periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

generalized voterlike model on activity driven networks with attractiveness we study the behavior of a generalized consensus dynamics on a temporal network of interactions the activity driven network with attractiveness in this temporal network model agents are endowed with an intrinsic activity a ruling the rate at which they generate connections and an intrinsic attractiveness b modulating the rate at which they receive connections the consensus dynamics considered is a mixed voter and moran dynamics each agent either in state 0 or 1 modifies his or her state when connecting with a peer thus an active agent copies his or her state from the peer with probability p or imposes his or her state to him or her with the complementary probability 1 ensuremath p applying a heterogeneous mean field approach we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1 which we use to evaluate the average time to reach consensus and the exit probability defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state we study a number of particular cases finding an excellent agreement with numerical simulations of the model interestingly we observe a symmetry between voter and moran dynamics in pure activity driven networks and their static integrated counterparts that exemplifies the strong differences that a time varying network can impose on dynamical processes

approach to an irregular time series on the basis of the fractal theory we present a technique to measure the fractal dimension of the set of points t f t forming the graph of a function f defined on the unit interval first we apply it to a fractional brownian function 1 which has a property of self similarity for all scales and we can get the stable and precise fractal dimension this technique is also applied to the observational data of natural phenomena it does not show self similarity all over the scale but has a different self similarity across the characteristic time scale the present method gives us a stable characteristic time scale as well as the fractal dimension

quantifying spatiotemporal chaos in rayleigh benard convection using large scale parallel numerical simulations we explore spatiotemporal chaos in rayleigh benard convection in a cylindrical domain with experimentally relevant boundary conditions we use the variation of the spectrum of lyapunov exponents and the leading order lyapunov vector with system parameters to quantify states of high dimensional chaos in fluid convection we explore the relationship between the time dynamics of the spectrum of lyapunov exponents and the pattern dynamics for chaotic dynamics we find that all of the lyapunov exponents are positively correlated with the leading order lyapunov exponent and we quantify the details of their response to the dynamics of defects the leading order lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased the spectrum of lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high dimensional strange attractor accommodates a range of different chaotic dynamics

a physical interpretation of the he non map abstract the orientation preserving henon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term the conservative dissipative and extremely dissipative logistic limit cases are each considered all can be derived from a single differential equation describing the kicked oscillator system in each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the henon map in the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function each is found to yield a henon like chaotic attractor

fractal dimension and degeneracy of the critical point for iterated maps variations of the three most commonly used definitions of fractal dimension namely the capacity the information dimension and the correlation exponent with the degeneracy of the critical point is studied both numerically and analytically the numerical results agree very well with analytical estimates it is found that these three fractal dimensions show quite different behaviour as the degeneracy of the critical point is varied therefore although they are sometimes used almost interchangeably as measures of fractal dimension they are actually very different concepts

chaotic motions near homoclinic manifolds and resonant tori in quasiperiodic perturbations of planar hamiltonian systems abstract we study chaotic dynamics of nonlinear oscillators with the form of a two frequency quasiperiodic perturbation of a planar hamiltonian system possessing a homoclinic orbit whose interior contains a one parameter family of periodic orbits in the extended phase space the unperturbed system has a three dimensional homoclinic manifold and a one parameter family of invariant 3 tori using melnikov s technique and the second order averaging method we show that chaotic motions may exist near the unperturbed homoclinic manifold and the unperturbed resonant tori these chaotic motions result from transverse intersection between the stable and unstable manifolds of normally hyperbolic invariant tori and are characterized by a generalization of the bernoulli shift we also give an example for the quasiperiodically forced duffing oscillator and demonstrate the existence chaotic motions by numerical simulation

characterization of sign singular measures abstract situations are considered in which a physical quantity varies rapidly in space from positive values to negative values down to some cutoff scale typically the cutoff scale is determined by some diffusive process e g viscosity and as the cutoff scale decreases e g viscosity decreases the physical quantity as a function of the spatial coordinate varies more and more rapidly approaching a generalized function as the cutoff scale approaches zero as one examines such a generalized function on smaller and smaller scale it is shown that a certain suitably defined quantity sensitive to the intensity of spatial oscillations from positive to negative can exhibit an algebraic divergence the exponent of this algebraic divergence with scale size is one characterization of this situation and is called the cancellation exponent in the spirit of the multifractal formalism we introduce a spectrum of cancellation exponents yielding a more complete characterization partition function and thermodynamic formalisms are employed we also discuss how to associate dq spectra with these signed generalized functions and discuss the relationship of dq to the cancellation exponent spectrum illustrative examples are presented

a unified control framework of the non regular dynamics of mechanical oscillators a method for controlling non linear dynamics and chaos based on avoiding homo heteroclinic bifurcations is considered in a unified framework various non linear oscillators both hardening and softening symmetric and asymmetric considered as archetypes of a more general class of single d o f systems are analyzed the melnikov s method is applied to analytically detect the homo heteroclinic bifurcations and the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space the generic character of the optimization problem is highlighted and the problem itself is discussed in detail various control strategies are proposed based on the elimination either of a single bifurcation one side controls or of all bifurcations global control the optimization problems are solved analytically and numerically under various forms by taking into account the physical admissibility of the related optimal excitation and the easiness of implementation in practical applications it is shown that the solutions of one side control problems are always system independent in turn the solutions of global control problems are system independent for the large class of symmetric systems considered in this work although there are other systems whose solutions are system dependent these considerations support the very generic nature of the optimal control

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

update rules and interevent time distributions slow ordering versus no ordering in the voter model we introduce a general methodology of update rules accounting for arbitrary interevent time iet distributions in simulations of interacting agents we consider in particular update rules that depend on the state of the agent so that the update becomes part of the dynamical model as an illustration we consider the voter model in fully connected random and scale free networks with an activation probability inversely proportional to the time since the last action where an action can be an update attempt an exogenous update or a change of state an endogenous update we find that in the thermodynamic limit at variance with standard updates and the exogenous update the system orders slowly for the endogenous update the approach to the absorbing state is characterized by a power law decay of the density of interfaces observing that the mean time to reach the absorbing state might be not well defined the iet distributions resulting from both update schemes show power law tails

degeneracy of the action frequency map a mechanism for homoclinic bifurcation of invariant tori a family of smooth invariant tori of a hamiltonian system can be parameterized by the values of the actions or the frequencies these parameterizations are related by the action frequency map the purpose of this paper is to show that when the action frequency map is degenerate it signals a homoclinic bifurcation remarkably the nonlinear properties of this homoclinic bifurcation to invariant tori are determined by the curvature of the action frequency map a homoclinic angle is also generated which is analogous to a hannay berry phase shift the theory is constructive and so can usefully be combined with computation some implications for quantization and the generation of solitary waves are also discussed

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

fractal distribution of floaters on a fluid surface and the transition to chaos for random maps abstract the long time spatial distribution of particles floating on the surface of a confined fluid whose flow velocity has complicated time dependence is considered it is shown that this distribution can be either a fractal or else can clump at several or one discrete points the transition from the latter type of distribution to the former occurs when the lyapunov exponent characterizing the particle motion passes through zero from negative values to positive values the characteristic features of this type of transition are investigated using random maps it is shown that near the transition there are extremely intermittent temporal fluctuations in the particle cloud and their scaling with a parameter is elucidated

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

convergence of the k 2 entropy for random noises with power law spectra abstract we discuss a counter example to the traditional expectation that stochastic processes lead to a non convergence of the k 2 entropy in computed or measured time series we consider a simple class of colored random noises with power law spectra which have already been shown to generate finite and predictable values of the correlation dimension we show that the k 2 entropy of these random noises converges as well these results imply that the observation of a finite or null value of the k 2 entropy in the analysis of data is not enough to infer that the system is dominated by a deterministic process such as low dimensional chaos

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

applications of singular systems analysis to baroclinic chaos abstract the method of singular systems analysis ssa is applied to the phase portrait analysis of high precision time series of measurements of temperature and total heat transport obtained in a high prandtl number pr 26 fluid contained in a rotating cylindrical annulus subject to a horizontal temperature gradient global ssa is found to be a highly effective means of separating weak high frequency oscillatory components from a signal dominated by strong oscillations at low frequency revealing the possible presence of weak inertia gravity oscillations within a large scale baroclinic wave flow ssa embeddings may fail in practice however for signals modulated in amplitude where the carrier and modulation frequencies differ widely unless a sufficiently long window is used a modification of global ssa with short windows consistent with takens method of delays is proposed and verified using data from a chaotic regime of the rotating annulus which may improve ssa embeddings for such cases local ssa is applied to the estimation of topological dimension in quasi periodic and chaotic flows and produces results consistent with more conventional dimension measures

determining the number of integrals of motion by an adapted correlation dimension method an adapted correlation dimension algorithm is used to numerically determine the number of integrals of motion in a variety of conservative classical systems the method is demonstrated on three sample systems that display various degrees of integrable and chaotic behavior the henon heiles hamiltonian an asymmetric top molecule in an electric field and the planetary system hd128311 two additional applications of the method emerge using the adapted correlation dimension algorithm a to study partial barriers and turnstiles in phase space and b to predict the long time stability of planetary systems using short time data

chaos transition of soliton motion on a microstructured lattice abstract soliton motion on a damped driven one dimensional lattice is proposed the lattice model is made of a one dimensional chain equipped at each point with a rotatory molecule the set of coupled difference differential equations for the molecule rotation is obtained for the discrete system using a semi discrete kink solution for which its center of mass is considered as a collective variable we derive the equation of motion for the kink location we show that the kink location is then governed by the equation of the damped sinusoidally driven pendulum when the first order corrections are dropped by means of numerical simulations carried out directly on the lattice model the complex dynamics of the kink location is investigated and compared to that of the well known forced pendulum for some ranges of the control parameters the numerical simulations exhibit a cascade of odd subharmonic components as well as strange attractors which characterizes the chaotic behavior of the kink motion these results agree fairly well with the main features of the transition to chaos for the damped driven pendulum

the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator abstract using recently developed techniques of wiggins 1988 we study the bifurcation to homoclinic tori in the quasiperiodically forced duffing oscillator we show how homoclinic tori give rise to chaotic dynamics for single degree of freedom quasiperiodically forced oscillators in much the same way as smale horseshoes in the periodically forced case we give a complete description of the bifurcation set in the five dimensional parameter space for the two frequency forced duffing oscillator and then show how the two frequency results can be used to understand the effects of an arbitrary but finite number of frequencies

differentiable generalized synchronization of chaos we consider simple lyapunov exponent based conditions under which the response of a system to a chaotic drive is a ital smooth function of the drive state we call this ital differentiable generalized synchronization dgs when dgs does not hold we quantify the degree of nondifferentiability using the h umlt o lder exponent we also discuss the consequences of dgs and give an illustrative numerical example copyright ital 1997 ital the american physical society

a nonlinear dynamical systems analysis of fricative consonants acoustic waveforms of the strident fricatives s z sh and yog spoken by two native american english speakers are analyzed using modern chaotic analysis techniques fricative data are extracted from both intervocalic and sustained utterances for comparison acoustic waveforms of the vowels a i and u are also analyzed for 44 of the unvoiced fricative tokens in vcv contexts and 59 of the sustained voiced fricatives indications of low dimensional dynamics could be found with the given limitations of stationarity the low dimensional chaotic behavior is exhibited by a correlation dimension d2 ranging between 3 and 7 2 and by positive maximum lyapunov exponents les for the remaining fricatives results suggest that the dimensional complexity therein is greater than the maximum d2 value that could be reliably estimated from the available data about 7 8 for the intervocalic cases and 9 for the sustained cases intervocalic voiced fricatives are excluded from the analysis due to stat

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

difficulties in detecting chaos in a complex system abstract the sequences given by a 7d map are analysed by means of the methods widely used to detect chaos in the real world in order to test their sensitivity to chaotic features of a non linear system determined by comparatively high number of parameters the same diagnostic approaches are applied to the 3d lorenz map for comparison the results show that for some of the sequences yielded from the 7d map the adopted methods are not able to give a straightforward answer to the question if the system is chaotic as in the 3d case since the sequences subject to the analysis are not contaminated by noise and are sufficiently long it could be assumed that such difficulties arise likely due to specific internal features of the more complex system it is found also that an increase from 0 01 to 0 5 of the sampling step determining the sequences obtained from the 7d map masks the chaos in some of them

obstructions and hypersurface sections minimally elliptic singularities we study the obstruction space t 2 for minimally elliptic surface singularities we apply the main lemma of our previous paper 3 which relates t 2 to deformations of hypersurface sections to use this we classify general hypersurface sections of minimally elliptic singularities as in the rational singularity case there is a simple formula for the minimal number of generators for t 2 as a module over the local ring this number is in many cases e g for cusps of hilbert modular surfaces equal to the vector space dimension of t 2

a dynamical meaning of fractal dimension when two atlractors of a dynamical system have a common basin boundary b small changes in initial conditions which lie near b can result in radically different long term behavior of the trajectory a quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary b

extreme phase sensitivity in systems with fractal isochrons abstract sensitivity to initial conditions is usually associated with chaotic dynamics and strange attractors however even systems with quasi periodic dynamics can exhibit it in this context we report on the fractal properties of the isochrons of some continuous time asymptotically periodic systems we define a global measure of phase sensitivity that we call the phase sensitivity coefficient and show that it is an invariant of the system related to the capacity dimension of the isochrons similar results are also obtained with discrete time systems as an illustration of the framework we compute the phase sensitivity coefficient for popular models of bursting neurons suggesting that some elliptic bursting neurons are characterized by isochrons of high fractal dimensions and exhibit a very sensitive unreliable phase response

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

topological dimension and local coordinates from time series data a method for the estimation of the topological dimension of a manifold from time series data is presented it is based on the approximation of the manifold near a point chi by its tangent space at chi the dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition the method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state

equivalence of nonequilibrium ensembles for simple maps abstract in this paper we discuss the equivalence of different approaches to the same transport problems with particular attention to recent works by breyman et al these suggest a singular equivalence between the stationary states of models from gaussian dynamics and those from chaotic scattering theory although the mentioned equivalence may concern but a limited set of the properties of those systems it is very important from a theoretical standpoint we argue that a special form of dissipativity in the evolution equations of the phase space distributions is to be invoked to understand those facts we discuss one application of periodic orbit theory to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained

bifurcations of lattice structures equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third neighbour harmonic interactions are related to orbits in a two or four dimensional space under a nonlinear symplectic mapping in this way a connection is given with dynamical systems with two or three degrees of freedom depending on the parameters the orbits are smooth or stochastic their characterisation in terms of their fixed points and their ljapunov exponents is given the mapping shows a large number of bifurcations of which one can distinguish various types and these are discussed

diagnosis of periodic and chaotic responses in vibratory systems this paper is devoted to the development of a new approach to distinguish between a chaotic and a periodic motion of a dynamical system a criterion namely periodicity ratio is introduced to distinguish a periodic motion from a nonperiodic motion and differentiate chaos from a regular motion by a numerical procedure without plotting any figures a duffing s equation and a nonlinear driven pendulum are analyzed by the periodicity ratio periodic chaotic region diagrams are presented which allow a simultaneous comparison of periodic and chaotic behavior of a system with varying system parameters and a variety of initial conditions

voter model with non poissonian interevent intervals recent analysis of social communications among humans has revealed that the interval between interactions for a pair of individuals and for an individual often follows a long tail distribution we investigate the effect of such a non poissonian nature of human behavior on dynamics of opinion formation we use a variant of the voter model and numerically compare the time to consensus of all the voters with different distributions of interevent intervals and different networks compared with the exponential distribution of interevent intervals i e the standard voter model the power law distribution of interevent intervals slows down consensus on the ring this is because of the memory effect in the power law case the expected time until the next update event on a link is large if the link has not had an update event for a long time on the complete graph the consensus time in the power law case is close to that in the exponential case regular graphs bridge these two results such that the slowing down of the consensus in the power law case as compared to the exponential case is less pronounced as the degree increases

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

measuring the fractal dimension of signals morphological covers and iterative optimization fractals can model many classes of time series data the fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales the covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal s graph in this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods it is theoretically estab lished that for the fractal dimension computation covering one dimensional signals with planar sets is equivalent to mor phologically transforming the signal by one dimensional func tions which reduces the computational complexity from quad ratic in the signal s length to linear then a morphological covering algorithm is developed and applied to discrete time signals synthesized from weierstrass functions fractal inter polation functions and fractional brownian motion further for deterministic parametric fractals depending on a single pa rameter related to their dimension we develop an optimization method that starts from an initial estimate and iteratively con verges to the true fractal dimension by searching in the param eter space and minimizing a distance between the original sig nal and all such signals from the same class experimental results are also provided to demonstrate the good performance of the developed methods

indications of nonlinearity intraindividual specificity and stability of human eeg the unfolding dimension abstract with the help of a parameter which we call the unfolding dimension ww provide a bivariate representation of dynamical eeg analysis applied to human eeg our approach successfully discriminates surrogate data from raw eeg and similarly shows human eeg to be both intraindividually specific and stable over time the heart of our approach is the grassberger procaccia algorithm for the determination of the correlation dimension d 2 within the context of the well known method of time delays to guarantee the reproducibility of results this algorithm as well as an estimate for the k 2 entropy and the determination of f spectra have been integrated into a computer program which encompasses an operator user independent automatic and reproducible specification of both an optimal time delay for calculating the correlation integral as well as an optimal scalar invariant plateau region for the extraction of d 2 and k 2 our embedding protocol applied to mathematical systems is shown to be consistent with findings based on other algorithms in particular with false nearest neighbor considerations and sauer s minimal embedding criteria particular intelligent features of the automation and optimization logic plus the inclusion of estimates of the inherent systematic error make the proposed algorithm especially appropriate for applications to biological psychological data

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

generalized master equations for non poisson dynamics on networks the traditional way of studying temporal networks is to aggregate the dynamics of the edges to create a static weighted network this implicitly assumes that the edges are governed by poisson processes which is not typically the case in empirical temporal networks accordingly we examine the effects of non poisson inter event statistics on the dynamics of edges and we apply the concept of a generalized master equation to the study of continuous time random walks on networks we show that this equation reduces to the standard rate equations when the underlying process is poissonian and that its stationary solution is determined by an effective transition matrix whose leading eigenvector is easy to calculate we conduct numerical simulations and also derive analytical results for the stationary solution under the assumption that all edges have the same waiting time distribution we discuss the implications of our work for dynamical processes on temporal networks and for the construction of network diagnostics that take into account their nontrivial stochastic nature

how to extract lyapunov exponents from short and noisy time series in this correspondence we discuss an algorithm for the extraction of lyapunov spectra from short and noisy time series initially we briefly describe an existing algorithm and include a method for improving the robustness to noise of this algorithm the problem of short time series is then addressed and a novel concatenation technique is presented and combined with the extraction algorithm previously utilized results of applying this extended algorithm to chaotic time series are presented and the effects of noise and data length are discussed

linear filters and non linear systems it has been asserted in the literature that the low pass filtering of time series data may lead to erroneous results when calculating attractor dimensions here we prove that finite order non recursive filters do not have this effect in fact a generic finite order non recursive filter leaves invariant all the quantities that can be estimated by using embedding techniques such as the method of delays

homoclinic motions and chaos in the quasiperiodically forced van der pol duffing oscillator with single well potential we study a two frequency quasiperiodically forced oscillator with single well po tential which reduces to the van der pol and duffing oscillators in certain special cases the unperturbed system without damping and forcing terms has a one parameter family of periodic orbits we concentrate on the dynamics near the unperturbed resonant periodic orbits using the second order averaging method and a version of melnikov s method we show that when double resonance occurs the stable and unstable manifolds of normally hyperbolic invariant tori intersect transversely i e transverse homoclinic motions exist near the unperturbed reso nant periodic orbits in certain parameter regions such homoclinic motions yield chaotic dynamics characterized by a generalization of the bernoulli shift numer ical simulation results are also given to demonstrate the theoretical results

some results on the behavior and estimation of the fractal dimensions of distributions on attractors the strong interest in recent years in analyzing chaotic dynamical systems according to their asymptotic behavior has led to various definitions of fractal dimension and corresponding methods of statistical estimation in this paper we first provide a rigorous mathematical framework for the study of dimension focusing on pointwise dimension x and the generalized renyi dimensionsd q and give a rigorous proof of inequalities first derived by grassberger and procaccia and hentschel and procaccia we then specialize to the problem of statistical estimation of the correlation dimension and information dimension it has been recognized for some time that the error estimates accompanying the usual procedures which generally involve least squares methods and nearest neighbor calculations grossly underestimate the true statistical error involved in least squares analyses of and we identify sources of error not previously discussed in the literature and address the problem of obtaining accurate error estimates we then develop an estimation procedure for which corrects for an important bias term the local measure density and provides confidence intervals for the general applicability of this method is illustrated with various numerical examples

full characterization of a strange attractor chaotic dynamics in low dimensional replicator system abstract two chaotic attractors observed in lotka volterra equations of dimension n 3 are shown to represent two different cross sections of one and the same chaotic regime the strange attractor is studied in the equivalent four dimensional catalytic replicator network analytical expressions are derived for the lyapunov exponents of the flow in the centre of the chaotic regime the strange attractor was characterized by the lyapunov dimension dl 2 06 0 02 and the renyi fractal dimensions the set corresponding to the attractor represents a multifractal its singularity spectrum is computed one route in parameter space leading into the chaotic regime and crossing it was studied in detail it leads to a feigenbaum type cascade of bifurcations before the chaos is fully developed the dynamic system passes an internal crisis through a period two tangent bifurcation then a second sequence of period doubling bifurcations leading to another chaotic regime is observed which eventually ends in a crisis and then after a rather complicated periodic regime the attractor finally disappears trajectories in the intermediate periodic regime show transient chaotic bursts of intermittency type a series of one dimensional maps is derived from a properly chosen poincare cross section which illustrates structural changes in the attractor mutations are included in the catalytic replicator network and the changes in the dynamics observed are compared with the predictions of an approach based on perturbation theory the most striking result is the gradual disappearance of complex dynamics with increasing mutation rates the parameter space spanned by the mutation rate and selected parameters of the replication network is split in regions of complex periodic behaviour near the chaotic regime fractal boundaries of these regions are likely to occur but the data available do not allow definitive conclusions yet

ordering dynamics in the voter model with aging abstract we study theoretically and numerically the voter model with memory dependent dynamics at the mean field level the internal age related to the time an individual spends holding the same state is added to the set of binary states of the population such that the activation probability p i for attempting a change of state has an explicit dependence on the internal age i we derive a closed set of integro differential equations describing the time evolution of the fraction of individuals with a given state and internal age and from it we obtain analytical results characterizing the behavior of the system close to the absorbing states in general different internal age dependent activation probabilities p i have different effects on the dynamics in the case of aging i e p i being a decreasing function of its argument i either the system reaches consensus or it gets trapped in a frozen state depending on whether the limiting value p is zero or positive and on the rate at which p i approaches p moreover when the system reaches consensus the fraction x t of nodes not holding the consensus state may decay exponentially with time or as a power law depending again on the specific details of the functional form of p i for the case of anti aging when the activation probability p i is an increasing function of i the system always reaches a steady state with coexistence of opinions exact conditions for having one or another behavior together with the equations and explicit expressions for the power law exponents are provided

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

an improved estimator of dimension and some comments on providing confidence intervals abstract an estimator of dimension is described that has many advantages over the conventional implementation of the grassberger procaccia method it can in some circumstances provide a confidence interval for the dimension estimate for high dimension sets greater than about 3 5 the dimension estimate has a bias which depends on the structure of the set if one has suitable knowledge of the structure of the set it is possible to make a bias adjustment some numerical calculations are presented which indicate the largest error in the dimension estimate that can be expected the problem of providing confidence intervals for dimension estimates in general is not solved completely but we do take a step away from a meaningless number

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

dimension of interaction dynamics a method allowing one to distinguish interacting from noninteracting systems based on available time series is proposed and investigated some facts concerning generalized renyi dimensions that form the basis of our method are proved we show that one can find the dimension of the part of the attractor of the system connected with interaction between its parts we use our method to distinguish interacting from noninteracting systems on the examples of logistic and henon maps a classification of all possible interaction schemes is given

nearest neighbor analysis of a family of fractal distributions we use a central limit theorem for entropy due to ibragimov to obtain limit theorems for linear normalizations of the log minimum distance when observations are sampled from measures belonging to a family of fractal distributions it is shown that in almost all cases the limit distribution is gaussian with parameters determined in part by the hausdorff dimension associated with the underlying measure exceptions to this rule include absolutely continuous measures which obey the classical extreme value limit laws

chaos for a damped and forced kdv equation abstract the attractor for a damped and forced korteweg de vries equation is studied for large damping it is shown that the attractor is trivial with all solutions converging to a unique fixed point for small damping however the dynamics can be complicated it is investigated numerically how the attractor changes when the damping term decreases in particular a period doubling cascade of periodic solutions is found culminating with a chaotic regime the ratio between consecutive points in this cascade converging to feigenbaum s constant beyond this cascade the frequency spectrum of the solution is continuous and one lyapunov exponent is positive this seems to be the first work to obtain the onset of temporal chaos for this equation

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

stochastic model for heart rate fluctuations a normal human heart rate shows complex fluctuations in time which is natural because the heart rate is controlled by a large number of different feedback control loops these unpredictable fluctuations have been shown to display fractal dynamics long term correlations and 1 f noise these characterizations are statistical and they have been widely studied and used but much less is known about the detailed time evolution dynamics of the heart rate control mechanism here we show that a simple one dimensional langevin type stochastic difference equation can accurately model the heart rate fluctuations in a time scale from minutes to hours the model consists of a deterministic nonlinear part and a stochastic part typical to gaussian noise and both parts can be directly determined from the measured heart rate data studies of 27 healthy subjects reveal that in most cases the deterministic part has a form typically seen in bistable systems there are two stable fixed points and one unstable one

geometrical exponents of contour loops on synthetic multifractal rough surfaces multiplicative hierarchical cascade p model in this paper we study many geometrical properties of contour loops to characterize the morphology of synthetic multifractal rough surfaces which are generated by multiplicative hierarchical cascading processes to this end two different classes of multifractal rough surfaces are numerically simulated as the first group singular measure multifractal rough surfaces are generated by using the p model the smoothened multifractal rough surface then is simulated by convolving the first group with a so called hurst exponent h the generalized multifractal dimension of isoheight lines contours d q correlation exponent of contours x l cumulative distributions of areas and perimeters are calculated for both synthetic multifractal rough surfaces our results show that for both mentioned classes hyperscaling relations for contour loops are the same as that of monofractal systems in contrast to singular measure multifractal rough surfaces h plays a leading role in smoothened multifractal rough surfaces all computed geometrical exponents for the first class depend not only on its hurst exponent but also on the set of p values but in spite of multifractal nature of smoothened surfaces second class the corresponding geometrical exponents are controlled by h the same as what happens for monofractal rough surfaces

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems

on determining the dimension of chaotic flows abstract we describe a method for determining the approximate fractal dimension of an attractor our technique fits linear subspaces of appropriate dimension to sets of points on the attractor the deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques we show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time varying quantity is available for analysis these methods are applied to several dissipative dynamical systems

pseudofractals and the box counting algorithm we show that for sets with the hausdorff besicovitch dimension equal to zero the box counting bc algorithm commonly used to calculate the renyi exponents dq can exhibit perfect scaling suggesting non zero dq the properties of these pathological sets pseudofractals are investigated numerical as well as analytical estimates for dq are obtained a simple indicator is given to distinguish pseudofractals and fractals in practical applications of the bc method histograms made of pseudofractal sets are shown to have pareto tails