measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

backlund transformations as exact integrable time discretizations for the trigonometric gaudin model we construct a two parameter family of backlund transformations for the trigonometric classical gaudin magnet the approach follows closely the one introduced by sklyanin and kuznetsov 1998 j phys a math gen 31 2241 51 in a number of seminal papers and takes advantage of the intimate relation between the trigonometric and the rational case as in the paper by hone kuznetsov and one of the authors or 2001 j phys a math gen 34 2477 90 the backlund transformations are presented as explicit symplectic maps starting from their lax representation the expected connection with the xxz heisenberg chain is established and the rational xxx case is recovered in a suitable limit it is shown how to obtain a physical transformation mapping real variables into real variables the interpolating hamiltonian flow is derived and some numerical iterations of the map are presented

telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

twenty five years of multifractals in fully developed turbulence a tribute to giovanni paladin the paper on the multifractal nature of fully developed turbulence and chaotic systems by benzi et al 1984 j phys a math gen 17 3521 has been a starting point of many investigations on the different faces of self similarity and intermittency in turbulent phenomena since then the multifractal model has become a useful tool for the study of small scale turbulence in particular for detailed predictions of different eulerian and lagrangian statistical properties on the occasion of the 50th birthday of our unforgettable friend and colleague giovanni paladin 1958 1996 we review here the basic concepts and some applications of the multifractal model for turbulence

desynchronization in diluted neural networks the dynamical behavior of a weakly diluted fully inhibitory network of pulse coupled spiking neurons is investigated upon increasing the coupling strength a transition from regular to stochasticlike regime is observed in the weak coupling phase a periodic dynamics is rapidly approached with all neurons firing with the same rate and mutually phase locked the strong coupling phase is characterized by an irregular pattern even though the maximum lyapunov exponent is negative the paradox is solved by drawing an analogy with the phenomenon of stable chaos i e by observing that the stochasticlike behavior is limited to an exponentially long with the system size transient remarkably the transient dynamics turns out to be stationary

error propagation in extended chaotic systems a strong analogy is found between the evolution of localized disturbances in extended chaotic systems and the propagation of fronts separating different phases a condition for the evolution to be controlled by nonlinear mechanisms is derived on the basis of this relationship an approximate expression for the nonlinear velocity is also determined by extending the concept of the lyapunov exponent to a growth rate of finite perturbations

quantum algorithmic integrability the metaphor of classical polygonal billiards we study the algorithmic complexity of motions in classical polygonal billiards which as the number of sides increases tend to curved billiards both regular and chaotic this study unveils the equivalence of this problem to the procedure of quantization the average complexity of symbolic trajectories in polygonal billiards features the same scaling relations with respect to the number of sides that govern quantum systems when a semiclassical parameter is varied two cases the polygonal approximations of the circle and of the stadium are examined in detail and are presented as paradigms of quantization of integrable and chaotic systems

backlund transformations and hamiltonian flows in this work we show that under certain conditions parametric backlund transformations for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non autonomous hamiltonian the two systems share the same constants of motion this observation leads to the identification of the hamiltonian interpolating the iteration of the discrete map defined by the transformations which indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the lax matrix an example with the periodic toda lattice is given

on baxter s q operator for the xxx spin chain we discuss the construction of baxter s q operator the suggested approach leads to the one parametric family of q operators satisfying wronskian type relations also we have found the generalization of baxter operators which defines the nondiagonal part of the monodromy

numerical stability analysis of an acceleration scheme for step size constrained time integrators time integration schemes with a fixed time step much smaller than the dominant slow time scales of the dynamics of the system arise in the context of stiff ordinary differential equations or in multiscale computations where a microscopic time stepper is used to compute macroscopic behaviour we discuss a method to accelerate such a time integrator by using extrapolation this method extends the scheme developed by sommeijer increasing the real stability boundary of explicit methods comput math appl 19 6 1990 37 49 and uses similar ideas as the projective integration method we analyse the stability properties of the method and we illustrate its performance for a convection diffusion problem

lagrangian coherent structures and mixing in two dimensional turbulence abstract we introduce a lagrangian definition for the boundaries of coherent structures in two dimensional turbulence the boundaries are defined as material lines that are linearly stable or unstable for longer times than any of their neighbors such material lines are responsible for stretching and folding in the mixing of passive tracers we derive an analytic criterion that can be used to extract coherent structures with high precision from numerical or experimental data sets the criterion provides a rigorous link between the lagrangian concept of hyperbolicity the okubo weiss criterion and vortex boundaries we apply the results to simulations of two dimensional barotropic turbulence

a high order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws we present a general high order fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions the scheme consists of two steps in a first relaxation step the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff bgk source term then this kinetic equation is integrated in time using a projective integration method after taking a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution the time derivative is estimated and used in an outer rungekutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the cfl condition for the hyperbolic conservation law moreover the number of inner time steps is also independent of the stiffness of the bgk source term we discuss stability and consistency and illustrate with numerical results linear advection burgers equation and the shallow water and euler equations in one and two spatial dimensions

chaos or noise difficulties of a distinction in experiments the dynamical behavior of systems is reflected in time series due to the finiteness of the observational data set it is not possible to reconstruct the invariant measure up to an arbitrarily fine resolution and an arbitrarily high embedding dimension these restrictions limit our ability to distinguish between signals generated by different systems such as regular chaotic or stochastic ones when analyzed from a time series point of view we propose to classify the signal behavior without referring to any specific model as stochastic or deterministic on a certain scale of the resolution ensuremath epsilon according to the dependence of the ensuremath epsilon ensuremath tau entropy h ensuremath epsilon ensuremath tau and the finite size lyapunov exponent ensuremath lambda ensuremath epsilon on ensuremath epsilon

the periodic toda chain and a matrix generalization of the bessel function recursion relations the authors obtain the quantization conditions of the periodic toda lattice in the baxter form lambda u q u inq u i i nq u i lambda is the transfer matrix containing the information about the spectrum and q is an integral operator commuting with lambda the logarithms of the matrix elements of q are the generating functions of the canonical backlund transformation the requirement that q is analytic and vanishes when u goes to infinity completely determines the spectrum of lambda

accuracy analysis of acceleration schemes for stiff multiscale problems in the context of multiscale computations techniques have recently been developed that enable microscopic simulators to perform macroscopic level tasks equation free multiscale computations the main tool is the so called coarse grained time stepper which implements an approximation of the unavailable macroscopic time stepper using only the microscopic simulator several schemes were developed to accelerate the coarse grained time stepper exploiting the smoothness in time of the macroscopic dynamics to date mainly the stability of these methods was analyzed in this paper we focus on their accuracy properties mainly in the context of parabolic problems we study the global error of the different methods compare with explicit stiff ode solvers and use the theoretical results to develop more accurate variants our theoretical results are confirmed by various numerical experiments

the predictability problem in systems with an uncertainty in the evolution law the problem of unpredictability in a physical system due to the incomplete knowledge of the evolution laws is addressed major interest is devoted to the analysis of error amplification in chaotic systems with many characteristic times and scales when the fastest scales are not resolved the parametrization of the unresolved scales introduces a non infinitesimal uncertainty with respect to the true evolution laws which affects the forecasting ability on the large resolved scales the evolution of non infinitesimal errors from the unresolved scales up to the large scales is analysed by means of the finite size lyapunov exponent it is shown that proper parametrization of the unresolved scales allows one to recover the maximal predictability of the system

a q difference baxter operator for the ablowitz ladik chain we construct the baxter operator and the corresponding baxter equation for a quantum version of the ablowitz ladik model the result is achieved in two different ways by using the well known bethe ansatz technique and by looking at the quantum analogue of the classical backlund transformations general results about integrable models governed by the same r matrix algebra will be given baxter s equation comes out to be a q difference equation involving both the trace and the quantum determinant of the monodromy matrix the spectrality property of the classical backlund transformations gives a trace formula representing the classical analogue of baxter s equation a q integral representation of the baxter operator is discussed

globally coupled maps phase transitions and synchronization abstract bifurcations in a system of coupled maps are investigated using symbolic dynamics it is shown that for coupled shift maps the well known space time mixing attractor becomes unstable at a critical coupling strength in favour of a synchronized state for coupled non hyperbolic maps analytical and numerical evidence is given that arbitrary small coupling changes the dynamical behaviour the anomalous dependence of fluctuations on the system size is attributed to these bifurcations

disturbance propagation in chaotic extended systems with long range coupling propagation of initially localized perturbations is investigated in chaotic coupled map lattices with long range couplings decaying as a power of the distance the initial perturbation propagates exponentially fast along the lattice with a rate given by the ratio of the maximal lyapunov exponent and the power of the coupling a complementary description in terms of a suitable comoving lyapunov exponent is also given

remarks on the mean field dynamics of networks of chaotic elements abstract fluctuations of the mean field of a globally coupled dynamical systems are discussed the origin of hidden coherence is related with the instability of the fixed point solution of the self consistent perron frobenius equation collective dynamics in globally coupled tent maps are re examined both with the help of direct simulation and the perron frobenius equation collective chaos in a single band state and bifurcation against initial conditions in a two band state are clarified with the return maps of the mean field lyapunov spectra and also the newly introduced lyapunov exponent for the perron frobenius equation future problems on the collective dynamics are discussed

diagonalization of the infinite q boson system we present a hierarchy of commuting operators in fock space con taining the q boson hamiltonian on z and show that the operators in question are simultaneously diagonalized by hall littlewood functions as an applica tion the n particle scattering operator is computed the q boson model constitutes a one dimensional exactly solvable particle system in fock space bik based on the q oscillator algebra ks ch 5 in the case of periodic boundary conditions i e with particles hopping on the finite lattice zm the integrability the spectrum and the eigenfunctions of the hamiltonian were analyzed by means of the algebraic bethe ansatz method bik remarkably these eigenfunctions turn out to be hall littlewood functions t k cf also j for an alternative construction of hall littlewood functions in fock space based on deformed vertex operator algebras with applications in the study of kp functions arising from generating functions of weighted plane partitions fw with the aid of explicit expressions for the commuting quantum integrals arising from an infinite dimensional solution of the yang baxter equation it was very recently demonstrated k that the eigenvalue problem for the q boson system on zm is in fact equivalent to that of an integrable discretization d of the celebrated delta bose gas on the circle ll the present work addresses the spectral problem and the integrability of the q boson system on the infinite lattice z specifically we demonstrate that the eigenfunctions of this infinite q boson system are again given by hall littlewood functions and provide explicit formulas for a complete hierarchy of operators com muting with the hamiltonian these formulas are natural infinite dimensional ana logues of the above mentioned expressions in k for the finite q boson system on zm finally the n particle scattering operator is computed as an application of ruijsenaars general scattering results in r2 2 the infinite q boson system

billiards in polygons survey of recent results in the ten or so years since the publication of ref 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature as a result our understanding of the subject although still far from being complete is much better than it was ten years ago this survey attempts to give a broad overview of the dynamics of billiards in polygons with an emphasis on the material that was not in the literature in 1985 this is not a survey of the publications on polygonal billiards and we apologize to authors whose work has not been included the selection of topics has been strongly influenced by the personal taste of the author and by space limitations thus we do not discuss quantum polygonal billiards in particular quantum chaos see e g refs 41 44 and 45 we hope that somebody will write a survey on this important subject in the near future a fascinating aspect of the subject is the interplay between the geometric shape of the billiard table i e a planar curve and the qualitative features of the billiard dynamics hence it is instructive to compare polygonal billiard tables with other classes of billiard tables in particular the smooth strictly convex tables and the dispersing or sinai tables produce strikingly different types of billiard dynamics we recommend refs 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables

finite size lyapunov exponent review on applications in dynamical systems the growth of infinitesimal perturbations is well characterized by the lyapunov exponents in many situations of interest however important phenomena involve finite amplitude perturbations which are ruled by nonlinear dynamics out of tangent space and thus cannot be captured by the standard lyapunov exponents we review the application of the finite size lyapunov exponent fsle for the characterization of non infinitesimal perturbations in a variety of systems in particular we illustrate their usage in the context of predictability of systems with multiple spatio temporal scales of geophysical relevance in the characterization of nonlinear instabilities and in some aspects of transport in fluid flows we also discuss the application of the fsle to more general aspects such as chaos noise detection and coarse grained descriptions of signals this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

synchronization and partial synchronization of linear maps abstract we study synchronization of low dimensional d 2 3 4 chaotic piecewise linear maps that are coupled bidirectionally for bernoulli maps we find lyapunov exponents and locate the synchronization transition which numerically is found to be discontinuous despite continuously vanishing lyapunov exponent s for tent maps a limit of stability of the synchronized state is used to locate the synchronization transition that numerically is found to be continuous for nonidentical tent maps at the partial synchronization transition the probability distribution of the synchronization error is shown to develop highly singular behavior we suggest that for nonidentical bernoulli maps and perhaps some other discontinuous maps partial synchronization is merely a smooth crossover rather than a well defined transition more subtle analysis in the d 4 case locates the point where the synchronized state becomes stable in some cases however a riddled basin attractor appears and synchronized and chaotic behaviors coexist

coarse projective kmc integration forward reverse initial and boundary value problems in equation free multiscale computation a dynamic model is given at a fine microscopic level yet we believe that its coarse grained macroscopic dynamics can be described by closed equations involving only coarse variables these variables are typically various low order moments of the distributions evolved through the microscopic model we consider the problem of integrating these unavailable equations by acting directly on kinetic monte carlo microscopic simulators thus circumventing their derivation in closed form in particular we use projective multi step integration to solve the coarse initial value problem forward in time as well as backward in time under certain conditions macroscopic trajectories are thus traced back to unstable source type and even sometimes saddle like stationary points even though the microscopic simulator only evolves forward in time we also demonstrate the use of such projective integrators in a shooting boundary value problem formulation for the computation of coarse limit cycles of the macroscopic behavior and the approximation of their stability through estimates of the leading coarse floquet multipliers

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

an algorithmic view of pseudochaos abstract the controversial concept of quantum chaos the dynamical chaos in bounded mesoscopic quantum systems is presented as the most important and universal instance of a new generic dynamical phenomenon pseudochaos the latter characterizes the irregular behaviour of dynamical systems with discrete energy and or frequency spectrum which include classical systems with discrete phase space the question of randomness is addressed in terms of the algorithmic theory of dynamical systems using the arnold cat map for illustration the relationship between the algorithmic theory and ergodic theory is discussed

nonlinearly driven transverse synchronization in coupled chaotic systems synchronization transitions are investigated in coupled chaotic maps depending on the relative weight of linear versus nonlinear instability mechanisms associated to the single map two different scenarios for the transition may occur when only two maps are considered we always find that the critical coupling el for chaotic synchronization can be predicted within a linear analysis by the vanishing of the transverse lyapunov exponent t however major differences between transitions driven by linear or nonlinear mechanisms are revealed by the dynamics of the transient toward the synchronized state as a representative example of extended systems a one dimensional lattice of chaotic maps with power law coupling is considered in this high dimensional model finite amplitude instabilities may have a dramatic effect on the transition for strong nonlinearities an exponential divergence of the synchronization times with the chain length can be observed above el notwithstanding the transverse dynamics is stable against infinitesimal perturbations at any instant therefore the transition takes place at a coupling enl definitely larger than el and its origin is intrinsically nonlinear the linearly driven transitions are continuous and can be described in terms of mean field results for non equilibrium phase transitions with long range interactions while the transitions dominated by nonlinear mechanisms appear to be discontinuous

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

a note on implementations of the boosting algorithm and heterogeneous multiscale methods we present improved convergence results for the boosting algorithm ba and demonstrate that an existing formulation of the heterogeneous multiscale methods hmm is accurate to first order only in the macro time step regardless of the order of the numerical solvers employed these results are obtained by considering the ba and two other formulations of hmm as special cases of a general formulation of hmm applied to dissipative stiff ordinary differential equations

deciding the nature of the coarse equation through microscopic simulations the baby bathwater scheme recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form the closure is obtained on demand through appropriately initialized bursts of microscopic simulation the effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation such decisions include a the highest spatial derivative active in the coarse equation b whether the equation satisfies certain conservation laws or c whether the coarse dynamics is hamiltonian or dissipative these decisions affect the number and type of boundary conditions as well as the algorithms employed in the absence of an explicit formula for the temporal derivative we propose implement and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation in the same spirit we show how to determine whether a certain coarse conservation law exists or not and we discuss plausibility tests for the existence of a coarse hamiltonian or integrability we believe that such schemes constitute an important part of the equation free approach to multiscale computation

synchronization and desynchronization of self sustained oscillators by common noise we consider the effect of external noise on the dynamics of limit cycle oscillators the lyapunov exponent becomes negative under influence of small white noise what means synchronization of two or more identical systems subject to common noise we analytically study the effect of small nonidentities in the oscillators and in the noise and derive statistical characteristics of deviations from the perfect synchrony large white noise can lead to desynchronization of oscillators provided they are nonisochronous this is demonstrated for the van der pol duffing system

on convergence of higher order schemes for the projective integration method for stiff ordinary differential equations we present a convergence proof for higher order implementations of the projective integration method pi for a class of deterministic multi scale systems in which fast variables quickly settle on a slow manifold the error is shown to contain contributions associated with the length of the microsolver the numerical accuracy of the macrosolver and the distance from the slow manifold caused by the combined effect of micro and macrosolvers respectively we also provide stability conditions for the pi methods under which the fast variables will not diverge from the slow manifold we corroborate our results by numerical simulations

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

increasing the real stability boundary of explicit methods abstract based on the simplest well known integration rules such as the forward euler scheme and the classical runge kutta method an extension is proposed to enlarge the real stability boundary the main characteristic of the resulting schemes is that the computational complexity is hardly increased

predictability in the large an extension of the concept of lyapunov exponent we investigate the predictability problem in dynamical systems with many degrees of freedom and a wide spectrum of temporal scales in particular we study the case of three dimensional turbulence at high reynolds numbers by introducing a finite size lyapunov exponent which measures the growth rate of finite size perturbations for sufficiently small perturbations this quantity coincides with the usual lyapunov exponent when the perturbation is still small compared to large scale fluctuations but large compared to fluctuations at the smallest dynamically active scales the finite size lyapunov exponent is inversely proportional to the square of the perturbation size our results are supported by numerical experiments on shell models we find that intermittency corrections do not change the scaling law of predictability we also discuss the relation between the finite size lyapunov exponent and information entropy

on an integrable discretisation of the ablowitz ladik hierarchy following the general results on the relationships about backlund transformations bts and exact discretisation given in a previous work zullo f backlund transformations and hamiltonian flows j phys a math theor 46 145203 2013 10 1088 1751 8113 46 14 145203 we consider the ablowitz ladik hierarchy and a corresponding family of bts after discussing the boundary conditions we show how to get explicit transformations the hamiltonian properties of the maps and of the discrete flows are examined the conditions on the parameters of the map giving exact discretisations are discussed finally analytical and numerical examples are given

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

a high order asymptotic preserving scheme for kinetic equations using projective integration we investigate a high order fully explicit asymptotic preserving scheme for a kinetic equation with linear relaxation both in the hydrodynamic and diffusive scalings in which a hyperbolic resp parabolic limiting equation exists the scheme first takes a few small inner steps with a simple explicit method such as direct forward euler to damp out the stiff components of the solution and estimate the time derivative of the slow components these estimated time derivatives are then used in an outer runge kutta method of arbitrary order we show that with an appropriate choice of inner step size the time step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation moreover the number of inner time steps is also independent of the scaling parameter we analyze stability and consistency and illustrate with numerical results

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

mean field theory for skewed height profiles in kpz growth processes we propose a mean field theory for interfaces growing according to the kardar parisi zhang kpz equation in 1 1 dimensions the mean field equations are formulated in terms of densities at different heights taking surface tension and the influence of the nonlinear term in the kpz equation into account although spatial correlations are neglected the mean field equations still reflect the spatial dimensionality of the system in the special case of edwards wilkinson growth our mean field theory correctly reproduces all features in the presence of a nonlinear term one observes a crossover to a kpz like behaviour with the correct dynamical exponent z 3 2 in particular we compute the skewed interface profile during roughening and we study the influence of a co moving reflecting wall which has been discussed recently in the context of nonequilibrium wetting and synchronization transitions also here the mean field approximation reproduces all qualitative features of the full kpz equation although with different values of the surface exponents

strong chaos without the butterfly effect in dynamical systems with feedback we discuss the predictability of a system that drives a chaotic system with a positive lyapunov exponent in the absence of feedback the driver is regular and fully predictable with a small feedback of strength the state of the driver can be predicted up to a time diverging with a power of although the total system is strongly chaotic the exponential amplification of the uncertainty on the initial conditions of the driver coexists with very long predictability times as illustrated in a model of coupled maps and of three point vortices in a disc

constructive effects of noise in homoclinic chaotic systems many chaotic oscillators have coherent phase dynamics but strong fluctuations in the amplitudes at variance with such a behavior homoclinic chaos is characterized by quite regular spikes but strong fluctuation in their time intervals due to the chaotic recurrence to a saddle point we study influences of noise on homoclinic chaos we demonstrate both numerically and experimentally on a co2 laser various constructive effects of noise including coherence resonance noise induced synchronization in uncoupled systems and noiseenhanced phase synchronization deterministic resonance with respect to signal frequency and stochastic resonance versus noise intensity in response to weak signals the peculiar sensitivity of the system along the weak unstable manifold of the saddle point underlines the unified mechanism of these nontrivial and constructive noise induced phenomena

conditions for common noise induced synchronization in time delay systems we studied synchronization induced by a common external noise in scalar time delay systems we have found a set of sufficient conditions for the synchronization this set of conditions shows that the synchronization occurs in a wide class of time delay systems numerical evidence for the analytically obtained conditions is also presented

projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum we show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters one containing the stiff or fast components and one containing the slow components these methods have large average step sizes relative to the fast components conventional implicit methods involve the solution of nonlinear equations at each step which for large problems requires significant communication between processors on a multiprocessor machine for such problems the methods proposed here have significant potential for speed improvement

geodesic theory of transport barriers in two dimensional flows abstract we introduce a new approach to locating key material transport barriers in two dimensional non autonomous dynamical systems such as unsteady planar fluid flows seeking transport barriers as minimally stretching material lines we obtain that such barriers must be shadowed by minimal geodesics under the riemannian metric induced by the cauchy green strain tensor as a result snapshots of transport barriers can be explicitly computed as trajectories of ordinary differential equations using this approach we locate hyperbolic barriers generalized stable and unstable manifolds elliptic barriers generalized kam curves and parabolic barriers generalized shear jets in temporally aperiodic flows defined over a finite time interval our approach also yields a metric geodesic deviation that determines the minimal computational time scale needed for a robust numerical identification of generalized lagrangian coherent structures lcss as we show an extension of our transport barrier theory to non euclidean flow domains such as a sphere follows directly we illustrate our main results by computing key transport barriers in a chaotic advection map and in a geophysical model flow with chaotic time dependence

generalized lyapunov exponents corresponding to higher derivatives abstract we generalize the notion of lyapunov exponents to higher order derivatives for fixed points and periodic orbits we derive a relationship for the higher order lyapunov spectrum in terms of the usual first order lyapunov spectrum based on numerical experiments as well as general arguments we conjecture that this relationship also holds for chaotic orbits this work is relevant to a priori error estimates for time series forecasting

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

macroscopic chaos in globally coupled maps abstract we study the coherent dynamics of globally coupled maps showing macroscopic chaos with this term we indicate the hydrodynamical like irregular behavior of some global observables with typical times much longer than the times related to the evolution of the single or microscopic elements of the system the usual lyapunov exponent is not able to capture the essential features of this macroscopic phenomenon using the recently introduced notion of finite size lyapunov exponent we characterize in a consistent way these macroscopic behaviors basically at small values of the perturbation we recover the usual microscopic lyapunov exponent while at larger values a sort of macroscopic lyapunov exponent emerges which can be much smaller than the former a quantitative characterization of the chaotic motion at hydrodynamical level is then possible even in the absence of the explicit equations for the time evolution of the macroscopic observables

relationship between directed percolation and the synchronization transition in spatially extended systems we study the nature of the synchronization transition in spatially extended systems by discussing a simple stochastic model an analytic argument is put forward showing that in the limit of discontinuous processes the transition belongs to the directed percolation dp universality class the analysis is complemented by a detailed investigation of the dependence of the first passage time for the amplitude of the difference field on the adopted threshold we find the existence of a critical threshold separating the regime controlled by linear mechanisms from that controlled by collective phenomena as a result of this analysis we conclude that the synchronization transition belongs to the dp class also in continuous models the conclusions are supported by numerical checks on coupled map lattices too

linear and nonlinear information flow in spatially extended systems infinitesimal and finite amplitude error propagation in spatially extended systems are numerically and theoretically investigated the information transport in these systems can be characterized in terms of the propagation velocity of perturbations vp a linear stability analysis is sufficient to capture all the relevant aspects associated to propagation of infinitesimal disturbances in particular this analysis gives the propagation velocity vl of infinitesimal errors if linear mechanisms prevail on the nonlinear ones vp vl on the contrary if nonlinear effects are predominant finite amplitude disturbances can eventually propagate faster than infinitesimal ones i e vp vl the finite size lyapunov exponent can be successfully employed to discriminate the linear or nonlinear origin of information flow a generalization of the finite size lyapunov exponent to a comoving reference frame allows us to state a marginal stability criterion able to provide vp both in the linear and in the nonlinear case strong analogies are found between information spreading and propagation of fronts connecting steady states in reaction diffusion systems the analysis of the common characteristics of these two phenomena leads to a better understanding of the role played by linear and nonlinear mechanisms for the flow of information in spatially extended systems

on backlund transformations for many body systems using the n particle periodic toda lattice and the relativistic generalization due to ruijsenaars of the elliptic calogero moser system as examples we revise the basic properties of the backlund transformations bts from the hamiltonian point of view the analogy between the bt and baxter s quantum q operator pointed out by pasquier and gaudin is exploited to produce a conjugated variable for the parameter of the bt such that belongs to the spectrum of the lax operator as a consequence the generating function of the composition of n bts gives rise to another canonical transformation separating variables for the model for the toda lattice the dual bt parametrized by is introduced

quantum backlund transformation for the integrable dst model for the integrable case of the discrete self trapping dst model we construct a backlund transformation the dual lax matrix and the corresponding dual backlund transformation are also found and studied the quantum analogue of the backlund transformation q operator is constructed as the trace of a monodromy matrix with an infinite dimensional auxiliary space we present the q operator as an explicit integral operator as well as describing its action on the monomial basis as a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable dst model these eigenfunctions are special functions of the heun class which is beyond the hypergeometric class the integral equations found are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions

the theory of individual based discrete time processes a general theory is developed to study individual based models which are discrete in time we begin by constructing a markov chain model that converges to a one dimensional map in the infinite population limit stochastic fluctuations are hence intrinsic to the system and can induce qualitative changes to the dynamics predicted from the deterministic map from the chapman kolmogorov equation for the discrete time markov process we derive the analogues of the fokker planck equation and the langevin equation which are routinely employed for continuous time processes in particular a stochastic difference equation is derived which accurately reproduces the results found from the markov chain model stochastic corrections to the deterministic map can be quantified by linearizing the fluctuations around the attractor of the map the proposed scheme is tested on stochastic models which have the logistic and ricker maps as their deterministic limits

transition to stochastic synchronization in spatially extended systems spatially extended dynamical systems namely coupled map lattices driven by additive spatio temporal noise are shown to exhibit stochastic synchronization in analogy with low dimensional systems synchronization can be achieved only if the maximum lyapunov exponent becomes negative for sufficiently large noise amplitude moreover noise can suppress also the nonlinear mechanism of information propagation which may be present in the spatially extended system an example of phase transition is observed when both the linear and the nonlinear mechanisms of information production disappear at the same critical value of the noise amplitude the corresponding critical properties cannot be estimated numerically with great accuracy but some general argument suggests that they could be ascribed to the kardar parisi zhang universality class conversely when the nonlinear mechanism prevails on the linear one another type of phase transition to stochastic synchronization occurs this one is shown to belong to the universality class of directed percolation

effect of noise on chaos synchronization in time delayed systems numerical and experimental observations abstract we discuss the constructive role of noise white and colored in chaos synchronization in time delayed systems we first numerically investigate noise induced synchronization nis between two identical uncoupled ikeda and mackey glass systems we find that synchronization occurs above a critical noise intensity that differs for different colors of noise synchronization onset is characterized by the value of the maximum transverse lyapunov exponent we then discuss the enhancement of chaos synchronization between two time delayed systems when they are coupled unidirectionally the effect of parameter mismatch for nis is described in detail we provide experimental evidence of nis for a mackey glass like system in an electronic circuit using different colors of noise an integration scheme for time delayed systems in the presence of additive white and colored noise is discussed

exit times and entropy for dynamical systems stochastic processes and turbulence abstract we present an investigation of entropy h in dynamical systems stochastic processes and turbulence this tool allows for a suitable characterization of dynamical behaviours arising in systems with many different scales of motion particular emphasis is put on a recently proposed approach to the calculation of the entropy based on the exit time statistics the advantages of this method are demonstrated in examples of deterministic diffusive maps intermittent maps stochastic self and multi affine signals and experimental turbulent data concerning turbulence the multifractal formalism applied to the exit time statistics allows us to predict that h 3 for velocity time measurement this power law is independent of the presence of intermittency and has been confirmed by the experimental data analysis moreover we show that the entropy density of a three dimensional velocity field is affected by the correlations induced by the sweeping of large scales

emergence of chaotic behaviour in linearly stable systems strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of lyapunov exponents this regime denoted as stable chaos has been so far mainly characterized by means of numerical studies in this paper we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries in order to clarify how linear stability can coexist with nonlinear instability we construct a suitable stochastic model in the absence of spatial coupling the model does not reveal the existence of any self sustained chaotic phase nevertheless already this simple regime reveals peculiar differences between the behaviour of finite size and that of infinitesimal perturbations a mean field analysis of the spatially extended case provides a semi quantitative description of the onset of irregular behaviour possible relations with directed percolation as a synchronization transition are also briefly discussed

backlund transformations for the elliptic gaudin model and a clebsch system a two parameters family of backlund transformations for the classical elliptic gaudin model is constructed the maps are explicit symplectic preserve the same integrals as for the continuous flows and are a time discretization of each of these flows the transformations can map real variables into real variables sending physical solutions of the equations of motion into physical solutions the starting point of the analysis is the integrability structure of the model it is shown how the analogue transformations for the rational and trigonometric gaudin model are a limiting case of this one an application to a particular case of the clebsch system is given

from synchronization to lyapunov exponents and back the goal of this paper is twofold in the first part we discuss a general approach to determine lyapunov exponents from ensemble rather than time averages the approach passes through the identification of locally stable and unstable manifolds the lyapunov vectors thereby revealing an analogy with generalized synchronization the method is then applied to a periodically forced chaotic oscillator to show that the modulus of the lyapunov exponent associated to the phase dynamics increases quadratically with the coupling strength and it is therefore different from zero already below the onset of phase synchronization the analytical calculations are carried out for a model the generalized special flow that we construct as a simplified version of the periodically forced rossler oscillator

q bosons toda lattice pieri rules and baxter q operator we use the pieri rules to recover the q boson model and show it is equivalent to a discretized version of the relativistic toda chain we identify its semi infinite transfer matrix and the corresponding baxter q matrix with half vertex operators related by an duality transformation we observe that the scalar product of two higher spin xxz wave functions can be expressed with a gaudin determinant

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed