random perturbations of heteroclinic attractors estimates are derived for the mean recurrence time of orbits in the neighborhood of an attracting homoclinic orbit or heteroclinic cycle in an ordinary differential equation subject to small additive random noise the theory presented is illustrated with numerical simulations of several systems including ones invariant under symmetry groups for which such heteroclinic attractors are structurally stable the physical implications of the work presented are briefly discussed

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

the spectrum of eigenparameter dependent discrete sturm liouville equations let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1y n 1 b ny n a ny n email protected n email protected n 0 2 c email protected email protected y 1 b email protected email protected y 0 0 where a n and b n email protected n are complex sequences c i b email protected c i 0 1 and l is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if email protected n exp en d 1 a n b n 0 and email protected email protected 1

volatility clustering and scaling for financial time series due to attractor bubbling a microscopic model of financial markets is considered consisting of many interacting agents spins with global coupling and discrete time heat bath dynamics similar to random ising systems the interactions between agents change randomly in time in the thermodynamic limit the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on off intermittency resembling the empirical financial time series with volatility clustering for a proper choice of the model parameters the probability distributions of returns exhibit power law tails with scaling exponents close to the empirical ones

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter abstract in this paper we establish new oscillation theorems for discrete symplectic eigenvalue problems with general boundary conditions we suppose that the symplectic coefficient matrix of the system and the boundary conditions are nonlinear functions of the spectral parameter and that they satisfy certain natural monotonicity assumptions in our new theory we admit possible oscillations in the coefficients of the symplectic system and the boundary conditions by incorporating their nonconstant rank with respect to the spectral parameter we also prove necessary and sufficient conditions for boundedness of the real part of spectrum of these eigenvalue problems

equivalent continuous and discrete realizations of levy flights a model of one dimensional motion of an inertial particle abstract the paper is devoted to the relationship between the continuous markovian description of levy flights developed previously see e g i a lubashevsky truncated levy flights and generalized cauchy processes eur phys j b 82 2011 189 195 and references therein and their equivalent representation in terms of discrete steps of a wandering particle a certain generalization of continuous time random walks to simplify understanding the key points of the technique to be created our consideration is confined to the one dimensional model for continuous random motion of a particle with inertia its dynamics governed by stochastic self acceleration is described as motion on the phase plane x v comprising the position x and velocity v d x d t of the given particle a notion of random walks inside a certain neighborhood l of the line v 0 the x axis and outside it is developed it enables us to represent a continuous trajectory of particle motion on the plane x v as a collection of the corresponding discrete steps each of these steps matches one complete fragment of the velocity fluctuations originating and terminating at the boundary of l as demonstrated the characteristic length of particle spatial displacement is mainly determined by velocity fluctuations with large amplitude which endows the derived random walks along the x axis with the characteristic properties of levy flights using the developed classification of random trajectories a certain parameter free core stochastic process is constructed its peculiarity is that all the characteristics of levy flights similar to the exponent of the levy scaling law are no more than the parameters of the corresponding transformation from the particle velocity v to the related variable of the core process in this way the previously found validity of the continuous markovian model for all the regimes of levy flights is explained based on the obtained results an efficient single peak approximation is constructed in particular it enables us to calculate the basic characteristics of levy flights using the probabilistic properties of extreme velocity fluctuations and the shape of the most probable trajectory of particle motion within such extreme fluctuations

harmonic oscillator with multiplicative noise nonmonotonic dependence on the strength and the rate of dichotomous noise the output signal of a undamped linear oscillator with a random frequency subject to a periodic force shows nonmonotonic dependence on the strength and the rate of color noise stochastic resonance the effect is absent for white noise

solution of the boundary value problem for optimal escape in continuous stochastic systems and maps topologies of invariant manifolds and optimal trajectories are investigated in stochastic continuous systems and maps a topological method is introduced that simplifies the solution of boundary value problems the activation energy is calculated as a function of a set of parameters characterizing the initial conditions of the escape path the method is applied explicitly to compute the optimal escape path and the activation energy for a variety of dynamical systems and maps

quadratic eigenparameter dependent discrete sturm liouville equations with spectral singularities abstract let us consider the boundary value problem bvp for the discrete sturm liouville equation 0 1 a n 1 y n 1 b n y n a n y n 1 y n n n 0 2 0 1 2 2 y 1 0 1 2 2 y 0 0 where a n and b n n n are complex sequences i i c i 0 1 2 and is a eigenparameter discussing the point spectrum we prove that the bvp 0 1 0 2 has a finite number of eigenvalues and spectral singularities with a finite multiplicities if sup n n exp e n 1 a n b n for some e 0 and 1 2 1

riddled basins of the optimal states in learning dynamical systems abstract a new phenomenon called riddled basin has been drawing attention of nonlinear scientists riddled basin is an extremely complicated domain of attraction usually accompanying invariant sets for dynamical systems driven by chaotic signal e g chaotic synchronizing systems in this paper the optimal state of a back propagation learning of chaotic time series is shown to have a locally riddled basin by a numerical experiment it is also rigorously proved that the optimal state of learning of the tent map by a gradient descent method can be an attractor with a riddled basin

coexistence of infinitely many attractors in a simple flow abstract asymptotic behavior of the orbits of game dynamics systems is analyzed and the existence of infinitely repetitive regular structure lying in the neighborhood of a network of a heteroclinic network with hierarchical structure is uncovered some novel and peculiar dynamical phenomena for example coexistence of infinitely many periodic chaotic attractors fractally interwoven tangled structure of infinitely many basin boundaries that are naturally derived from the structure are also reported

analysis of a finite element pml approximation for the three dimensional time harmonic maxwell problem in our paper math comp 76 2007 597 614 we considered the acoustic and electromagnetic scattering problems in three spatial dimensions in particular we studied a perfectly matched layer pml approximation to an electromagnetic scattering problem we demonstrated both the solvability of the continuous pml approximations and the exponential convergence of the resulting solution to the solution of the original acoustic or electromagnetic problem as the layer increased in this paper we consider finite element approximation of the truncated pml electromagnetic scattering problem specifically we consider approximations which result from the use of nedelec edge finite elements we show that the resulting finite element problem is stable and gives rise to quasi optimal convergence when the mesh size is sufficiently small

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

synchronization and desynchronization under the influence of quasiperiodic forcing we study the influence of quasiperiodic forcing on synchronization and desynchronization using two coupled quasiperiodically forced logistic maps as a paradigm we show that due to the forcing the synchronization region in parameter space shrinks the loss of transverse stability of the synchronized attractors leads to desynchronization two types of such blowout bifurcations are described namely the blowout bifurcations of synchronized quasiperiodic motion on invariant curves and synchronized strange nonchaotic attractors both yielding desynchronized chaotic attractors

simple estimation of synchronization threshold in ensembles of diffusively coupled chaotic systems in this paper we define a simple criterion of the synchronization threshold in the set of coupled chaotic systems flows or maps with diagonal diffusive coupling the condition of chaotic synchronization is determined only by two parameters of order i e the largest lyapunov exponent and the coupling coefficient our approach can be applied for both regular chaotic networks and arrays or lattices of chaotic oscillators with irregular arbitrarily assumed structure of coupling the main idea of the synchronization stability criterion is based on linear analysis of the ensembles of simplest dynamical systems numerical simulations confirm that such a linear approach approximates the synchronization threshold with high precision

on the unfolding of a blowout bifurcation abstract suppose a chaotic attractor a in an invariant subspace loses stability on varying a parameter at the point of loss of stability the most positive lyapunov exponent of the natural measure on a crosses zero at what has been called a blowout bifurcation we introduce the notion of an essential basin of an attractor a this is the set of points x such that accumulation points of the sequence of measures 1 n n 1 k 0 f k x are supported on a we characterise supercritical and subcritical scenarios according to whether the lebesgue measure of the essential basin of a is positive or zero we study a drift diffusion model and a model class of piecewise linear mappings of the plane in the supercritical case we find examples where a lyapunov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace for the mappings we find asymptotically linear scaling of lyapunov exponents average distance from the subspace and basin size on varying a parameter we conjecture that these are general characteristics of blowout bifurcations

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

dynamic numerical models of stock market price from microscopic determinism to macroscopic randomness abstract a variant of threshold dynamics is introduced to model the behaviors of a large assembly of dealers in a stock market although the microscopic evolution dynamics is deterministic the collective behaviors such as market prices show seemingly stochastic fluctuations the statistical properties of market price change can be well approximated by a simple discrete langevin type equation with random amplification the macroscopic stochastic equation is solved both numerically and analytically showing that the market price change generally follow power law distributions in the steady state the reason for the appearance of rapid decay in the distribution tails are discussed

pseudo riddling in chaotic systems abstract previous investigations of riddling have focused in the case where the dynamical invariant set in the symmetric invariant subspace of the system is a chaotic attractor a situation expected to arise commonly in dynamical systems however is that the dynamics in the invariant subspace is in a periodic window in such a case there are both a stable periodic attractor and a coexisting non attracting chaotic saddle in the invariant subspace we show that riddling can still occur in a generalized sense in particular we argue that the basin of the periodic attractor in the invariant subspace contains both an open set and a set of measure zero with riddled holes that belong to the basin of another attractor we call such basin pseudo riddled and we argue that pseudo riddling can be more pervasive than riddling of chaotic attractors because it can occur regardless of whether the chaotic saddle in the invariant subspace is transversely stable or unstable we construct an analyzable model derive a scaling law for pseudo riddling and provide numerical support we also show that the chaotic saddle in the invariant subspace can undergo an enlargement into the full phase space when it becomes transversely unstable

dynamical singularities in adaptive delayed feedback control we demonstrate the dynamical characteristics of adaptive delayed feedback control systems exploiting a discrete time adaptive control method derived for carrying out detailed analysis in particular the systems exhibit singularities such as power law decay of the distribution of transient times and almost zero finite time lyapunov exponents we can explain these results by characterizing such systems as having 1 a jacobian matrix with unity eigenvalue in the whole phase space and 2 parameters approaching a stability boundary proven to be identical with that of nonadaptive delayed feedback control

non extensive random walks stochastic variables whose addition leads to q gaussian distributions gq x 1 q 1 bx 2 1 1 q with b40 1pqo3 and where f x maxff x 0g as limit law for a large number of terms are investigated random walk sequences related to this problem possess a simple additive multiplicative structure commonly found in several contexts thus justifying the ubiquity of those distributions a characterization of the statistical properties of the random walk step lengths is performed moreover a connection with non linear stochastic processes is exhibited q gaussian distributions have special relevance within the framework of non extensive statistical mechanics a generalization of the standard boltzmann gibbs formalism introduced by tsallis over one decade ago therefore the present findings may give insights on the domain of applicability of such generalization r 2005 elsevier b v all rights reserved

fluctuation controlled transient below the on off intermittency transition on off intermittency is an ubiliquitous phenomenon observed beyond the instability of a particular chaotic motion the system below the instability reveals a typical transient chaos it is found that near the instability point the inverse characteristic first passage time satisfies the scaling law with the transverse lyapunov exponent tle and being respectively an initial condition dependent characteristic tle and the fluctuation intensity of local transverse expansion rate numerical simulation for several systems suggests that the scaling function g z is a universal function

crises and hysteresis in coupled logistic maps a new type of crisis is found in coupled logistic maps in this crisis several chaotic attractors undergo cyclic collisions with their basin boundaries and merge into a bigger one whose chaotic dynamical behavior exhibits cyclic transitions among its components with a well defined overall period

characterization of blowout bifurcation by unstable periodic orbits blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor lying in some invariant subspace becomes transversely unstable we establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor we argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced s1063 651x 97 50902 0 pacs number s 05 45 1b recently a novel type of bifurcation has been discovered in chaotic dynamical systems 1 2 this is the so called blowout bifurcation that occurs in dynamical systems with a symmetric invariant subspace let s be the invariant subspace in which there is a chaotic attractor since s is invariant initial conditions in s result in trajectories that remain in s forever whether the chaotic attractor in s is also an attractor in the full phase space depends on the sign of the largest lyapunov exponent l computed for trajectories in s with respect to perturbations in the subspace t which is transverse to s when l is negative s attracts trajectories transversely in the vicinity of s and hence the chaotic attractor in s is an attractor in the full phase space if l is positive trajectories in the neighborhood of s are repelled away from it and consequently the attractor in s is transversely unstable and it is hence not an attractor in the full phase space blowout bifurcation occurs when l changes from negative to positive values there are distinct physical phenomena associated with the blowout bifurcation for example near the bifurcation point where l is negative if there are other attractors in the phase space then typically the basin of the chaotic attractor in s is riddled 3 when l is slightly positive if there are no other attractors in the phase space the dynamics in the transverse subspace t exhibits an extreme type of temporally intermittent bursting behavior the on off intermittency 4 5 recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems 6 in the study of chaos theory it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics 7 9 periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems for example the period doubling bifurcation 10 and the saddlenode bifurcation are bifurcations of periodic orbits catastrophic events in chaotic systems such as crises 11 and basin boundary metamorphoses 12 are triggered by collision of periodic orbits usually of low period embedded in different dynamical invariant sets the birth of wada basin boundaries meaning common boundaries of more than two basins of attraction is caused by a saddle node bifurcation on the basin boundary 13 more recent study indicates that the riddling bifurcation bifurcation that gives birth to a riddled basin is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in s 14 in view of the role of periodic orbits played in these major bifurcations it is desirable to study the blowout bifurcation by periodic orbits in this regard ashwin buescu and stewart have noticed that as a system parameter changes towards the blowout bifurcation point more and more atypical invariant measures become transversely unstable 2 at the bifurcation the natural measure of the chaotic attractor in s becomes unstable in this paper we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace s i n particular we argue that near the bifurcation there exist two groups of periodic orbits s s and s u each having an infinite number of members one transversely stable and another transversely unstable respectively the sign of the largest transverse lyapunov exponent l is determined by the relative weights of s s and s u l is negative positive when s s s u weighs over s u s s a precise definition of the weights will be described in the sequel at the bifurcation the weights of s s and s u are balanced in contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits 10 14 blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits the num

asymptotic and numerical techniques for resonances of thin photonic structures we consider the problem of calculating resonance frequencies and radiative losses of an optical resonator the optical resonator is in the form of a thin membrane with variable dielectric properties this work provides two very different approaches for doing such calculations the first is an asymptotic method which exploits the small thickness and high index of the membrane we derive a limiting resonance problem as the thickness goes to zero and for the case of a simple resonance find a first order correction the limiting problem and the correction are in one less space dimension which can make the approach very efficient convergence estimates are proved for the asymptotics the second approach based on the finite element method with a truncated perfectly matched layer is not restricted to thin structures we demonstrate the use of these methods in numerical calculations which further illustrate their differences the asymptotic method finds resonance by solving a dense but small nonlinear eige

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

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

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

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

border collision bifurcations snap back repellers and chaos the normal form for codimension 1 border collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems is considered in the unstable case we show that in appropriate parameter regions there is a snap back repeller immediately after the bifurcation and hence that the bifurcation creates chaos although the chaotic solutions are repellers they may explain observations and this is illustrated through an example

on riddling and weak attractors abstract we propose general definitions for riddling and partial riddling of a subset v of r m with non zero lebesgue measure and show that these properties are invariant for a large class of dynamical systems we introduce the concept of a weak attractor a weaker notion than a milnor attractor and use this to re examine and classify riddled basins of attractors we find that basins of attraction can be partially riddled but if this is the case then any partial riddling must be evident near the attractor we use these concepts to aid classification of bifurcations of attractors from invariant subspaces in particular our weak attractor is a generalisation of the absorbing area investigated by other authors and we suggest that a transition of a basin to riddling is usually associated with loss of stability of a weak attractor

exact invariant measures how the strength of measure settles the intensity of chaos the aim of this paper is to show how to extract dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures on the one hand we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure then we show how to identify ergodic properties by means of transitions along the phase space via exact measures on the other hand we discuss quantitatively how infinite measures imply maps having subexponential lyapunov instability weakly chaotic as opposed to finite measure ergodic maps which are fully chaotic in addition we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval a demand from this field finally we give a simple proof that infinite measure implies universal mittag leffler statistics of observables rather than narrow distributions typically observed in finite measure ergodic maps

minimal attractors and bifurcations of random dynamical systems we consider attractors for certain types of random dynamical systems these are skew product systems whose base transformations preserve an ergodic invariant measure we discuss definitions of invariant sets attractors and invariant measures for deterministic and random dynamical systems under assumptions that include for example iterated function systems but that exclude stochastic differential equations we demonstrate how random attractors can be seen as examples of milnor attractors for a skew product system we discuss the minimality of these attractors and invariant measures supported by them as a further connection between random dynamical systems and deterministic dynamical systems we show how dynamical or d bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations and we relate the issue of branching at such d bifurcations to branching at blowout bifurcations

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

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

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

extensive numerical investigations on the ergodic properties of two coupled pomeau manneville maps abstract we present extensive numerical investigations on the ergodic properties of two identical pomeau manneville maps interacting on the unit square through a diffusive linear coupling the system exhibits anomalous statistics as expected but with strong deviations from the single intermittent map such differences are characterized by numerical experiments with densities which do not have singularities in the marginal fixed point escape and poincare recurrence time statistics that share a power law decay exponent modified by a clear dimensional scaling while the rate of phase space filling and the convergence of ensembles of lyapunov exponents show a stretched instead of pure exponential behavior in spite of the lack of rigorous results about this system the dependence on both the intermittency and the coupling parameters appears to be smooth paving the way for further analytical development we remark that dynamical exponents appear to be independent of the nonzero coupling strength

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

a simple way to compute the existence region of 1d chaotic attractors in 2d maps abstract we study a new approach to computing the existence region of self synchronized states in the error feedback synchronization scheme for discrete time chaotic systems the overall system is modelled by a triangular 2d map in which the nonlinearity is a symmetric tent map we compute the stability region d c of one dimensional chaotic attractors using the boundaries in the bifurcation diagram of the 1d symmetric tent map also called critical lines we show that with a modest number of these critical lines about 16 the approximation of d c is quite accurate

periodic orbit theory of the blowout bifurcation this paper presents a theory for characterization of the blowout bifurcation by periodic orbits blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric invariant subspace becomes transversely unstable we present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits

scaling laws for noise induced temporal riddling in chaotic systems recent work has considered the situation of riddling where when a chaotic attractor lying in an invariant subspace is transversely stable the basin of the attractor can be riddled with holes that belong to the basin of another attractor the existence of invariant subspace often relies on certain symmetry of the system which is however a nongeneric property as system defects and small random noise can destroy the symmetry this paper addresses the influence of noise on riddling we show that riddling can actually be induced by arbitrarily small noise even in parameter regimes where one expects no riddling in the absence of noise specifically we argue that when there are attractors located off the invariant subspace the basins of these attractors can be temporally riddled even when the chaotic attractor in the invariant subspace is transversely unstable w e investigate universal scaling laws for noise induced temporal riddling our results imply that the phenomenon of riddling is robust and it can be more prevalent than expected before as noise is practically inevitable in physical systems s1063 651x 97 01410 4

synchronization threshold in coupled logistic map lattices abstract we consider regular lattices of coupled chaotic maps whose local dynamics is ruled by the logistic map x 4 x 1 x a usual paradigm of chaotic systems through finite time exponents we scrutinize the lattice dynamics in the vicinity of the threshold of complete synchronization we connect dynamical features such as relaxation to the coherent state and intermittency with the statistics of finite time exponents focusing on the implications of the particular statistics related to the logistic map although numerical examples are given for lattice couplings decaying with distance as a power law our results are expected to be valid for a wider class of schemes coupling logistic maps

the dimension of the maximal measure for a polynomial map let f c c be a polynomial of degree d 2 the most interesting dynamics of f takes place on its julia set j f an invariant compact perfect set which can be defined for example as the closure of the repelling periodic points brolin constructed the equilibrium distribution m on j f an ergodic measure which is also the weak limit as n x of a probability measure distributed equally over the dn nth inverse images of almost any point we shall prove the following

an approximate method for scattering by thin structures scattering of waves by a thin structure is considered in this work the helmholtz equation with variable coefficient models the wave phenomena the scatterer is assumed to have a high index of refraction while at the same time it is very small in one of the dimensions we show that if the index scales as o 1 h where h is the thickness of the scatterer then an approximate solution based on perturbation analysis can be obtained the approximate solution consists of a leading order term plus a corrector each of which solves an integral equation in two dimensions for a three dimensional problem we provide error analysis on the approximation the approximate method can be viewed as an efficient computational approach since it can potentially greatly simplify scattering calculations numerical results provide an assessment of the accuracy of the approximate solution

nonlinear eigenvalue approximation for compact operators in the work of osborn math comput 29 712 725 1975 a general spectral approximation theory was developed for compact operators on a banach space which does not require that the operators be self adjoint and also provides a first order correction term here we extend some of the results of that paper to nonlinear eigenvalue problems we present examples of its application that arise in electromagnetics and numerical analysis

phase coherence in an ensemble of uncoupled limit cycle oscillators receiving common poisson impulses an ensemble of uncoupled limit cycle oscillators receiving common poisson impulses shows a range of nontrivial behavior from synchronization desynchronization to clustering the group behavior that arises in the ensemble can be predicted from the phase response of a single oscillator to a given impulsive perturbation we present a theory based on phase reduction of a jump stochastic process describing a poisson driven limit cycle oscillator and verify the results through numerical simulations and electric circuit experiments we also give a geometrical interpretation of the synchronizing mechanism a perturbative expansion to the stationary phase distribution and the diffusion limit of our jump stochastic model

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

dynamics of coupled nonlinear maps and its application to ecological modeling this paper investigates the behavior of coupled nonlinear dynamical systems we take two such dynamical systems or units couple them together and study the effect of coupling on the dynamics we demonstrate that coupling two chaotic units can indeed stabilize both of them several results describing the global dynamics of the coupled nonlinear system are established using them we show that by and large the presence of coupling appears to increase the orderliness of the coupled system s response producing periodicity synchronicity and quasi symmetry using exponential maps which are commonly used to simulate population dynamics we establish the applicability of our results to population dynamics

periodic orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system chaotic dynamical systems with two or more attractors lying on invariant subspaces may provided certain mathematical conditions are fulfilled exhibit intermingled basins of attraction each basin is riddled with holes belonging to basins of the other attractors in order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest two species competition with extinction we have characterized quantitatively the intermingled basins using periodic orbit theory and scaling laws the latter results agree with a theoretical prediction from a stochastic model and also with an exact result for the scaling exponent we derived for the specific class of models investigated we discuss the consequences of the scaling laws in terms of the predictability of a final state extinction of either species in an ecological experiment

transverse instability and riddled basins in a system of two coupled logistic maps riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space an example is the synchronized motion of two identical chaotic oscillators the paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps we determine the regions in parameter plane where the transverse lyapunov exponent is negative the bifurcation curves for the transverse destabilization of low periodic orbits embedded in the chaotic attractor are obtained and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations it is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin riddling and that the loss of weak stability when the transverse lyapunov exponent becomes positive does not necessarily destroy the basin of attraction instead the symmetry of the synchronized state may break and the attractor may spread into two dimensional phase space

noise induced unstable dimension variability and transition to chaos in random dynamical systems results are reported concerning the transition to chaos in random dynamical systems in particular situations are considered where a periodic attractor coexists with a nonattracting chaotic saddle which can be expected in any periodic window of a nonlinear dynamical system under noise the asymptotic attractor of the system can become chaotic as characterized by the appearance of a positive lyapunov exponent generic features of the transition include the following 1 the noisy chaotic attractor is necessarily nonhyperbolic as there are periodic orbits embedded in it with distinct numbers of unstable directions unstable dimension variability and this nonhyperbolicity develops as soon as the attractor becomes chaotic 2 for systems described by differential equations the unstable dimension variability destroys the neutral direction of the flow in the sense that there is no longer a zero lyapunov exponent after the noisy attractor becomes chaotic and 3 the largest lyapunov exponent becomes positive from zero in a continuous manner and its scaling with the variation of the noise amplitude is algebraic formulas for the scaling exponent are derived in all dimensions numerical support using both low and high dimensional systems is provided

transcritical riddling in a system of coupled maps the transition from fully synchronized behavior to two cluster dynamics is investigated for a system of n globally coupled chaotic oscillators by means of a model of two coupled logistic maps an uneven distribution of oscillators between the two clusters causes an asymmetry to arise in the coupling of the model system while the transverse period doubling bifurcation remains essentially unaffected by this asymmetry the transverse pitchfork bifurcation is turned into a saddle node bifurcation followed by a transcritical riddling bifurcation in which a periodic orbit embedded in the synchronized chaotic state loses its transverse stability we show that the transcritical riddling transition is always hard for this we study the sequence of bifurcations that the asynchronous point cycles produced in the saddle node bifurcation undergo and show how the manifolds of these cycles control the magnitude of asynchronous bursts in the case where the system involves two subpopulations of oscillators with a small mismatch of the parameters the transcritical riddling will be replaced by two subsequent saddle node bifurcations or the saddle cycle involved in the transverse destabilization of the synchronized chaotic state may smoothly shift away from the synchronization manifold in this way the transcritical riddling bifurcation is substituted by a symmetry breaking bifurcation which is accompanied by the destruction of a thin invariant region around the symmetrical chaotic state

statistical properties of the noisy on off intermittency the dependence of the escape time on the deviation from the critical point is investigated for the noisy on off intermittency the power law scaling behaviour is destroyed by the noise in a wide region above the critical point in the general case the width of the region decreases logarithmically with the decrease in the noise amplitude below the critical point the noise induced escape time is sensitive to the statistical properties of the chaotic driving signal universal exponential dependence holds only for the gaussian approximation

classical harmonic oscillator with multiplicative noise abstract the first two moments of an underdamped harmonic oscillator were found for additive and multiplicative noises a detailed analysis is performed of the case of a random damping which supplements the case of a random frequency which has received the most study in both cases the second moments are unstable for the sufficient large noise strength the difference appears in the stability conditions of the first moments which can diverge already for white noise for the random damping and only for color noise for the random frequency the difference between these two types of multiplicative noises shows up also in the response to an external periodic force namely white noise influences the first moment only in the case of a random damping however the second moments in both cases show the non monotonic dependence on the noise strength and correlation rate stochastic resonance in linear systems

on off intermittency in oscillatory media abstract on off intermittency observed in three types of spatially extended dynamical systems is reported this is done by examining the linear stability of spatially synchronized state under a spatially inhomogeneous fluctuation when the system size slightly exceeds the critical size and a single inhomogeneous mode becomes unstable systems exhibit typical on off intermittency whose statistical characteristics such as distribution of burst amplitude fourier spectrum and distribution of laminar duration turn out to be in agreement with those known in the on off intermittency for small degrees of freedom system

basins of attraction in piecewise smooth hamiltonian systems piecewise smooth hamiltonian systems arise in physical and engineering applications for such a system typically an infinite number of quasiperiodic attractors coexist here we use the term attractors to indicate invariant sets to which typically initial conditions approach as a result of the piecewise smoothness of the underlying system these attractors are therefore characteristically different from the attractors in dissipative dynamical systems we find that the basins of attraction of different attractors exhibit a riddledlike feature in that they mix with each other on arbitrarily fine scales this practically prevents prediction of attractors from specific initial conditions and parameters the mechanism leading to the complicated basin structure is found to be characteristically different from those reported previously for similar basin structure in smooth dynamical systems we demonstrate the phenomenon using a class of electronic relaxation oscillators with voltage protection and provide a theoretical explanation

invariant manifolds and collective coordinates we introduce suitable coordinate systems for interacting many body systems with invariant manifolds these are cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold in this sense these coordinates are collective we make a connection to zickendraht s collective coordinates and present certain configurations of few body systems where rotations and vibrations decouple from single particle motion these configurations do not depend on details of the interaction

influence of noise near blowout bifurcation we consider effects of zero mean additive noise on systems that are undergoing supercritical blowout bifurcation i e where a chaotic attractor in an invariant subspace loses transverse stability to a nearby on off intermittent attractor we concentrate on the low noise limit and two statistical properties of the trajectories the variance of the normal component and the mean first crossing time of the invariant subspace before blowout we find that the asymptotic variance scales algebraically with the noise level and exponentially with the lyapunov exponent after blowout it is limited to the nonzero variance of the associated on off intermittent state surprisingly for a large enough lyapunov exponent the effect of added noise can be to decrease rather than increase the variance the mean crossing time becomes infinite at and after the blowout in the limit of small noise after the blowout there is exponential dependence on the lyapunov exponent and algebraic dependence on the noise level the results are obtained using a drift diffusion model of venkataramani et al the results are confirmed in numerical experiments on a smooth mapping we observe that although there are qualitative similarities between bubbling noise driven and on off intermittency dynamics driven these can be differentiated using the statistical properties of the variance of the normal dynamics and the mean crossing time of the invariant subspace in the limit of low noise

spectral properties of non selfadjoint difference operators abstract let l denote the operator generated in l2 z by the difference expression ly n an 1yn 1 bnyn anyn 1 n z 0 1 2 where an n z and bn n z are complex sequences in this paper we investigated the spectrum the spectral singularities and the properties of the principal vectors corresponding to the spectral singularities of l we also studied similar problems for the discrete dirac operator m generated in l2 z c 2 by the system of difference expression y n 1 y n 2 y n y 2 n p n y 1 n y 1 n 1 q n y 2 n n z where y y 1 n y 2 n n z is the forward difference operator i e y i n y i n 1 y i n i 1 2 and pn n z qn n z are complex sequences

synchronizing multiple chaotic maps with a randomized scalar coupling abstract synchronization between chaotic systems has recently been a topic of great interest among physicists and engineers in addition to theoretical results a number of applications in communications and control have also been proposed we have previously shown that identical chaotic maps can under certain conditions be synchronized by a common noise like input this raises the question whether the chaotic output from a map can synchronize other maps to itself in this paper we present results on the synchronization of two identical maps where the output of one map is used to drive both maps we then apply this method to synchronizing identical but internally inhomogeneous populations of chaotic oscillators using a randomly constructed scalar coupling signal which tends to white noise as the number of oscillators increases such a system may have applications in secure communication

distribution of the first return time in fractional brownian motion and its application to the study of on off intermittency herein the term fractional brownian motion is used to refer to a class of random walks with long range correlated steps where the mean square displacement of the walker at large time t is proportional to mathit t 2 mathit h with 0h1 for ordinary brownian motion we obtain h 1 2 let t denote the time at which the random walker starting at the origin first returns to the origin the purpose of this paper is to show that the probability distribution of t scales with t as p t ensuremath sim mathit t mathit h mathrm ensuremath 2 theoretical arguments and numerical simulations are presented to support the result additional issues explored include modification to the power law distribution when the random walk is biased and the application of the result to the characterization of on off intermittency a recently proposed mechanism for bursting

unpredictability of the asymptotic attractors in phase coupled oscillators an array of phase coupled oscillators may exhibit multiple coexisting chaotic and nonchaotic attractors the system of coupled circle maps is such an example we demonstrate that it is common for this type of system to exhibit an extreme type of final state sensitivity in both parameter and phase space numerical computations reveal that there exist substantial regions of the parameter space where arbitrarily small perturbations in parameters or initial conditions can alter the asymptotic attractor of the system completely consequently asymptotic attractors of the system cannot be predicted reliably for specific parameter values and initial conditions

symmetry breaking bifurcation with on off intermittency in chaotic dynamical systems when a dynamical system possesses certain symmetry there can be an invariant subspace in the phase space in the invariant subspace there can be a chaotic attractor as a parameter changes through a critical value the chaotic attractor can lose stability with respect to perturbations transverse to the invariant subspace we show that the loss of the transverse stability can lead to a symmetry breaking bifurcation characterized by lack of the system symmetry in the asymptotic attractor an accompanying physical phenomenon is an extreme type of temporally intermittent bursting behavior the mechanism for this type of symmetry breaking bifurcation is elucidated s1063 651x 96 51505 9

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

power law distributions in random multiplicative processes with non gaussian colored multipliers abstract one class of universal mechanisms that generate power law probability distributions is that of random multiplicative processes in this paper we consider a multiplicative langevin equation driven by non gaussian colored multipliers we analytically derive a formula that relates the power law exponent to the statistics of the multipliers and numerically confirm its validity using multiplicative noise generated by chaotic dynamical systems and by a two valued markov process we also investigate the relationship between our treatment and the large deviation analysis of time series and demonstrate the appearance of log periodic fluctuations superimposed on the power law distribution due to the non gaussian nature of the multipliers

on off intermittency power spectrum and fractal properties of time series abstract some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic this situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators the invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow out value this could result in an extreme form of temporally intermittent bursting called on off intermittency we propose a model that incorporates the universal features of systems that display on off intermittency we study this model both with and without additive noise and we derive scaling results for the power spectral density of the on off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold we then present numerical simulations realizing these results

maximum likelihood estimators for generalized cauchy processes maximum likelihood estimator mle for a generalized cauchy process gcp is studied with the aid of the method of information geometry in statistics our gcp is described by the langevin equation with multiplicative and additive noises the exact expressions of mles are given for the two cases that the two types of noises are uncorrelated and mutually correlated it is shown that the mles for these two gcps are free from divergence even in the parameter region wherein the ordinary moments diverge the mle relations can be regarded as a generalized fluctuation dissipation theorem for the present langevin equation availability of them and of some other higher order statistics is demonstrated theoretically and numerically

discrete chaotic maps obtained by symmetric integration abstract chaotic return maps are widely used to model various dynamical systems such as charged particle movement laser beam dynamic celestial body orbiting and many others return maps are commonly obtained by discretization of continuous equations using the euler cromer operator with the only motivation that it is the simplest symplectic operator recent progress in geometric integration raised considerable interest to symmetric operators due to their ability to preserve some geometric properties of continuous flows this way yielding better agreement between discrete and continuous dynamical systems here we compare symmetric and asymmetric discretization approaches applied to several examples of hamiltonian systems in particular we suggest symmetric modifications of chirikov and henon maps and show explicitly that the implied symmetric integration procedure yields reflectional symmetry in the phase space for verification we show that a smooth even perturbation function used instead of a discontinuous delta pulse provides asymptotically similar results numerical experiments using several statistical methods show that symmetric and asymmetric maps while yielding similar asymptotic behavior often exhibit considerably different statistical properties for intermediate regimes providing smoother transitions that are more reminiscent to those observed in various natural phenomena we believe that the proposed approach may be useful for modeling empirical systems by preserving their keynote physical properties