fractal dimension of higher dimensional chaotic repellors abstract using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in typical dynamical systems in terms of the lyapunov exponents and the characteristic escape time from the repellor our main example a three dimensional chaotic scattering billiard yields a new structure for its invariant manifolds this system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae

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

an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

varieties of periodic attractor in cellular automata we apply three alternate definitions of attractor to cellular automata examples are given to show that using the different definitions can give different answers to the question does this cellular automaton have a periodic attractor the three definitions are the topological notion of attractor as used by c conley a more measure theoretic version given by j milnor and a variant of milnor s deenition that is based on the concept of the center of attraction of an orbit restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described with any of these definitions a cellular automaton has at most one periodic attractor additionally if conley s definition is used then a periodic attractor must be a fixed point using milnor s definition each point on a periodic attractor must be fixed by all shifts so the number of symbols used is an upper bound on the period whether the actual upper bound is 1 is unknown with the third definition this restriction is removed and examples are given of onedimensional cellular automata on three symbols that have finite attractors of arbitrarily large size with the third definition a finite attractor is not necessarily a single periodic orbit the purpose of this paper is to describe the types of periodic orbits that can be attractors of a cellular automaton we will consider three different definitions of attractor one that is based upon topological dynamics a second that is based upon a mixture of topological and measure theoretic dynamics and a third that is even more measure theoretic in nature examples will be given to show that these three definitions can lead to different answers to the question of the existence of a periodic attractor for cellular automata we begin with an informal discussion of some of the results details and precise definitions are given in later sections a cellular automaton is a type of endomorphism of a certain function space let z x zm s where zm is the integer lattice in rm and s is a finite set called the symbol set a shift on z is a map ct z z of the form vtx n x n t for some t e zm a cellular automaton is a continuous map f z z that commutes with all of these shifts cellular automata have received much attention in recent years one reason for this attention is the fact that especially for m 1 or 2 cellular automata are readily accessible received by the editors july 20 1989 1980 mathematics subject classi fication 1985 revision primary 58f12 58f21 research supported in part by nsf grant dms 880758 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems we investigate mechanisms for the hard bubbling transition in symmetrically coupled one dimensional maps a transition from strong to weak synchronization occurs via a first period doubling or pitchfork transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor the consequence of such transverse bifurcations depends on the existence of an absorbing area controlling the global dynamics in the presence of an absorbing area a subcritical transverse bifurcation is found to give rise to abrupt appearance of transient intermittent bursts with large amplitude for this case any small parameter mismatch that is unavoidable in real systems induces a hard bubbling transition via an interior crisis through a detailed numerical analysis we present explicit examples for the subcritical period doubling and pitchfork transverse bifurcations leading to hard bubbling

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

delayed decision feedback sequence estimation the authors delayed decision feedback sequence estimation ddfse for uncoded pam signals is considered estimates on the performance of the algorithm are given and simulation results are provided for several examples a more general form of ddfse applicable to coded modulation systems is also presented as an example detection of trellis coded qpsk quadrature phase shift keyed signals over intersymbol interference channels is discussed

a least squares model for fuzzy rules of inference abstract in this paper the method of least squares which uses the power function is applied to the fuzzy inference rules these rules in fuzzy logic are known as the generalized modus ponens and the generalized modus tollens the proposed technique gives reasonable evaluations for fuzzy consequences and is simple for computer applications

expectation propagation detection for high order high dimensional mimo systems modern communications systems use multiple input multiple output mimo and high order qam constellations for maximizing spectral efficiency however as the number of antennas and the order of the constellation grow the design of efficient and low complexity mimo receivers possesses big technical challenges for example symbol detection can no longer rely on maximum likelihood detection or sphere decoding methods as their complexity increases exponentially with the number of transmitters receivers in this paper we propose a low complexity high accuracy mimo symbol detector based on the expectation propagation ep algorithm ep allows approximating iteratively at polynomial time the posterior distribution of the transmitted symbols we also show that our ep mimo detector outperforms classic and state of the art solutions reducing the symbol error rate at a reduced computational complexity

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

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

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

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

riddling and chaotic synchronization of coupled piecewise linear lorenz maps abstract we investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise linear lorenz maps riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale which has serious consequences on the predictability of the final state for such a coupled system we found that there are wide parameter intervals for which two piecewise linear lorenz maps exhibit riddled basins globally or locally which indicates that there are riddled basins in coupled lorenz equations as previously suggested by numerical experiments the use of piecewise linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins

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

tree structured expectation propagation for ldpc decoding over bms channels in this paper we put forward the tree structured expectation propagation tep algorithm for decoding block and convolutional low density parity check codes over any binary channel we have already shown that tep improves belief propagation bp over the binary erasure channel bec by imposing marginal constraints over a set of pairs of variables that form a tree or a forest the tep decoder is a message passing algorithm that sequentially builds a tree forest of erased variables to capture additional information disregarded by the standard bp decoder which leads to a noticeable reduction of the error rate for finite length codes in this paper we show how the tep can be extended to any channel specifically to binary memoryless symmetric bms channels we particularly focus on how the tep algorithm can be adapted for any channel model and more importantly how to choose the tree forest to keep the gains observed for block and convolutional ldpc codes over the bec

the boundary constraint method for constructing vortex surface fields we develop a boundary constraint method for constructing the vortex surface field vsf in a three dimensional fluid velocityvorticity field the isosurface of the vsf is a vortex surface consisting of vortex lines which can be used to characterize the evolution of vortical structures in a lagrangian sense the evolution equation with pseudo time is solved under the vsf boundary constraint to obtain a numerical solution of the vsf compared with the existing two time method the boundary constraint method constructs the vsf from a single velocity dataset at a given time instead of a time series of velocity fields starting from a simple condition this improvement significantly increases the applicability of the vsf method and reduces the demanding computational cost and required velocity data size using the boundary constraint method we construct the vsfs in taylorgreen flow and transitional channel flow in addition the uniqueness of the vsf solution is discussed and the convergence of the error in the calculation of vsfs is analyzed

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

some considerations about a least squares model for fuzzy rules of inference abstract in this paper the method of least squares is applied to the fuzzy inference rules we begin studying the conditions in which from a fuzzy set we can build another through the method of least squares then we apply this technique in order to evaluate the conclusions of the generalized modus ponens and the generalized modus tollens we present different theorems and examples that demonstrate the fundamental advantages of the method studied we conclude presenting a method for obtaining the conclusion when working with fuzzy conditional propositions that contain two fuzzy propositions joined by the connective and

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

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

reduced state sequence estimation with set partitioning and decision feedback a reduced state sequence estimator for linear intersymbol interference channels is described the estimator uses a conventional viterbi algorithm with decision feedback to search a reduced state subset trellis that is constructed using set partitioning principles the complexity of maximum likelihood sequence estimation mlse due to the length of the channel memory and the size of the signal set is systematically reduced an error probability analysis shows that a good performance complexity tradeoff can be obtained in particular the results indicate that the required complexity to achieve the performance of mlse is independent of the size of the signal set for large enough signal sets simulation results are provided for two partial response systems a simple technique for quadrature partial response signaling qprs is described that eliminates the quasicatastrophic nature of the ml trellis

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

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

intermingled basins in coupled lorenz systems we consider a system of two identical linearly coupled lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength we verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa we investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system

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

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

tree structure expectation propagation for ldpc decoding over the bec we present the tree structure expectation propagation tree ep algorithm to decode low density parity check ldpc codes over discrete memoryless channels dmcs expectation propagation generalizes belief propagation bp in two ways first it can be used with any exponential family distribution over the cliques in the graph second it can impose additional constraints on the marginal distributions we use this second property to impose pairwise marginal constraints over pairs of variables connected to a check node of the ldpc code s tanner graph thanks to these additional constraints the tree ep marginal estimates for each variable in the graph are more accurate than those provided by bp we also reformulate the tree ep algorithm for the binary erasure channel bec as a peeling type algorithm tep and we show that the algorithm has the same computational complexity as bp and it decodes a higher fraction of errors we describe the tep decoding process by a set of differential equations that represents the expected residual graph evolution as a function of the code parameters the solution of these equations is used to predict the tep decoder performance in both the asymptotic regime and the finite length regimes over the bec while the asymptotic threshold of the tep decoder is the same as the bp decoder for regular and optimized codes we propose a scaling law for finite length ldpc codes which accurately approximates the tep improved performance and facilitates its optimization

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

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

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

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

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

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

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

bayesian equalization for ldpc channel decoding we describe the channel equalization problem and its prior estimate of the channel state information csi as a joint bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder our approach takes into consideration not only the uncertainty due to the noise in the channel but also the uncertainty in the csi estimate however this solution cannot be computed in linear time because it depends on all the transmitted symbols hence we also put forward an approximation for each symbol s posterior using the expectation propagation algorithm which is optimal from the kullback leibler divergence viewpoint and yields an equalization with a complexity identical to the bcjr algorithm we also use a graphical model representation of the full posterior in which the proposed approximation can be readily understood the proposed posterior estimates are more accurate than those computed using the ml estimate for the csi in order to illustrate this point we measure the error rate at the output of a low density parity check decoder which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates for example for qpsk modulation and a channel with three taps we can expect gains over 0 5 db with same computational complexity as the ml receiver

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

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

fractal basins of attraction associated with a damped newton s method an intriguing and unexpected result for students learning numerical analysis is that newton s method applied to the simple polynomial z 3 1 0 in the complex plane leads to intricately interwoven basins of attraction of the roots as an example of an interesting open question that may help to stimulate student interest in numerical analysis we investigate the question of whether a damping method which is designed to increase the likelihood of convergence for newton s method modifies the fractal structure of the basin boundaries the overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting

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

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

expectation propagation as turbo equalizer in isi channels in probabilistic equalization of channels with inter symbol interference the bcjr algorithm and its approximations become intractable for high order modulations even for moderate channel dispersions in this paper we introduce a novel soft equalizer to approximate the symbol a posteriori probabilities app where the expectation propagation ep algorithm is used to provide an accurate estimation this new soft equalizer is presented as a block solution denoted as block ep bep where the structure of the matrices involved is exploited to reduce the complexity order to mathcal o l n 2 i e linear in the length of the channel l and quadratic in the frame length n the solution is presented in complex valued formulation within a turbo equalization scheme this algorithm can be cast as a linear minimum mean squared error lmmse turbo equalization with double feedback architecture where constellations being discrete is a restriction exploited by the ep that provides a first refinement of the app in the experiments included the bep exhibits a robust performance regardless of the channel response with gains in the range 1 5 5 db compared with the lmmse equalization

inaccessibility and undecidability in computation geometry and dynamical systems non self similar sets defined by a decision procedure are numerically investigated by introducing the notion of inaccessibility to ideal decision procedure that is connected with undecidability a halting set of a universal turing machine utm the mandelbrot set and a riddled basin are mainly investigated as non self similar sets with a decision procedure by encoding a symbol sequence to a point in a euclidean space a halting set of a utm is shown to be geometrically represented as a non self similar set having different patterns and different fine structures on arbitrarily small scales the boundary dimension of this set is shown to be equal to the space dimension implying that the ideal decision procedure is inaccessible in the presence of error this property is shown to be invariant under application of fractal code transformations thus a characterization of undecidability is given by the inaccessibility to the ideal decision procedure and its invariance against the code transformations it is also shown that the distribution of halting time of the utm decays with a power law or slower and that this characteristic is also unchanged under code transformation the mandelbrot set is shown to have these features including the invariance against the code transformation in common and is connected with undecidable sets in contrast although a riddled basin as a geometric representation of a certain context free language has the boundary dimension equal to the space dimension and a power law halting time distribution these properties are not invariant against the code transformation thus the riddled basin is ranked as middle between an ordinary fractal and a halting set of a utm or the mandelbrot set last we propose undecidability for analog computation and discuss the appropriate condition for coding 2001 elsevier science

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

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

on the application of factor graphs and the sum product algorithm to isi channels in this paper based on the application of the sum product sp algorithm to factor graphs fgs representing the joint a posteriori probability app of the transmitted symbols we propose new iterative soft input soft output siso detection schemes for intersymbol interference isi channels we have verified by computer simulations that the sp algorithm converges to a good approximation of the exact marginal apps of the transmitted symbols if the fg has girth at least 6 for isi channels whose corresponding fg has girth 4 the application of a stretching technique allows us to obtain an equivalent girth 6 graph for sparse isi channels the proposed algorithms have advantages in terms of complexity over optimal detection schemes based on the bahl cocke jelinek raviv bcjr algorithm they also allow a parallel implementation of the receiver and the possibility of a more efficient complexity reduction the application to joint detection and decoding of low density parity check ldpc codes is also considered and results are shown for some partial response magnetic channels also in these cases we show that the proposed algorithms have a limited performance loss with respect to that can be obtained when the optimal serial bcjr algorithm is used for detection therefore for their parallel implementation they represent a favorable alternative to the modified parallel bcjr algorithm proposed in the literature for the application to magnetic channels

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

detection of symmetry of attractors from observations ii an experiment with s 4 symmetry abstract in part i of this paper we adapt the detectives of barany et al 1993 for symmetries of attractors of symmetric dynamical systems to apply to experiments as an illustration of this theory we consider the dynamics of a system of four forced coupled electronic oscillators with full permutation symmetry s4 on varying parameters in the circuit we obtain attractors with many different symmetry types and detect these using the methods described in part i

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

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

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

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

comparison of fuzzy reasoning methods l a zadeh e h mamdani and m mizumoto et al have proposed methods for fuzzy reasoning in which the antecedent involves a fuzzy conditional proposition if x is a then y is b with a and b being fuzzy concepts mizumoto et al have investigated the properties of their methods in the case of generalized modus ponens this paper deals with the properties of their methods in the case of generalized modus tollens and investigates the other new fuzzy reasoning methods obtained by introducing the implication rules of many valued logic systems finally the properties of syllogism and contrapositive are investigated under each fuzzy reasoning method

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

universality for the parameter mismatching effect on weak synchronization in coupled chaotic systems to examine the universality for the parameter mismatching effect on weak chaotic synchronization we study coupled multidimensional invertible systems such as the coupled henon maps and coupled pendula by generalizing the method proposed in coupled one dimensional 1d noninvertible maps we introduce the parameter sensitivity exponent to measure the degree of the parameter sensitivity of a weakly stable synchronous chaotic attractor in terms of the parameter sensitivity exponents we characterize the effect of the parameter mismatch on the intermittent bursting and the basin riddling occurring in the regime of weak synchronization it is thus found that the scaling exponent for the average characteristic time i e the average interburst time and the average chaotic transient lifetime for both the bubbling and riddling cases is given by the reciprocal of the parameter sensitivity exponent as in the simple system of coupled 1d maps hence the reciprocal relation i e 1 seems to be universal in the sense that it holds in typical coupled chaotic systems of different nature

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

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

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

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

riddling bifurcations in coupled piecewise linear maps abstract a mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described we give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold it is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

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

symmetric functions and exact lyapunov exponents abstract newton s method applied to the elementary symmetric functions of polynomials generates a class of dynamical systems these systems have invariant lines on which the lyapunov exponents can be found analytically thus predicting the exact parameter values for which these structures are chaotic attractors in the sense of milnor and precisely when bifurcations such as blowout destroy their stability often blowout bifurcations lead to a behavior called on off intermittency we also present evidence that the bursting mechanism of on off intermittency frequently occurs in neighborhoods focal points a common feature of maps that have singularities finally because of an embedding property this new class of examples can be extended to construct systems having this bifurcation in higher dimensions

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