.


title: Integration of the mKdV hierarchy with integral type of source

abstract:
  We investigate the mKdV hierarchy with integral type of source (mKdVHWS),
which consist of the reduced AKNS eigenvalue problem with $r=q$ and the mKdV
hierarchy with extra term of the integration of square eigenfunction. First we
propose a method to find the explicit evolution equation for eigenfunction of
the auxiliary linear problems of the mKdVHWS. Then we determine the evolution
equations of scattering data corresponding to the mKdVHWS which allow us to
solve the equation in the mKdVHWS by inverse scattering transformation.


title: Self-averaging in time reversal for the parabolic wave equation

abstract:
  We analyze the self-averaging properties of time-reversed solutions of the
paraxial wave equation with random coefficients, which we take to be Markovian
in the direction of propagation. This allows us to construct an approximate
martingale for the phase space Wigner transform of two wave fields. Using a
priori $L^2$-bounds available in the time-reversal setting, we prove that the
Wigner transform in the high frequency limit converges in probability to its
deterministic limit, which is the solution of a transport equation.


title:
Dynamical model of sequential spatial memory: winnerless competition of
  patterns

abstract:
  We introduce a new biologically-motivated model of sequential spatial memory
which is based on the principle of winnerless competition (WLC). We implement
this mechanism in a two-layer neural network structure and present the learning
dynamics which leads to the formation of a WLC network. After learning, the
system is capable of associative retrieval of pre-recorded sequences of spatial
patterns.


title: The Limit Cycles of Lienard Equations in the Strongly Non-Linear Regime

abstract:
  Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an
even function, are studied in the strongly nonlinear regime
($\epsilon\to\infty$). A method for obtaining the number, amplitude and loci of
the limit cycles of these equations is derived. The accuracy of this method is
checked in several examples. Lins-Melo-Pugh conjecture for the polynomial case
is true in this regime.


title: Bifurcation Curves of Limit Cycles in some Lienard Systems

abstract:
  Lienard systems of the form $\ddot{x}+\epsilon f(x)\dot{x}+x=0$, with f(x) an
even continous function, are considered. The bifurcation curves of limit cycles
are calculated exactly in the weak ($\epsilon\to 0$) and in the strongly
($\epsilon\to\infty$) nonlinear regime in some examples. The number of limit
cycles does not increase when $\epsilon$ increases from zero to infinity in all
the cases analyzed.


title: Chaotic Synchronization Via Minimum Information Transmission

abstract:
  Chaotic synchronization is generally extremely sensitive to the presence of
noise and other inference in the channel. Is this sensitivity a fundamental
property of chaotic synchronization or is it related to the choice of
synchronization method and can be suppressed by a modification of the method?
If the answer is positive, then what are the relationships between the
properties of a dynamical system and the level of noise at which the
suppression of this sensitivity is still possible? What are particular methods
to achieve synchronization that is stable to the presence of noise? In this
paper we present the analysis of this issue from the standpoint of the
information theory. The fundamental reason of this sensitivity is the fact that
chaotic signal contains information and that requires a certain minimal
threshold signal-to-noise ratio for this information to be transmitted. Only in
this case high quality synchronization is achievable and only if the required
information is transmitted (coded) optimally. Otherwise the threshold level can
be much higher.


title: Initial-Boundary Value Problems for Linear and Soliton PDEs

abstract:
  Evolution PDEs for dispersive waves are considered in both linear and
nonlinear integrable cases, and initial-boundary value problems associated with
them are formulated in spectral space. A method of solution is presented, which
is based on the elimination of the unknown boundary values by proper
restrictions of the functional space and of the spectral variable complex
domain. Illustrative examples include the linear Schroedinger equation on
compact and semicompact n-dimensional domains and the nonlinear Schroedinger
equation on the semiline.


title:
Resonant flux motion and I-V -characteristics in frustrated Josephson
  junctions

abstract:
  We describe the dynamics of fluxons moving in a frustrated Josephson junction
with p, d, and f-wave symmetry and calculate the I-V characteristics. The
behavior of fluxons is quite distinct in the long and short length junction
limit. For long junctions the intrinsic flux is bound at the center and the
moving integer fluxon or antifluxon interacts with it only when it approaches
the junction's center. For small junctions the intrinsic flux can move as a
bunched type fluxon introducing additional steps in the I-V characteristics.
Possible realization in quantum computation is presented.


title: Analysis of the chaos dynamics in(Xn,Xn+1)plane

abstract:
  in the last decade, studies of chaotic system are more often used for
classical choatic system than for quantum chaotic system, there are many ways
of observing the chaotic system such us analyzing the frequency with Fourier
transform or analyzing initial condition distance with Liapunov Exponent, this
paper explains dynamic chaotic process by observing trajectory of dynamic
system in (Xn,Xn+1)


title: A Statistical Measure of Complexity

abstract:
  A measure of complexity based on a probabilistic description of physical
systems is proposed. This measure incorporates the main features of the
intuitive notion of such a magnitude. It can be applied to many physical
situations and to different descriptions of a given system. Moreover, the
calculation of its value does not require a considerable computational effort
in many cases of physical interest.


title: A Binary Approach to the Lorenz Model

abstract:
  The order of orbit generation in one-dimensional Lorenz-like maps is
presented within a two letter symbolics scheme. This order is derived from the
natural order of a set of fractions associated to the binary sequences. Its
relation to the universal sequence of unimodal maps is explained.


title: Driven response of time delay coupled limit cycle oscillators

abstract:
  We study the periodic forced response of a system of two limit cycle
oscillators that interact with each other via a time delayed coupling. Detailed
bifurcation diagrams in the parameter space of the forcing amplitude and
forcing frequency are obtained for various interesting limits using numerical
and analytical means. In particular, the effects of the coupling strength, the
natural frequency spread of the two oscillators and the time delay parameter on
the size and nature of the entrainment domain are delineated. The system is
found to display a nonlinear response on certain critical contours in the space
of the coupling strength and time delay. Time delay offers a novel tuning knob
for controlling the system response over a wide range of frequencies and this
may have important practical applications.


title: Integrable deformations of systems on graphs with loops

abstract:
  We find necessary and sufficient conditions for an operator of the fourth
order on a graph with loops to admit the (L,A,B)-triple Krichever-Novikov
deformation of one energy level.


title:
Quantum relativistic Toda chain at root of unity: isospectrality,
  modified Q-operator and functional Bethe ansatz

abstract:
  We investigate an N-state spin model called quantum relativistic Toda chain
and based on the unitary finite dimensional representations of the Weyl algebra
with q being N-th primitive root of unity. Parameters of the finite dimensional
representation of the local Weyl algebra form the classical discrete integrable
system. Nontrivial dynamics of the classical counterpart corresponds to
isospectral transformations of the spin system. Similarity operators are
constructed with the help of modified Baxter's Q-operators. The classical
counterpart of the modified Q-operator for the initial homogeneous spin chain
is a Baecklund transformation. This transformation creates an extra Hirota-type
soliton in a parameterization of the chain structure. Special choice of values
of solitonic amplitudes yields a degeneration of spin eigenstates, leading to
the quantum separation of variables, or the functional Bethe ansatz. A
projector to the separated eigenstates is constructed explicitly as a product
of modified Q-operators.


title:
Liouville integrability of the finite dimensional Hamiltonian systems
  derived from principal chiral field

abstract:
  For finite dimensional Hamiltonian systems derived from 1+1 dimensional
integrable systems, if they have Lax representations, then the Lax operator
creates a set of conserved integrals. When these conserved integrals are in
involution, it is believed quite popularly that there will be enough
functionally independent ones among them to guarantee the Liouville
integrability of the Hamiltonian systems, at least for those derived from
physical problems. In this paper, we give a counterexample based on the U(2)
principal chiral field. It is proved that the finite dimensional Hamiltonian
systems derived from the U(2) principal chiral field are Liouville integrable.
Moreover, their Lax operator gives a set of involutive conserved integrals, but
they are not enough to guarantee the integrability of the Hamiltonian systems.


title: Nonlinear Interaction of Transversal Modes in a CO2 Laser

abstract:
  We show the possibility of achieving experimentally a Takens-Bogdanov
bifurcation for the nonlinear interaction of two transverse modes ($l = \pm 1$)
in a $CO_2$ laser. The system has a basic O(2) symmetry which is perturbed by
some symmetry-breaking effects that still preserve the $Z_2$ symmetry. The
pattern dynamics near this codimension two bifurcation under such symmetries is
described. This dynamics changes drastically when the laser properties are
modified.


title:
Remarks on NonHamiltonian Statistical Mechanics: Lyapunov Exponents and
  Phase-Space Dimensionality Loss

abstract:
  The dissipation associated with nonequilibrium flow processes is reflected by
the formation of strange attractor distributions in phase space. The
information dimension of these attractors is less than that of the equilibrium
phase space, corresponding to the extreme rarity of nonequilibrium states. Here
we take advantage of a simple model for heat conduction to demonstrate that the
nonequilibrium dimensionality loss can definitely exceed the number of
phase-space dimensions required to thermostat an otherwise Hamiltonian system.


title:
Statistics and Characteristics of Spatio-Temporally Rare Intense Events
  in Complex Ginzburg-Landau Models

abstract:
  We study the statistics and characteristics of rare intense events in two
types of two dimensional Complex Ginzburg-Landau (CGL) equation based models.
Our numerical simulations show finite amplitude collapse-like solutions which
approach the infinite amplitude solutions of the nonlinear Schr\"{o}dinger
(NLS) equation in an appropriate parameter regime. We also determine the
probability distribution function (PDF) of the amplitude of the CGL solutions,
which is found to be approximately described by a stretched exponential
distribution, $P(|A|) \approx e^{-|A|^\eta}$, where $\eta < 1$. This
non-Gaussian PDF is explained by the nonlinear characteristics of individual
bursts combined with the statistics of bursts. Our results suggest a general
picture in which an incoherent background of weakly interacting waves,
occasionally, `by chance', initiates intense, coherent, self-reinforcing,
highly nonlinear events.


title: Transition between Two Oscillation Modes

abstract:
  A model for the symmetric coupling of two self-oscillators is presented. The
nonlinearities cause the system to vibrate in two modes of different
symmetries. The transition between these two regimes of oscillation can occur
by two different scenarios. This might model the release of vortices behind
circular cylinders with a possible transition from a symmetric to an
antisymmetric Benard-von Karman vortex street.


title:
Locating Pollicott-Ruelle resonances in chaotic dynamical systems: A
  class of numerical schemes

abstract:
  A class of numerical methods to determine Pollicott-Ruelle resonances in
chaotic dynamical systems is proposed. This is achieved by relating some
existing procedures which make use of Pade approximants and interpolating
exponentials to both the memory function techniques used in the theory of
relaxation and the filter diagonalization method used in the harmonic inversion
of time correlation functions. This relationship leads to a theoretical
framework in which all these methods become equivalent and which allows for new
and improved numerical schemes.


title:
Internal modes of sine-Gordon solitons in the presence of spatiotemporal
  perturbations

abstract:
  We investigate the dynamics of the sine-Gordon solitons perturbed by
spatiotemporal external forces. We prove the existence of internal (shape)
modes of sine-Gordon solitons when they are in the presence of inhomogeneous
space-dependent external forces, provided some conditions (for these forces)
hold. Additional periodic time-dependent forces can sustain oscillations of the
soliton width. We show that, in some cases, the internal mode even can become
unstable, causing the soliton to decay in an antisoliton and two solitons. In
general, in the presence of spatiotemporal forces the soliton behaves as a
deformable (non-rigid) object. A soliton moving in an array of inhomogeneities
can also present sustained oscillations of its width. There are very important
phenomena (like the soliton-antisoliton collisions) where the existence of
internal modes plays a crucial role.


title:
Solitons in Triangular and Honeycomb Dynamical Lattices with the Cubic
  Nonlinearity

abstract:
  We study the existence and stability of localized states in the discrete
nonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-square
lattices. The model includes both the nearest-neighbor and long-range
interactions. For the fundamental strongly localized soliton, the results
depend on the coordination number, i.e., on the particular type of the lattice.
The long-range interactions additionally destabilize the discrete soliton, or
make it more stable, if the sign of the interaction is, respectively, the same
as or opposite to the sign of the short-range interaction. We also explore more
complicated solutions, such as twisted localized modes (TLM's) and solutions
carrying multiple topological charge (vortices) that are specific to the
triangular and honeycomb lattices. In the cases when such vortices are
unstable, direct simulations demonstrate that they turn into zero-vorticity
fundamental solitons.


title:
Renormalization group in the statistical theory of turbulence: Two-loop
  approximation

abstract:
  The field theoretic renormalization group is applied to the stochastic
Navier--Stokes equation that describes fully developed fluid turbulence. The
complete two-loop calculation of the renormalization constant, the beta
function and the fixed point is performed. The ultraviolet correction exponent,
the Kolmogorov constant and the inertial-range skewness factor are derived to
second order of the $\epsilon$ expansion.


title: Linearisable hierarchy of evolution equations in (1+1) dimensions

abstract:
  In our article "A tree of linearisable second-order evolution equations by
generalised hodograph transformations" [J. Nonlin. Math. Phys. {\bf 8} (2001),
342-362] we presented a tree of linearisable (C-integrable) second-order
evolution equations in (1+1) dimensions. Expanding this result we report here
the complete set of recursion operators for this tree and present several
linearisable (C-integrable) hierarchies in (1+1) dimensions.


title: Julia set describes quantum tunneling in the presence of chaos

abstract:
  We find that characteristics of quantum tunneling in the presence of chaos
can be regarded as a manifestation of the Julia set of the complex dynamical
system. Several numerical evidences for the standard map together with a
rigorous statement for the H\'enon map are presented demonstrating that the
complex classical paths which contribute to the semiclassical propagator are
dense in the Julia set. Chaotic tunneling can thus be characterized by the
transitivity of the dynamics and high density of the trajectories on the Julia
set.


title:
A note on the extension of the polar decomposition for the
  multidimensional Burgers equation

abstract:
  It is shown that the generalizations to more than one space dimension of the
pole decomposition for the Burgers equation with finite viscosity and no force
are of the form u = -2 viscosity grad log P, where the P's are explicitly known
algebraic (or trigonometric) polynomials in the space variables with polynomial
(or exponential) dependence on time. Such solutions have polar singularities on
complex algebraic varieties.


title:
Breather initial profiles in chains of weakly coupled anharmonic
  oscillators

abstract:
  A systematic correlation between the initial profile of discrete breathers
and their frequency is described. The context is that of a very weakly
harmonically coupled chain of softly anharmonic oscillators. The results are
structurally stable, that is, robust under changes of the on-site potential and
are illustrated numerically for several standard choices. A precise genericity
theorem for the results is proved.


title:
Deformations of the Monge/Riemann hierarchy and approximately integrable
  systems

abstract:
  Dispersive deformations of the Monge equation u_u=uu_x are studied using
ideas originating from topological quantum field theory and the deformation
quantization programme. It is shown that, to a high-order, the symmetries of
the Monge equation may also be appropriately deformed, and that, if they exist
at all orders, they are uniquely determined by the original deformation. This
leads to either a new class of integrable systems or to a rigorous notion of an
approximate integrable system. Quasi-Miura transformations are also constructed
for such deformed equations.


title:
Renormalization group, operator product expansion and anomalous scaling
  in models of passive turbulent advection

abstract:
  The field theoretic renormalization group is applied to Kraichnan's model of
a passive scalar quantity advected by the Gaussian velocity field with the pair
correlation function $\propto\delta(t-t')/k^{d+\epsilon}$. Inertial-range
anomalous scaling for the structure functions and various pair correlators is
established as a consequence of the existence in the corresponding operator
product expansions of ``dangerous'' composite operators (powers of the local
dissipation rate), whose {\it negative} critical dimensions determine anomalous
exponents. The latter are calculated to order $\epsilon^3$ of the $\epsilon$
expansion (three-loop approximation).


title:
Exact Solutions and hypothesis on Phase Transition in the
  Polyelectrolite Model of DNA

abstract:
  The two-dimensional generalization of the Polyelectolite model of DNA is
proposed. It is reduced to the boundary problem for nonlinear completely
integrable equation $\sh$-Gordon. In the linearisible version the exact
solution is constructed and its asymptotic is found. The soliton solutions of
nonlinear equation are calculated that allowed tells about the possibility of
the structural phase transition in the considered system (DNA+polyelectrolite)
on the temperature.


title: Resonance-Assisted Tunneling

abstract:
  We present evidence that tunneling processes in near-integrable systems are
enhanced due to the manifestation of nonlinear resonances and their respective
island chains in phase space. A semiclassical description of this
"resonance-assisted" mechanism is given, which is based on a local perturbative
description of the dynamics in the vicinity of the resonances. As underlying
picture, we obtain that the quantum state is coupled, via a succession of
classically forbidden transitions across nonlinear resonances, to high
excitations within the well, from where tunneling occurs with a rather large
rate. The connection between this description and the complex classical
structure of the underlying integrable dynamics is furthermore studied, giving
ground to the general coherence of the description as well as guidelines for
the identification of the dominant tunneling paths. The validity of this
mechanism is demonstrated within the kicked Harper model, where good agreement
between quantum and semiclassical (resonance-assisted) tunneling rates is
found.


title:
Scaling and the prediction of energy spectra in decaying hydrodynamic
  turbulence

abstract:
  Few rigorous results are derived for fully developed turbulence. By applying
the scaling properties of the Navier-Stokes equation we have derived a relation
for the energy spectrum valid for unforced or decaying isotropic turbulence. We
find the existence of a scaling function $\psi$. The energy spectrum can at any
time by a suitable rescaling be mapped onto this function. This indicates that
the initial (primordial) energy spectrum is in principle retained in the energy
spectrum observed at any later time, and the principle of permanence of large
eddies is derived. The result can be seen as a restoration of the determinism
of the Navier-Stokes equation in the mean. We compare our results with a
windtunnel experiment and find good agreement.


title: Decomposition of higher order equations of Monge-Ampere type

abstract:
  Given a function f(x, t), its fourth (symmetric) differential is a quartic
form in dx, dt. It is well-known that any quartic form in two variables can be
represented as a sum of three 4th powers of linear forms. The particular case
of two 4th powers is characterized by the vanishing of a catalecticant
determinant, which is a fourth order PDE for the function f. We demonstrate
that this PDE decouples in a direct sum of two Monge-Ampere equations. A higher
order generalization of this decomposition is also proposed.


title:
Effect of Phase Shift in Shape Changing Collision of Solitons in Coupled
  Nonlinear Schroedinger Equations

abstract:
  Soliton interactions in systems modelled by coupled nonlinear Schroedinger
(CNLS) equations and encountered in phenomena such as wave propagation in
optical fibers and photorefractive media possess unusual features : shape
changing intensity redistributions, amplitude dependent phase shifts and
relative separation distances. We demonstrate these properties in the case of
integrable 2-CNLS equations. As a simple example, we consider the stationary
two-soliton solution which is equivalent to the so-called partially coherent
soliton (PCS) solution discussed much in the recent literature.


title: Clustering dynamics of Lagrangian tracers in free-surface flows

abstract:
  We study the formation of clusters of passive Lagrangian tracers in a
non-smooth turbulent flow in a flat free-slip surface as a model for particle
dynamics on free surfaces. Single particle and pair dispersion show different
behavior for short and large times: on short times particles cluster
exponentially rapidly until patches of the size of the divergence correlation
length are depleted; on larger times the pair dispersion is dominated by almost
ballistic hopping between clusters. We also find that the distribution of
particle density is close to algebraic and can trace this back to the
exponential distribution of the divergence field of the surface flow.


title:
Fluctuation formula and non-Gaussian distribution of the entropy
  production rate in the thermostated Lorentz gas

abstract:
  In this paper we numerically examine the connection of the Gallavotti-Cohen
fluctuation formula and the functional form of the corresponding probability
density function in the field driven Lorentz gas thermostated by the Gaussian
isokinetic thermostat. We analyze the moments of the entropy production rate
fluctuations and show that all the central moments of the averaged fluctuations
exhibit power law dependence on the length of the averaging time interval,
indicating that this density deviates from a Gaussian. Furthermore the obtained
exponents are found to obey a special pairing rule showing that the
corresponding probability density function can not be scaled in the averaging
time interval.


title: Closure of two dimensional turbulence: the role of pressure gradients

abstract:
  Inverse energy cascade regime of two dimensional turbulence is investigated
by means of high resolution numerical simulations. Numerical computations of
conditional averages of transverse pressure gradient increments are found to be
compatible with a recently proposed self-consistent Gaussian model. An
analogous low order closure model for the longitudinal pressure gradient is
proposed and its validity is numerically examined. In this case numerical
evidence for the presence of higher order terms in the closure is found. The
fundamental role of conditional statistics between longitudinal and transverse
components is highlighted.


title:
Darboux Covariant Equations of von Neumann Type and their
  Generalizations

abstract:
  Lax pairs with operator valued coefficients, which are explicitly connected
by means of an additional condition, are considered. This condition is proved
to be covariant with respect to the Darboux transformation of a general form.
Nonlinear equations arising from the compatibility condition of the Lax pairs
in the matrix case include, in particular, Nahm equations, Volterra,
Bogoyavlenskii and Toda lattices. The examples of another one-, two- and
multi-field lattice equations are also presented.


title: Stabilization of Hydrodynamic Flows by Small Viscosity Variations

abstract:
  Motivated by the large effect of turbulent drag reduction by minute
concentrations of polymers we study the effects of a weakly space-dependent
viscosity on the stability of hydrodynamic flows. In a recent Letter [Phys.
Rev. Lett. {\bf 87}, 174501, (2001)] we exposed the crucial role played by a
localized region where the energy of fluctuations is produced by interactions
with the mean flow (the "critical layer"). We showed that a layer of weakly
space-dependent viscosity placed near the critical layer can have a very large
stabilizing effect on hydrodynamic fluctuations, retarding significantly the
onset of turbulence. In this paper we extend these observation in two
directions: first we show that the strong stabilization of the primary
instability is also obtained when the viscosity profile is realistic (inferred
from simulations of turbulent flows with a small concentration of polymers).
Second, we analyze the secondary instability (around the time-dependent primary
instability) and find similar strong stabilization. Since the secondary
instability develops around a time-dependent solution and is three-dimensional,
this brings us closer to the turbulent case. We reiterate that the large effect
is {\em not} due to a modified dissipation (as is assumed in some theories of
drag reduction), but due to reduced energy intake from the mean flow to the
fluctuations. We propose that similar physics act in turbulent drag reduction.


title: On a q-difference Painlev\'e III equation: II. Rational solutions

abstract:
  Rational solutions for a $q$-difference analogue of the Painlev\'e III
equation are considered. A Determinant formula of Jacobi-Trudi type for the
solutions is constructed.


title:
Four dimensional Lie symmetry algebras and fourth order ordinary
  differential equations

abstract:
  Realizations of four dimensional Lie algebras as vector fields in the plane
are explicitly constructed. Fourth order ordinary differential equations which
admit such Lie symmetry algebras are derived. The route to their integration is
described.


title: Comment on "Ehrenfest times for classically chaotic systems"

abstract:
  In a recent Phys. Rev. E Rapid Communication, the authors, Silvestrov and
Beenakker, introduce a way to lengthen the Ehrenfest time, $\tau$, for fully
chaotic systems. We disagree with several statements made in their paper, and
address the following points essential to their conclusions: 1) it is not true
that all semiclassical approximations for chaotic systems fail at a so-called
`logtime', $\tau \propto - \ln (\hbar)$, differing only by a numerical
coefficient; and 2) the limitation of the semiclassical approximation as
expressed in the authors' Eq. (8) is not limited by their argument leading to
Eq. (12).


title:
Extended discrete KP hierarchy and its reductions from a geometric
  viewpoint

abstract:
  We interpret the recently suggested extended discrete KP (Toda lattice)
hierarchy from a geometrical point of view. We show that the latter corresponds
to the union of invariant submanifolds $S_0^n$ of the system which is a chain
of infinitely many copies of Darboux-KP hierarchy, while the intersections
$S_0^n\cap S_{l-1}^{ln-r}$ yields a number of reductions to $l$-field lattices.


title:
Emergence of cooperation, network formation and selection of
  interactions

abstract:
  We propose a model of emergence of cooperation in evolutionary games that
high- lights the role of network formation and effect of network structure. In
line with empirical data, the model proposes a mechanism that explains the
persistence of heterogeneous types (heterogeneity in rules for changing
behavior) within a population, and in particular the sustainability of altruism
(presence of unconditional cooperators) even in case of strong social dilemma.
This explanation constitutes an alternative to the choice and refusal mechanism
that is often presented as an explanation for cooperation on dynamic networks.
We also exhibit a minimal set of strategies for emergence of cooperation
(altruists, reciprocators and selfish agents) and sketch a two steps scenario
for this emergence. We adopt an hybrid methodology with both analytical and
computational insights.


title: A New Kind of Science?

abstract:
  Book Review for: "A New Kind of Science", by Stephen Wolfram (Wolfram Media,
Inc. Champaign IL 2002).


title:
A Study on the Relevance of Information in Discriminative and
  Non-Discriminative Media

abstract:
  In this paper we compare the relevance of information obtained from
"discriminative" media and from "non-discriminative" media. Discriminative
media are the ones which accumulate and deliver information using a heuristic
selection of it. This can be made by humans, or by artificial intelligent
systems, exhibiting some form of "knowledge". Non-discriminative media just
collect and return information without any distinction. This can also be made
by humans or by artificial systems, but there is no "knowledge" involved in the
process. We ranked the words occurring in an edited electronic publication
specialized in complex systems research, and we found that they approximate a
modified Zipf distribution. We compared occurrences of representative words
from the distribution with the occurrences in non-discriminative media. We
found that a non-discriminative medium (Google) has a higher variance from of
our original distribution than a semi-discriminative one (NEC Research Index),
even when both appear to have their own modified Zipf distribution. We conclude
that discriminative media have a higher efficiency rating, at least in the area
in which they specialize, than non-discriminative media. Using the same search
method, the discriminative media should deliver more relevant information. This
relevancy also depends on the skills of the user, but non-discriminative media
are more sensitive to poor searching skills, as there is a higher probability
of delivering irrelevant information. This leads us to suggest the
incorporation of intelligent classifications in different media (such as the
ones suggested by the Semantic Web project), in order to increase the relevance
of the delivered information.


title:
Drastic facilitation of the onset of global chaos in a periodically
  driven Hamiltonian system due to an extremum in the dependence of
  eigenfrequency on energy

abstract:
  The Chirikov resonance-overlap criterion predicts the onset of global chaos
if nonlinear resonances overlap in energy, which is conventionally assumed to
require a non-small magnitude of perturbation. We show that, for a
time-periodic perturbation, the onset of global chaos may occur at unusually
{\it small} magnitudes of perturbation if the unperturbed system possesses more
than one separatrix. The relevant scenario is the combination of the overlap in
the phase space between resonances of the same order and their overlap in
energy with chaotic layers associated with separatrices of the unperturbed
system. One of the most important manifestations of this effect is a drastic
increase of the energy range involved into the unbounded chaotic transport in
spatially periodic system driven by a rather {\it weak} time-periodic force,
provided the driving frequency approaches the extremal eigenfrequency or its
harmonics. We develop the asymptotic theory and verify it in simulations.


title:
Numerical studies on chaoticity of a classical hard-wall billiard with
  openings

abstract:
  Using fractal analysis, we investigate how the size of openings affects the
chaotic behavior of a classical closed billiard when two openings are made on
the boundary of the billiard. This kind of open billiards retains chaotic
properties of original closed billiards when openings are small compared to the
size of the billiard. We calculate the fractal dimension using an
one-dimensional subset of all possible initial conditions that will produce
trajectories of a particle injected from an opening for the billiard, and then
observe how the opening size can change the classical chaotic properties of
this open billiard.


title:
Jacobi's last multiplier and the complete symmetry group of the
  Euler-Poinsot system

abstract:
  The symmetry approach to the determination of Jacobi's last multiplier is
inverted to provide a source of additional symmetries for the Euler-Poinsot
system. These additional symmetries are nonlocal. They provide the symmetries
for the representation of the complete symmetry group of the system.


title: Targeting and control in simple dynamical systems

abstract:
  A simple algorithm is described to target any desired operation point for
simple one-dimensional and two-dimensional dynamical systems. What makes the
algorithm unique is the fact that it targets any desired point, not merely a
stable/unstable fixed point of the dynamics. It is shown how the method may be
applied to the logistic map and the H\'enon map. Generalisations to the case of
n-dimensional dynamical systems are discussed.


title:
Riddled-like Basin in Two-Dimensional Map for Bouncing Motion of an
  Inelastic Particle on a Vibrating Board

abstract:
  Motivated by bouncing motion of an inelastic particle on a vibrating board, a
simple two-dimensional map is constructed and its behavior is studied
numerically. In addition to the typical route to chaos through a periodic
doubling bifurcation, we found peculiar behavior in the parameter region where
two stable periodic attractors coexist. A typical orbit in the region goes
through chaotic motion for an extended transient period before it converges
into one of the two periodic attractors. The basin structure in this parameter
region is almost riddling and the fractal dimension of the basin boundary is
close to two, {\it i.e.}, the dimension of the phase space.


title:
Stable three-dimensional spinning optical solitons supported by
  competing quadratic and cubic nonlinearities

abstract:
  We show that the quadratic interaction of fundamental and second harmonics in
a bulk dispersive medium, combined with self-defocusing cubic nonlinearity,
give rise to completely localized spatiotemporal solitons (vortex tori) with
vorticity s=1. There is no threshold necessary for the existence of these
solitons. They are found to be stable against small perturbations if their
energy exceeds a certain critical value, so that the stability domain occupies
about 10% of the existence region of the solitons. We also demonstrate that the
s=1 solitons are stable against very strong perturbations initially added to
them. However, on the contrary to spatial vortex solitons in the same model,
the spatiotemporal solitons with s=2 are never stable.


title: Non-Equilibrium Statistical Mechanics of Classical and Quantum Systems

abstract:
  We study the statistical mechanics of classical and quantum systems in
non-equilibrium steady states. Emphasis is placed on systems in strong thermal
gradients. Various measures and functional forms of observables are presented.
The quantum problem is set up using random matrix techniques, which allows for
the construction of the master equation. Special solutions are discussed.


title: Computing Riemann Theta Functions

abstract:
  The Riemann theta function is a complex-valued function of g complex
variables. It appears in the construction of many (quasi-) periodic solutions
of various equations of mathematical physics. In this paper, algorithms for its
computation are given. First, a formula is derived allowing the pointwise
approximation of Riemann theta functions, with arbitrary, user-specified
precision. This formula is used to construct a uniform approximation formula,
again with arbitrary precision.


title:
Integrable non-commutative equations on quad-graphs. The consistency
  approach

abstract:
  We extend integrable systems on quad-graphs, such as the Hirota equation and
the cross-ratio equation, to the non-commutative context, when the fields take
values in an arbitrary associative algebra. We demonstrate that the
three-dimensional consistency property remains valid in this case. We derive
the non-commutative zero curvature representations for these systems, based on
the latter property. Quantum systems with their quantum zero curvature
representations are particular cases of the general non-commutative ones.


title: The tortuous behavior of lightning

abstract:
  The complex branched structure of lightning induce scientists to think that
dielectric breakdown is a very complicated phenomena, we will show that this is
not true and that simulating the structure of lightning is an easy task, but
depends strongly on boundary conditions. In this work we will introduce a new
way of understanding the origin of this tortuous path that relies on minimizing
the total energy stored in the system.


title:
A discrete Schrodinger spectral problem and associated evolution
  equations

abstract:
  A recently proposed discrete version of the Schrodinger spectral problem is
considered. The whole hierarchy of differential-difference nonlinear evolution
equations associated to this spectral problem is derived. It is shown that a
discrete version of the KdV, sine-Gordon and Liouville equations are included
and that the so called `inverse' class in the hierarchy is local. The whole
class of related Darboux and Backlund transformations is also exhibited.


title: Interaction of a Discrete Breather with a Lattice Junction

abstract:
  We study the scattering of a moving discrete breather (DB) on a junction in a
Fermi-Pasta-Ulam (FPU) chain consisting of two segments with different masses
of the particles. We consider four distinct cases: (i) a light-heavy (abrupt)
junction in which the DB impinges on the junction from the segment with lighter
mass, (ii) a heavy-light junction, (iii) an up mass-ramp in which the mass in
the heavier segment increases continuously as one moves away from the junction
point, and (iv) a down mass-ramp. Depending on the mass difference and DB
characteristics (frequency and velocity), the DB can either reflect from, or
transmit through, or get trapped at the junction or on the ramp. For the
heavy-light junction, the DB can even split at the junction into a reflected
and a transmitted DB. The latter is found to subsequently split into two or
more DBs. For the down mass-ramp the DB gets accelerated in several stages,
with accompanying radiation (phonons). These results are rationalized by
calculating the Peierls-Nabarro barrier for the various cases. We also point
out implications of our results in realistic situations such as electron-phonon
coupled chains.


title: Algebraic and Analytic Aspects of Soliton Type Equations

abstract:
  This is a review of two of the fundamental tools for analysis of soliton
equations: i) the algebraic ones based on Kac-Moody algebras, their central
extensions and their dual algebras which underlie the Hamiltonian structures of
the NLEE; ii) the construction of the fundamental analytic solutions of the Lax
operator and the Riemann-Hilbert problem (RHP) which they satisfy. The fact
that the inverse scattering problem for the Lax operator can be viewed as a RHP
gave rise to the dressing Zakharov-Shabat, one of the most effective ones for
constructing soliton solutions. These two methods when combined may allow one
to prove rigorously the results obtained by the abstract algebraic methods.
They also allow to derive spectral decompositions for non-self-adjoint Lax
operators.


title:
Transition from Gaussian-orthogonal to Gaussian-unitary ensemble in a
  microwave billiard with threefold symmetry

abstract:
  Recently it has been shown that time-reversal invariant systems with discrete
symmetries may display in certain irreducible subspaces spectral statistics
corresponding to the Gaussian unitary ensemble (GUE) rather than to the
expected orthogonal one (GOE). A Kramers type degeneracy is predicted in such
situations. We present results for a microwave billiard with a threefold
rotational symmetry and with the option to display or break a reflection
symmetry. This allows us to observe the change from GOE to GUE statistics for
one subset of levels. Since it was not possible to separate the three
subspectra reliably, the number variances for the superimposed spectra were
studied. The experimental results are compared with a theoretical and numerical
study considering the effects of level splitting and level loss.


title:
Dynamic Relationship between Diversity and Plasticity of Cell types in
  Multi-cellular State

abstract:
  Dynamics maintaining diversity of cell types in a multi-cellular system are
studied in relationship with the plasticity of cellular states. First, we
introduce a new theoretical framework, reaction-diffusion system on `chemical
species space' to model intra-cellular chemical reaction dynamics. Then, by
considering the cell division and death of such cells, developmental process
from a single cell is studied. Cell differentiation process is found to occur
through instability in transient dynamics and cell-cell interaction. In a long
time behavior, extinction of multiple cells is repeated, which leads to
itinerancy over successive quasi-stable multi-cellular states consisting of
different types of cells. By defining the plasticity of a cellular state, it is
shown that the plasticity of cells decreases before the large extinction, from
which diversity and plasticity are recovered. After this switching, decrease of
plasticity again occurs, leading to the next extinction of multiple cells. This
cycle is repeated. Relevance of our results to the development, evolution is
briefly discussed.


title:
Coexistence of Anticipating Synchronization and Lag Synchronization in
  an Optical System

abstract:
  The chaotic synchronization between two bi-directionally coupled external
cavity single-mode semiconductor lasers is investigated. Numerical simulation
shows that anticipating synchronization and lag synchronization coexist in
certain parameter regime. The anticipating time with different effects that
were discussed quite differently in the previous theoretical analysis and
experimental observation is determined by the involved parameters in the
system.


title: Fractional Logistic Map

abstract:
  A new type of an integrable mapping is presented. This map is equipped with
fractional difference and possesses an exact solution, which can be regarded as
a discrete analogue of the Mittag-Leffler function.


title:
Chaotic behaviour of the solutions to a perturbed Korteweg-de Vries
  equation

abstract:
  Stationary wave solutions of the perturbed Korteweg-de Vries equation are
considered in the presence of external hamiltonian perturbations. Conditions of
their chaotic behaviour are studied with the help of Melnikov theory. For the
homoclinic chaos Poincar\'e sections are constructed to demonstrate the
complicated behaviour, and Lyapunov exponents are also numerically calculated.


title: Melnikov analysis for multi-symplectic PDEs

abstract:
  In this work the Melnikov method for perturbed Hamiltonian wave equations is
considered in order to determine possible chaotic behaviour in the systems. The
backbone of the analysis is the multi-symplectic formulation of the unperturbed
PDE and its further reduction to travelling waves. In the multi-symplectic
approach two separate symplectic operators are introduced for the spatial and
temporal variables, which allow one to generalise the usual symplectic
structure. The systems under consideration include perturbations of generalised
KdV equation, nonlinear wave equation, Boussinesq equation. These equations are
equivariant with respect to abelian subgroups of Euclidean group. It is assumed
that the external perturbation preserves this symmetry. Travelling wave
reduction for the above-mentioned systems results in a four-dimensional system
of ODEs, which is considered for Melnikov type chaos. As a preliminary for the
calculation of a Melnikov function, we prove the persistence of a fixed point
for the perturbed Poincare map by using Lyapunov-Schmidt reduction. The
framework sketched will be applied to the analysis of possible chaotic
behaviour of travelling wave solutions for the above-mentioned PDEs within the
multi-symplectic approach.


title: Normal form and solitons

abstract:
  We present a review of the normal form theory for weakly dispersive nonlinear
wave equations where the leading order phenomena can be described by the KdV
equation. This is an infinite dimensional extension of the well-known
Poincar\'e-Dulac normal form theory for ordinary differential equations. We
also provide a detailed analysis of the interaction problem of solitary wavesas
an important application of the normal form theory. Several explicit examples
are discussed based on the normal form theory, and the results are compared
with their numerical simulations. Those examples include the ion acoustic wave
equation, the Boussinesq equation as a model of the shallow water waves, the
regularized long wave equation and the Hirota bilinear equation having a 7th
order linear dispersion.


title: Cusp-scaling behavior in fractal dimension of chaotic scattering

abstract:
  A topological bifurcation in chaotic scattering is characterized by a sudden
change in the topology of the infinite set of unstable periodic orbits embedded
in the underlying chaotic invariant set. We uncover a scaling law for the
fractal dimension of the chaotic set for such a bifurcation. Our analysis and
numerical computations in both two- and three-degrees-of-freedom systems
suggest a striking feature associated with these subtle bifurcations: the
dimension typically exhibits a sharp, cusplike local minimum at the
bifurcation.


title:
Statistics of Self-Crossings and Avoided Crossings of Periodic Orbits in
  the Hadamard-Gutzwiller Model

abstract:
  Employing symbolic dynamics for geodesic motion on the tesselated
pseudosphere, the so-called Hadamard-Gutzwiller model, we construct extremely
long periodic orbits without compromising accuracy. We establish criteria for
such long orbits to behave ergodically and to yield reliable statistics for
self-crossings and avoided crossings. Self-encounters of periodic orbits are
reflected in certain patterns within symbol sequences, and these allow for
analytic treatment of the crossing statistics. In particular, the distributions
of crossing angles and avoided-crossing widths thus come out as related by
analytic continuation. Moreover, the action difference for Sieber-Richter pairs
of orbits (one orbit has a self-crossing which the other narrowly avoids and
otherwise the orbits look very nearly the same) results to all orders in the
crossing angle. These findings may be helpful for extending the work of Sieber
and Richter towards a fuller understanding of the classical basis of quantum
spectral fluctuations.


title:
Localization and Coherent Structures in Wave Dynamics via
  Multiresolution

abstract:
  We apply variational-wavelet approach for constructing multiscale
high-localized eigenmodes expansions in different models of nonlinear waves. We
demonstrate appearance of coherent localized structures and stable pattern
formation in different collective dynamics models.


title: The Mermin fixed point

abstract:
  The most efficient known method for solving certain computational problems is
to construct an iterated map whose fixed points are by design the problem's
solution. Although the origins of this idea go back at least to Newton, the
clearest expression of its logical basis is an example due to Mermin. A
contemporary application in image recovery demonstrates the power of the
method.


title: Regularization modeling for large-eddy simulation

abstract:
  A new modeling approach for large-eddy simulation (LES) is obtained by
combining a `regularization principle' with an explicit filter and its
inversion. This regularization approach allows a systematic derivation of the
implied subgrid-model, which resolves the closure problem. The central role of
the filter in LES is fully restored, i.e., both the interpretation of LES
predictions in terms of direct simulation results as well as the corresponding
subgrid closure are specified by the filter. The regularization approach is
illustrated with `Leray-smoothing' of the nonlinear convective terms. In
turbulent mixing the new, implied subgrid model performs favorably compared to
the dynamic eddy-viscosity procedure. The model is robust at arbitrarily high
Reynolds numbers and correctly predicts self-similar turbulent flow
development.


title:
Fractal structures of normal and anomalous diffusion in nonlinear
  nonhyperbolic dynamical systems

abstract:
  A paradigmatic nonhyperbolic dynamical system exhibiting deterministic
diffusion is the smooth nonlinear climbing sine map. We find that this map
generates fractal hierarchies of normal and anomalous diffusive regions as
functions of the control parameter. The measure of these self-similar sets is
positive, parameter-dependent, and in case of normal diffusion it shows a
fractal diffusion coefficient. By using a Green-Kubo formula we link these
fractal structures to the nonlinear microscopic dynamics in terms of fractal
Takagi-like functions.


title:
Experimental Test of a Trace Formula for a Chaotic Three Dimensional
  Microwave Cavity

abstract:
  We have measured resonance spectra in a superconducting microwave cavity with
the shape of a three-dimensional generalized Bunimovich stadium billiard and
analyzed their spectral fluctuation properties. The experimental length
spectrum exhibits contributions from periodic orbits of non-generic modes and
from unstable periodic orbit of the underlying classical system. It is well
reproduced by our theoretical calculations based on the trace formula derived
by Balian and Duplantier for chaotic electromagnetic cavities.


title: Spatial semiconductor-resonator solitons

abstract:
  We demonstrate experimentally and numerically the existence spatial solitons
in multiple-quantum-well semiconductor microresonators driven by an external
coherent optical field. We discuss stability of the semiconductor-resonator
solitons over a wide spectral range around the band edge. We demonstrate the
manipulation of such solitons: switching solitons on and off by coherent as
well as incoherent light; reducing the light power necessary to sustain and
switch a soliton, by optical pumping.


title: The Joint Cascade of Energy and Helicity in Three-Dimensional Turbulence

abstract:
  Three-dimensional (3D) turbulence has both energy and helicity as inviscid
constants of motion. In contrast to two-dimensional (2D) turbulence, where a
second inviscid invariant--the enstrophy--blocks the energy cascade to small
scales, in 3D there is a joint cascade of both energy and helicity
simultaneously to small scales. The basic cancellation mechanism which permits
a joint cascade of energy and helicity is illuminated by means of the helical
decomposition of the velocity into positively and negatively polarized waves.
This decomposition is employed in the present study both theoretically and also
in a numerical simulation of homogeneous and isotropic 3D turbulence. It is
shown that the transfer of energy to small scales produces a tremendous growth
of helicity separately in the + and - helical modes at high wavenumbers,
diverging in the limit of infinite Reynolds number. However, because of a
tendency to restore reflection invariance at small scales, the net helicity
from both modes remains finite in that limit. The net helicity flux is shown to
be constant all the way up to the Kolmogorov wavenumber: there is no shorter
inertial-range for helicity cascade than for energy cascade. The transfer of
energy and helicity between + and - modes, which permits the joint cascade, is
shown to be due to two distinct physical processes, advection and vortex
stretching.


title:
Sinai billiards, Ruelle zeta-functions and Ruelle resonances: microwave
  experiments

abstract:
  We discuss the impact of recent developments in the theory of chaotic
dynamical systems, particularly the results of Sinai and Ruelle, on microwave
experiments designed to study quantum chaos. The properties of closed Sinai
billiard microwave cavities are discussed in terms of universal predictions
from random matrix theory, as well as periodic orbit contributions which
manifest as `scars' in eigenfunctions. The semiclassical and classical Ruelle
zeta-functions lead to quantum and classical resonances, both of which are
observed in microwave experiments on n-disk hyperbolic billiards.


title:
The Hunt Hypothesis and the Dividend Policy of the Firm. The Chaotic
  Motion of the Profits

abstract:
  We have carried out simulations of a financial model of the firm to analyse
the validity of the concept of Trade on Equity in dynamics. The results exhibit
the ability of the borrowing policy connected to a cautious dividend
distribution to inject chaos into the profit motion. The 3D system built with
the van der Pol's oscillator produces a new class of strange attractors.


title: Kinetic boundary conditions in the lattice Boltzmann method

abstract:
  Derivation of the lattice Boltzmann method from the continuous kinetic theory
[X. He and L. S. Luo, {\it Phys. Rev. E} {\bf 55}, R6333 (1997); X. Shan and X.
He, {\it Phys. Rev. Lett.} {\bf 80}, 65 (1998)] is extended in order to obtain
boundary conditions for the method. For the model of a diffusively reflecting
moving solid wall, the boundary condition for the discrete set of velocities is
derived, and the error of the discretization is estimated. Numerical results
are presented which demonstrate convergence to the hydrodynamic limit. In
particular, the Knudsen layer in the Kramers' problem is reproduced correctly
for small Knudsen numbers.


title: Integrability of the Bakirov system: a zero-curvature representation

abstract:
  For the Bakirov system, which is known to possess only one higher-order local
generalized symmetry, we explicitly find a zero-curvature representation
containing an essential parameter.


title: Invariant manifolds and collective motion in many-body systems

abstract:
  Collective modes of interacting many-body systems can be related to the
motion on classically invariant manifolds. We introduce suitable coordinate
systems. These coordinates are Cartesian in position and momentum space. They
are collective since several components vanish for motion on the invariant
manifold. We make a connection to Zickendraht's collective coordinates and also
obtain shear modes. The importance of collective configurations depends on the
stability of the manifold. We present an example of quantum collective motion
on the manifold


title: The Phase Information Associated to Synchronized Electronic Fireflies

abstract:
  An electronic implementation referring to fireflies ensembles flashing in
synchrony in a self-organization mode, shows the details of the phase-locking
mechanism and how the phases between the electronic oscillators are related to
their common period. Quantitative measurements of the timing signals link the
limits of a steadily established synchronization to the physics of the
electronic circuit. Preliminary observations suggest the existence of
bifurcation-like phenomena.


title:
Recursion operators of the N=2 supersymmetric unconstrained matrix GNLS
  hierarchies

abstract:
  A super-algebraic formulation of the N=2 supersymmetric unconstrained matrix
(k|n,m)-MGNLS hierarchies (nlin.SI/0201026) is established. Recursion
operators, fermionic and bosonic symmetries as well as their superalgebra are
constructed for these hierarchies.


title: On a puzzle posed by the Kurien-Sreenivasan boundary layer experiment

abstract:
  The Kolmogorov inertial range ratio of the mixed- to- longitudinal
third-order structure functions of isotropic and homogeneous turbulence is
$\frac{S_{1,2}}{S_{3,0}}=1/3$. Recent measurements by Kurien and Sreenivasan
(Phys. Rev. E {\bf 64}, 056302 (2001)), showed that, while the longitudinal
structure function was extremely close to Kolmogorov's, the measured ratio was
very far: $\frac{S_{1,2}}{S_{3,0}}\approx 0.43$. Explanation of this puzzle is
presented in this letter.


title: Understanding the complex patterns of snow crystals

abstract:
  We will show that the complex shapes of snow crystals can be explained from a
simple basic mechanism that is also responsible for the appearance of many
others structures in nature. We expect that this new physical mechanism, that
follows from minimizing the total stored energy, will permit to explain most of
the features of snow crystal growth.


title: Statistics of branched flow in a weak correlated random potential

abstract:
  Recent images of electron flow through a two-dimensional electron gas (2DEG)
device show branching behavior that is reproduced in numerical simulations of
motion in a correlated random potential [cond-mat/0010348]. We show how such
branching naturally arises from caustics in the classical flow and find a
simple scaling behavior of the branching under variation of the random
potential strength. Analytic results describing the statistical properties of
the branching are confirmed by classical and quantum numerical tests.


title: The modified nonlinear Schroedinger equation: Facts and artefacts

abstract:
  It is argued that the integrable modified nonlinear Schroedinger equation
with the nonlinearity dispersion term is the true starting point to
analytically describe subpicosecond pulse dynamics in monomode fibers. Contrary
to the known assertions, solitons of this equation are free of self-steepining
and the breather formation is possible.


title: Headway distribution of the asymmetric simple exclusion model

abstract:
  We present an exact solution of headway distribution of the asymmetric simple
exclusion model with open boundary conditions and compare it to the headway
distributions of the highway traffic.


title: Headway statistics of public transport in Mexican cities

abstract:
  We present a cellular automaton simulating the behavior of public bus
transport in several Mexican cities. The headway statistics obtained from the
model is compared to the measured time intervals between subsequent bus
arrivals to a given bus stop and to a spacing distribution resulting from a
random matrix theory.


title: N=(1|1) supersymmetric dispersionless Toda lattice hierarchy

abstract:
  Generalizing the graded commutator in superalgebras, we propose a new bracket
operation on the space of graded operators with an involution. We study
properties of this operation and show that the Lax representation of the
two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized
via the generalized bracket operation; this is important in constructing the
semiclassical (continuum) limit of this hierarchy. We construct the continuum
limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda
hierarchy. In this limit, we obtain the Lax representation, with the
generalized graded bracket becoming the corresponding Poisson bracket on the
graded phase superspace. We find bosonic symmetries of the dispersionless
N=(1|1) supersymmetric Toda equation.


title: Limitations of the Method of Multiple-Time-Scales

abstract:
  In the Method of Multiple-Time-Scales (MMTS), the introduction of independent
time scales and the elimination of secular terms in the fast time variable, T0
= t, lead to the well-known solvability conditions. Starting from first order,
free terms (solutions of the unperturbed equations) emerge in every order in
the expansion of the approximate solution. In orders higher than first, the
amplitudes of these free terms appear in the solvability conditions. Contrary
to the common belief, in the MMTS analysis, these free terms play a role above
and beyond the satisfaction of initial conditions: They make feasible mutual
consistency among solvability conditions that arise in different orders. In
general, this consistency may not be ensured if the free terms are chosen
arbitrarily (e.g., set to zero, as is commonly done in many applications). If
consistency is not ensured, the analysis may lead to wrong results, or allow
only trivial solutions.
  The solvability conditions constitute a system of PDE's for the dependence of
the amplitudes that appear in the expansion on the slow time scales. However,
whenever the free amplitudes must be included to ensure a consistent expansion,
these PDE's cannot determine the dependence of the solution on slow time
variables beyond the first one, T1=epsilo*t. The dependence on slower time
scales, Tn=epsilon^n*t, n > 1, must be imposed either through initial data at,
say, T1 = 0, or through requirements on the structure of the approximate
solution (based, for example, on physical intuition) that are not related to
the validity of the perturbative scheme.
  These claims are illustrated through several simple examples, and then
discussed in the general case.


title:
Integrability of Kersten-Krasil'shchik coupled KdV-mKdV equations:
  singularity analysis and Lax pair

abstract:
  The integrability of a coupled KdV-mKdV system is tested by means of
singularity analysis. The true Lax pair associated with this system is obtained
by the use of prolongation technique.


title: Self-sustained oscillations in homogeneous shear flow

abstract:
  Generation of the large-scale coherent vortical structurs in homogeneous
shear flow couples dynamical processes of energy and enstrophy production. In
the large rate of strain limit, the simple estimates of the contributions to
the energy and enstrophy equations result in a dynamical system, describing
experimentally and numerically observed self-sustained non-linear oscillations
of energy and enstrophy. It is shown that the period of these oscilaltions is
independent upon the box size and the energy and enstrophy fluctuations are
strongly correlated.


title: The myth about nonlinear differential equations

abstract:
  Taking the example of Koretweg--de Vries equation, it is shown that soliton
solutions need not always be the consequence of the trade-off between the
nonlinear terms and the dispersive term in the nonlinear differential equation.
Even the ordinary one dimensional linear partial differential equation can
produce a soliton.


title:
Spectral curve, Darboux coordinates and Hamiltonian structure of
  periodic dressing chains

abstract:
  A chain of one-dimensional Schr\"odinger operators connected by successive
Darboux transformations is called the ``Darboux chain'' or ``dressing chain''.
The periodic dressing chain with period $N$ has a control parameter $\alpha$.
If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a
generalization of the fourth or fifth (depending on the parity of $N$)
Painlev\'e equations . The $N$-periodic dressing chain has two different Lax
representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax
pair can be used to construct a transition matrix around the periodic lattice.
One can thereby define an associated ``spectral curve'' and a set of Darboux
coordinates called ``spectral Darboux coordinates''. The equations of motion of
the dressing chain can be converted to a Hamiltonian system in these Darboux
coordinates. The symplectic structure of this Hamiltonian formalism turns out
to be consistent with a Poisson structure previously studied by Veselov,
Shabat, Noumi and Yamada.


title:
Stationary and moving breathers in a simplified model of curved
  alpha--helix proteins

abstract:
  The existence, stability and movability of breathers in a model for
alpha-helix proteins is studied. This model basically consists a chain of
dipole moments parallel to it. The existence of localized linear modes brings
about that the system has a characteristic frequency, which depends on the
curvature of the chain. Hard breathers are stable, while soft ones experiment
subharmonic instabilities that preserve, however the localization. Moving
breathers can travel across the bending point for small curvature and are
reflected when it is increased. No trapping of breathers takes place.


title: Breathers in a system with helicity and dipole interaction

abstract:
  Recent papers that have studied variants of the Peyrard-Bishop model for DNA,
have taken into account the long range interaction due to the dipole moments of
the hydrogen bonds between base pairs. In these models the helicity of the
double strand is not considered. In this particular paper we have performed an
analysis of the influence of the helicity on the properties of static and
moving breathers in a Klein--Gordon chain with dipole-dipole interaction. It
has been found that the helicity enlarges the range of existence and stability
of static breathers, although this effect is small for a typical helical
structure of DNA. However the effect of the orientation of the dipole moments
is considerably higher with transcendental consequences for the existence of
mobile breathers.


title: Discrete symmetry's chains and links between integrable equations

abstract:
  The discrete symmetry's dressing chains of the nonlinear Schrodinger equation
(NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS)
equation and the modified DS (mDS) equations are obtained. The explicitly
reversible Backlund auto-transformations for the mNLS and mDS equations are
constructed. We demonstrate discrete symmetry's conjugate chains of the KP and
DS models. The two-dimensional generalization of the P4 equation are obtained.


title:
Hypersensitivity to perturbations of quantum-chaotic wave-packet
  dynamics

abstract:
  We re-examine the problem of the "Loschmidt echo", which measures the
sensitivity to perturbation of quantum chaotic dynamics. The overlap squared
$M(t)$ of two wave packets evolving under slightly different Hamiltonians is
shown to have the double-exponential initial decay $\propto \exp(-{\rm
constant}\times e^{2\lambda_0 t})$ in the main part of phase space. The
coefficient $\lambda_0$ is the self-averaging Lyapunov exponent. The average
decay $\bar{M}\propto e^{-\lambda_1 t}$ is single exponential with a different
coefficient $\lambda_1$. The volume of phase space that contributes to
$\bar{M}$ vanishes in the classical limit $\hbar\to 0$ for times less than the
Ehrenfest time $\tau_E=\frac{1}{2}\lambda_0^{-1}|\ln \hbar|$. It is only after
the Ehrenfest time that the average decay is representative for a typical
initial condition.


title: Active vs passive scalar turbulence

abstract:
  Active and passive scalars transported by an incompressible two-dimensional
conductive fluid are investigated. It is shown that a passive scalar displays a
direct cascade towards the small scales while the active magnetic potential
builds up large-scale structures in an inverse cascade process. Correlations
between scalar input and particle trajectories are found to be responsible for
those dramatic differences as well as for the behavior of dissipative
anomalies.


title:
Complex Ginzburg-Landau equation for self-oscillatory systems under
  amplitude modulated forcing in a 1:1 resonance

abstract:
  The dynamics of self-oscillatory extended systems, resonantly forced at a
frequency close to that of the natural oscillations (1:1 resonance), is shown
to be universally described by a complex Ginzburg-Landau equation containing an
inhomogeneous term. The case of amplitude modulated forcing is considered,
which generalizes previous studies. Application to the two-frequency forcing in
a 1:1 resonance is considered as an example.


title:
Active and Passive Fields in Turbulent Transport: the Role of
  Statistically Preserved Structures

abstract:
  We have recently proposed that the statistics of active fields (which affect
the velocity field itself) in well-developed turbulence are also dominated by
the Statistically Preserved Structures of auxiliary passive fields which are
advected by the same velocity field. The Statistically Preserved Structures are
eigenmodes of eigenvalue 1 of an appropriate propagator of the decaying
(unforced) passive field, or equivalently, the zero modes of a related
operator. In this paper we investigate further this surprising finding via two
examples, one akin to turbulent convection in which the temperature is the
active scalar, and the other akin to magneto-hydrodynamics in which the
magnetic field is the active vector. In the first example, all the even
correlation functions of the active and passive fields exhibit identical
scaling behavior. The second example appears at first sight to be a
counter-example: the statistical objects of the active and passive fields have
entirely different scaling exponents. We demonstrate nevertheless that the
Statistically Preserved Structures of the passive vector dominate again the
statistics of the active field, except that due to a dynamical conservation law
the amplitude of the leading zero mode cancels exactly. The active vector is
then dominated by the sub-leading zero mode of the passive vector. Our work
thus suggests that the statistical properties of active fields in turbulence
can be understood with the same generality as those of passive fields.


title: Field theoretic renormalization group for a nonlinear diffusion equation

abstract:
  The paper is an attempt to relate two vast areas of the applicability of the
renormalization group (RG): field theoretic models and partial differential
equations. It is shown that the Green function of a nonlinear diffusion
equation can be viewed as a correlation function in a field-theoretic model
with an ultralocal term, concentrated at a spacetime point. This field theory
is shown to be multiplicatively renormalizable, so that the RG equations can be
derived in a standard fashion, and the RG functions (the $\beta$ function and
anomalous dimensions) can be calculated within a controlled approximation. A
direct calculation carried out in the two-loop approximation for the
nonlinearity of the form $\phi^{\alpha}$, where $\alpha>1$ is not necessarily
integer, confirms the validity and self-consistency of the approach. The
explicit self-similar solution is obtained for the infrared asymptotic region,
with exactly known exponents; its range of validity and relationship to
previous treatments are briefly discussed.


title:
Renormalization-group approach to the stochastic Navier--Stokes
  equation: Two-loop approximation

abstract:
  The field theoretic renormalization group is applied to the stochastic
Navier--Stokes equation that describes fully developed fluid turbulence. The
complete two-loop calculation of the renormalization constant, the $\beta$
function, the fixed point and the ultraviolet correction exponent is performed.
The Kolmogorov constant and the inertial-range skewness factor, derived to
second order of the $\eps$ expansion, are in a good agreement with the
experiment. The possibility of the extrapolation of the $\eps$ expansion beyond
the threshold where the sweeping effects become important is demonstrated on
the example of a Galilean-invariant quantity, the equal-time pair correlation
function of the velocity field. The extension to the $d$-dimensional case is
briefly discussed.


title: Spectra of turbulence in dilute polymer solutions

abstract:
  We investigate turbulence in dilute polymer solutions when polymers are
strongly stretched by the flow. We establish power-law spectrum of velocity,
which is not associated with a flux of a conserved quantity, in two cases. The
first case is the elastic waves range of high Reynolds number turbulence of
polymer solutions above the coil-stretch transition. The second case is the
elastic turbulence, where chaotic flow is excited due to elastic instabilities
at small Reynolds numbers.


title:
Dispersive and friction-induced stabilization of an inverse cascade. The
  theory for the Kolmogorov flow in the slightly supercritical regime

abstract:
  We discuss the stabilisation of the inverse cascade in the large scale
instability of the Kolmogorov flow described by the complete Cahn-Hilliard
equation with inclusion of $\beta$ effect, large-scale friction and deformation
radius. The friction and the $\beta$values halting the inverse cascade at the
various possible intermediate states are calculated by means of singular
perturbation techniques and compared to the values resulting from numerical
simulation of the complete Cahn-Hilliard equation. The excellent agreement
validates the theory. Our main result is that the critical values of friction
or $\beta$ halting the inverse cascade scale exponentially as a function of the
jet separation in the final flow, contrary to previous theories and
phenomenological approach.


title: On generalized Lotka-Volterra lattices

abstract:
  Generalized matrix Lotka-Volterra lattice equations are obtained in a
systematic way from a "master equation" possessing a bicomplex formulation.


title: Scaling Exponents in Anisotropic Hydrodynamic Turbulence

abstract:
  In anisotropic turbulence the correlation functions are decomposed in the
irreducible representations of the SO(3) symmetry group (with different
"angular momenta" $\ell$). For different values of $\ell$ the second order
correlation function is characterized by different scaling exponents
$\zeta_2(\ell)$. In this paper we compute these scaling exponents in a Direct
Interaction Approximation (DIA). By linearizing the DIA equations in small
anisotropy we set up a linear operator and find its zero-modes in the inertial
interval of scales. Thus the scaling exponents in each $\ell$-sector follow
from solvability condition, and are not determined by dimensional analysis. The
main result of our calculation is that the scaling exponents $\zeta_2(\ell)$
form a strictly increasing spectrum at least until $\ell=6$, guaranteeing that
the effects of anisotropy decay as power laws when the scale of observation
diminishes. The results of our calculations are compared to available
experiments and simulations.


title:
The Decay of Passive Scalars Under the Action of Single Scale Smooth
  Velocity Fields in Bounded 2D Domains : From non self similar pdf's to self
  similar eigenmodes

abstract:
  We examine the decay of passive scalars with small, but non zero, diffusivity
in bounded 2D domains. The velocity fields responsible for advection are smooth
(i.e., they have bounded gradients) and of a single large scale. Moreover, the
scale of the velocity field is taken to be similar to the size of the entire
domain. The importance of the initial scale of variation of the scalar field
with respect to that of the velocity field is strongly emphasized. If these
scales are comparable and the velocity field is time periodic, we see the
formation of a periodic scalar eigenmode. The eigenmode is numerically realized
by means of a deterministic 2D map on a lattice. Analytical justification for
the eigenmode is available from theorems in the dynamo literature. Weakening
the notion of an eigenmode to mean statistical stationarity, we provide
numerical evidence that the eigenmode solution also holds for aperiodic flows
(represented by random maps). Turning to the evolution of an initially small
scale scalar field, we demonstrate the transition from an evolving (i.e., {\it
non} self similar) pdf to a stationary (self similar) pdf as the scale of
variation of the scalar field progresses from being small to being comparable
to that of the velocity field (and of the domain). Furthermore, the {\it non}
self similar regime itself consists of two stages. Both the stages are examined
and the coupling between diffusion and the distribution of the Finite Time
Lyapunov Exponents is shown to be responsible for the pdf evolution.


title: Integrable initial boundary value problems

abstract:
  The problem of searching boundary value problems for soliton equations
consistent with the integrability property is discussed. A method of describing
integrals of motion for the integrable initial boundary value problems for the
Kadomtsev-Petviashvili equation is suggested via Green identity.


title: Travelling Wave Solutions of a Simple Non-Local Burgers-Like Equation

abstract:
  A Lax pair consisting of the Forsyth-Hopf-Cole transformation and a linear
differential (in time) - difference (in continuous space) equation generates
the whole hierarchy of the Burgers equation and admits exact moving front
solutions. This model mimics finite-distance spatial correlations or nearest
neighbor interactions in a physical situation


title:
Negative and Nonlinear Response in an Exactly Solved Dynamical Model of
  Particle Transport

abstract:
  We consider a simple model of particle transport on the line defined by a
dynamical map F satisfying F(x+1) = 1 + F(x) for all x in R and F(x) = ax + b
for |x| < 0.5. Its two parameters a (`slope') and b (`bias') are respectively
symmetric and antisymmetric under reflection x -> R(x) = -x. Restricting
ourselves to the chaotic regime |a| > 1 and therein mainly to the part a>1 we
study not only the `diffusion coefficient' D(a,b), but also the `current'
J(a,b). An important tool for such a study are the exact expressions for J and
D as obtained recently by one of the authors. These expressions allow for a
quite efficient numerical implementation, which is important, because the
functions encountered typically have a fractal character. The main results are
presented in several plots of these functions J(a,b) and D(a,b) and in an
over-all `chart' displaying, in the parameter plane, all possibly relevant
information on the system including, e.g., the dynamical phase diagram as well
as invariants such as the values of topological invariants (kneading numbers)
which, according to the formulas, determine the singularity structure of J and
D. Our most significant findings are: 1) `Nonlinear Response': The parameter
dependence of these transport properties is, throughout the `ergodic' part of
the parameter plane (i.e. outside the infinitely many Arnol'd tongues)
fractally nonlinear. 2) `Negative Response': Inside certain regions with an
apparently fractal boundary the current J and the bias b have opposite signs.


title: Numerical and Theoretical Studies of Noise Effects in the Kauffman Model

abstract:
  In this work we analyze the stochastic dynamics of the Kauffman model
evolving under the influence of noise. By considering the average crossing time
between two distinct trajectories, we show that different Kauffman models
exhibit a similar kind of behavior, even when the structure of their basins of
attraction is quite different. This can be considered as a robust property of
these models. We present numerical results for the full range of noise level
and obtain approximate analytic expressions for the above crossing time as a
function of the noise in the limit cases of small and large noise levels.


title:
Microscopic/stochastic timesteppers and coarse control: a kinetic Monte
  Carlo example

abstract:
  Coarse timesteppers provide a bridge between microscopic / stochastic system
descriptions and macroscopic tasks such as coarse stability/bifurcation
computations. Exploiting this computational enabling technology, we present a
framework for designing observers and controllers based on microscopic
simulations, that can be used for their coarse control. The proposed
methodology provides a bridge between traditional numerical analysis and
control theory on the one hand and microscopic simulation on the other.


title:
The reflection-antisymmetric counterpart of the K\'arm\'an-Howarth
  dynamical equation

abstract:
  We study the isotropic, helical component in homogeneous turbulence using
statistical objects which have the correct symmetry and parity properties.
Using these objects we derive an analogue of the K\'arm\'an-Howarth equation,
that arises due to parity violation in isotropic flows. The main equation we
obtain is consistent with the results of O. Chkhetiani [JETP, 63, 768, (1996)]
and
  V.S. L'vov et al. [chao-dyn/9705016,
  (1997)] but is derived using only velocity correlations, with no direct
consideration of the vorticity or helicity. This alternative formulation offers
an advantage to both experimental and numerical measurements. We also
postulate, under the assumption of self-similarity, the existence of a
hierarchy of scaling exponents for helical velocity correlation functions of
arbitrary order, analogous to the
  Kolmogorov 1941 prediction for the scaling exponents of velocity structure
function.


title:
Models and Metaphors. Part I: From the Boundaries of Formalization to
  the Visualization of the Non-Formalized

abstract:
  The explicit relationship among perception, communication, and design is
being discussed in some detail, in order to relate it to characteristic details
of the modeling of the world which defines the scientific and artistic
activities of humans, in the first place, but also the philosophical grasping
of the world and its foundations in order to position humankind within this
worldly framework, at the same time. In particular, the structure and
constitutional role of language are being analyzed with a view to the latter's
universality. For the special case of synergetics (within the recent context of
theories of self-organization and the formation of structure), this cognitive
background is being inspected as to possible consequences for aspects of
complexity and nonlinearity.


title:
Models and Metaphors. Part II: The Aesthetical Conception of Natural
  Laws

abstract:
  Following the results and the general methodological framework of a
preceeding paper (Models and Metaphors, part I, henceforth: MM), we develop
some results on the epistemological state of natural laws and ask for explicit
possibilities of mediating the contexts of nature (in terms of the sciences)
and aesthetics.


title: Determination of fractal dimensions of solar radio bursts

abstract:
  We present a dimension analysis of a set of solar type I storms and type IV
events with different kind of fine structures, recorded at the Trieste
Astronomical Observatory. The signature of such types of solar radio events is
highly structured in time. However, periodicities are rather seldom, and linear
mode theory can provide only limited interpretation of the data. Therefore, we
performed an analysis based on methods of the nonlinear dynamics theory.
Additionally to the commonly used correlation dimension, we also calculated
local pointwise dimensions. This alternative approach is motivated by the fact
that astrophysical time series represent real-world systems, which cannot be
kept in a controlled state and which are highly interconnected with their
surroundings. In such systems pure determinism is rather unlikely to be
realized, and therefore a characterization by invariants of the dynamics might
probably be inadequate. In fact, the outcome of the dimension analysis does not
give hints for low-dimensional determinism in the data, but we show that,
contrary to the correlation dimension method, local dimension estimations can
give physical insight into the events even in cases in which pure determinism
cannot be established. In particular, in most of the analyzed radio events
nonlinearity in the data is detected, and the local dimension analysis provides
a basis for a quantitative description of the time series, which can be used to
characterize the complexity of the related physical system in a comparative and
non-invariant manner.