computing two dimensional poincar maps for hyperchaotic dynamics poincar map pm is one of the felicitous discrete approximation of the continuous dynamics to compute pm the discrete relation s between the successive point of interactions of the trajectories on the suitable poincar section ps are found out these discrete relations act as an amanuensis of the nature of the continuous dynamics in this article we propose a computational scheme to find a hyperchaotic pm hpm from an equivalent three dimensional 3d subsystem of a 4d or higher hyperchaotic model for the experimental purpose a standard four dimensional 4d hyperchaotic lorenz stenflo system hlss and a five dimensional 5d hyperchaotic laser model hlm is considered equivalent 3d subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3d subsystems the quantitative measurement of this comparison is made promising by recurrence quantification analysis rqa various two dimensional 2d poincar mas are computed for several suitable poincar sections for both the systems but only some of them are hyperchaotic in nature the hyperchaotic behavior is verified by positive values of both one dimensional 1d lyapunov exponent le i and 2d lyapunov exponent le ii at the end similarity of the dynamics between the hyperchaotic systems and their 2d hyperchaotic poincar maps hpm has been established through mean recurrence time mrt statistics for both of 4d hlss and 5d hlm and the best approximated discrete dynamics for both the hyperchaotic systems are found out

nonlinear structures in electroencephalogram signals abstract we apply a nonlinear prediction algorithm to investigate the presence of nonlinear structure in electroencephalogram eeg recordings the eeg signal could be modeled as a realization of a nonlinear model plus a residual noise uncorrelated gaussian noise using linear and nonlinear models we analyze the statistical nature of these residual noises in the case of epileptic patients and normal subjects we found that the residual noise presents gaussian distribution for epileptic patients if a nonlinear model is used whereas in the case of normal subjects the residual noise will exhibit a gaussian distribution only if a linear model autoregressive is used these results provide another evidence of the nonlinear character of the epileptic seizure recordings while the normal eeg seems to be better described as linearly correlated noise

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

dynamical complexity detection in short term physiological series using base scale entropy physiological systems generate complex fluctuations in their output signals that reflect the underlying dynamics the base scale entropy method was proposed as a complexity measure to investigate the complexity of time series the advantages of this method are simplicity and extremely fast calculation for very short data sets this method enables analyzing very short nonstationary and noisy data sets we employed this method for short term physiological time series for analysis of heart rate variability signals the results show that the simple and easily calculated measure can effectively detect the complexity dissimilarity of physiological time series in different physiological or pathological states which is convenient for clinical applications

chaos hyperchaos transition the chaos hyperchaos transition occurs when the second lyapunov exponent becomes positive we argue that this transition is mediated by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor bifurcations of unstable periodic orbits occur in the neighborhood of the chaos hyperchaos transition point where we observe unstable variable dimensionality we give evidence that the chaos hyperchaos transition is initiated by i the saddle repeller bifurcation of a particular unstable periodic orbit usually of low period ii the appearance of a repelling node in the saddle node bifurcation after which the chaotic attractor becomes riddled or iii the absorption of the repeller unstable node or focus originally located out of the attractor by the growing attractor

modelling and simulating higher dimensional chaotic data abstract we consider time series data whose estimated lyapunov exponents are positive this data is represented by a probability density function on a suitably fine partition of the phase space an algorithm is presented for constructing a transformation which has a unique absolutely continuous invariant measure whose density function is and which has the same lyapunov exponents as the original time series an example is given

controlling chaos in high dimensions theory and experiment naval surface warfare center carderock division silver spring maryland 20903 received 27 november 1995 the main contribution of this work is the development of a high dimensional chaos control method that iseffective robust against noise and easy to implement in experiment assuming no knowledge of the modelequations the method achieves control by stabilizing a desired unstable periodic orbit with any number ofunstable directions using small time dependent perturbations of a single system parameter speci cally ourmajor results are as follows first we derive explicit control laws for time series produced by discrete maps second we show how to apply this control law to continuous time problems by introducing straightforwardways to extract from a continuous time series a discrete time series that measures the dynamics of somepoincare map of the original system third we illustrate our approach with two examples of high dimensionalordinary differential equations one autonomous and the other periodically driven fourth we present the resulton our successful control of chaos in a high dimensional experimental system demonstrating the viability ofthe method in practical applications pacs number s 05 45 1bi introduction

lyapunov exponents for the miles spherical pendulum equations abstract lyapunov exponents and hence the lyapunov dimension have been evaluated for the equations developed by miles quart appl math 20 1962 21 physica d 11 1984 309 as an approximation to the equations of motion of a spherical pendulum forced at a frequency close to its natural frequency computations have been performed throughout the frequency range in which there are no stable fixed points and a little on either side of this range for various values of the damping parameter the results include strong dependence on the damping of both the probability of finding chaotic motion and the typical lyapunov dimension of such motion the co existence of alternative types of attractor in regions both inside and outside the range with no stable fixed points and the occurrence of metastable chaos the transitions to chaos at either end of a limit cycle window chosen as one that has also been observed in the laboratory have been explored in some detail the results are compared with other computations and with observations with laboratory pendulums and the implications of the results for future laboratory studies are discussed some results are presented for the mathematically related problem of liquid surface waves in a vibrating cylinder

symplectic topology and hamiltonian dynamics on etudie des applications symplectiques non lineaires capacites symplectiques construction d une capacite symplectique problemes de plongement problemes de rigidite

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

non linear dynamic analysis and control of chaos for a two degrees of freedom rigid body with vibrating support abstract the dynamic behaviors of a two degrees of freedom rigid body with vibrating support are studied in this paper both analytical and computational results are employed to obtain the characteristics of the system by using the lyapunov direct method the conditions of stability of the relative equilibrium position can be determined the incremental harmonic balance method ihb is used to find the stable and unstable periodic solutions for the strongly non linear system by applying various numerical results such as phase plane poincare map time history and power spectrum analysis a variety of periodic solutions and the phenomena of the chaotic motion is presented the effects of the changes of parameters in the system can be found in the bifurcation diagrams further the chaotic behavior is verified by using lyapunov exponents the modified interpolated cell mapping method micm is used to study the basins of attraction of periodic attractors and the fractal structure besides additions of a constant torque a periodic torque addition of dither signals delayed feedback control adaptive control and bang bang control are used to control the chaos phenomena effectively

response of nonlinear oscillators with random frequency of excitation revisited periodically driven oscillators of low frequency random excitations are analyzed computer simulation which was carried out for the duffing equation and forced vibrations of a pendulum indicated that in these cases noise has a stabilizing effect computation of lyapunov exponents showed that by adding noise to chaotic motion the largest lyapunov exponent as a rule turns to negative and consequently the chaotic motion is annihilated

invariant random graphs with iid degrees in a general geography let d be a non negative integer valued random variable and let g v e be an infinite transitive finite degree graph continuing the work of deijfen and meester adv appl probab 38 287 298 and deijfen and jonasson electron comm probab 11 336 346 we seek an aut g invariant random graph model with v as vertex set iid degrees distributed as d and finite mean connections i e the sum of the edge lengths in the graph metric of g of a given vertex has finite expectation it is shown that if g has either polynomial growth or rapid growth then such a random graph model exists if and only if double strok e sign d r d 0 all known transitive graphs have either polynomial or rapid growth it is believed that no other growth rates are possible when g has rapid growth the result holds also when the degrees form an arbitrary invariant process a counter example shows that this is not the case when g grows polynomially for this case we provide other quite sharp conditions under which the stronger statement does and does not hold respectively our work simplifies and generalizes the results for g z in 4 and proves e g that with g zd there exists an invariant model with finite mean connections if and only if e d d 1 d when g has exponential growth e g when g is a regular tree the condition becomes double strok e sign d log d

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

hyperchaotic states in the parameter space abstract in this paper we propose a numerical method to characterize hyperchaotic points in the parameter space of continuous time dynamical systems the method considers the second largest lyapunov exponent value as a measure of hyperchaotic motion to construct two dimensional parameter space color plots different levels of hyperchaos in these plots are represented by a continuously changing yellow red scale as an example a particular system modeled by a set of four nonlinear autonomous first order ordinary differential equations is considered practical applications of these plots include by instance walking in the parameter space of hyperchaotic systems along desirable paths

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

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

intrinsic chaos and external noise in population dynamics abstract we address the problem of the relative importance of the intrinsic chaos and the external noise in determining the complexity of population dynamics we use a recently proposed method for studying the complexity of nonlinear random dynamical systems the new measure of complexity is defined in terms of the average number of bits per time unit necessary to specify the sequence generated by the system this measure coincides with the rate of divergence of nearby trajectories under two different realizations of the noise in particular we show that the complexity of a nonlinear time series model constructed from sheep populations comes completely from the environmental variations however in other situations intrinsic chaos can be the crucial factor this method can be applied to many other systems in biology and physics

biomechanics in posture space properties and relevance of principal accelerations for characterizing movement control abstract human movements recorded through kinematic data can be described by means of principal component analysis pca through a small set of variables representing correlated segment movements the pc eigenvectors then form a basis in the associated vector space of postural changes similar to 3d movements the kinematics in this posture space can be quantified through principal positions pps velocities pvs and accelerations pas the pas represent a novel set of variables characterizing neuro muscular control the aim of the current technical note was to i compare the variance explained by pas with the variance explained by pps ii clarify the relationship between pas and segment accelerations and iii compare variability of the first principal acceleration pa1 with the local dynamic stability largest lyapunov exponent lye of the first principal position pp1 a pca was applied on 3d upper body positions collected by an xsens inertial sensor system as nineteen volunteers performed a bimanual repetitive tapping task the main finding revealed that the pp explained variance considerably differed from the pa explained variance indicating that the latter should be considered when reducing the dimensionality in postural movement analysis through a pca further the current study formally established that the acceleration curves obtained from differentiating segment positions and from linear combinations of pas are identical finally a strong correlation r 17 0 92 p

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

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

recurrence plot analysis of irregularly sampled data irregularly sampled time series usually require data preprocessing before a desired time series analysis can be applied we propose an approach for distance measuring of pairs of data points which is directly applicable to irregularly sampled time series in order to apply recurrence plot analysis to irregularly sampled time series we use this approach and detect regime transitions in prototypical models and for an application from palaeoclimatatology this approach might be useful for any method that is based on distance measuring e g correlation sum or lyapunov exponent estimation

delimiting hyperchaotic regions in parameter planes of a 5d continuous time dynamical system different ways to numerically characterize hyperchaotic regions in parameter planes of a 5d continuous time nonlinear dynamical system are utilized in this work the method considers the three largest lyapunov exponents to construct two dimensional parameter planes colorful plots in some these plots different levels of hyperchaos are represented by a continuously changing yellow to red scale while in other different colors mean different number of positive lyapunov exponents

chaos in a variable mass relaxation oscillator model for the leaky tap abstract a variable mass relaxation oscillator model for the behaviour of the leaky tap is numerically investigated different regions of its three dimensional parameter space have been searched and interesting dynamical behaviour has been found the dynamics of the system is thoroughly characterized using projections of phase space trajectories 2 and 3 dimensional time delay plots power spectra lyapunov exponents hausdorff and correlation dimensions and bifurcation diagrams dynamical variables obtained from the model like drop masses velocities and drip intervals are used for reconstructing attractors we found several periodic intermittent and chaotic attractors we conclude that the rich dynamics of the model investigated represents an improvement in describing the experimentally observed behaviour of the actual dripping faucet system

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

a statistical framework for testing chaotic dynamics via lyapunov exponents abstract one common criticism of the existing lyapunov exponents algorithms is the absence of a distributional theory which provides a framework for the statistical hypothesis testing for the calculated lyapunov exponents this paper presents a methodology to calculate the empirical distributions of lyapunov exponents by using a bootstrapping technique the methodology of the paper provides a formal test of chaos under the null hypothesis the numerical examples show that the method works well with small data sets

on the estimation of topological entropy we study a method for estimating the topological entropy of a smooth dynamical system our method is based on estimating the logarithmic growth rates of suitably chosen curves in the system we present two algorithms for this purpose and we analyze each according to its strengths and pitfalls we also contrast these with a method based on the definition of topological entropy using n spanning sets

a benchmark test of accuracy and precision in estimating dynamical systems characteristics from a time series characteristics of dynamical systems are often estimated to describe physiological processes for instance lyapunov exponents have been determined to assess the stability of the cardio vascular system respiration and more recently human gait and posture however the systematic evaluation of the accuracy and precision of these estimates is problematic because the proper values of the characteristics are typically unknown we fill this void with a set of standardized time series with well defined dynamical characteristics that serve as a benchmark estimates ought to match these characteristics at least to good approximation we outline a procedure to employ this generic benchmark test and illustrate its capacity by examining methods for estimating the maximum lyapunov exponent in particular we discuss algorithms by wolf and co workers and by rosenstein and co workers and evaluate their performances as a function of signal length and signal to noise ratio in all scenarios the precision of rosenstein s algorithm was found to be equal to or greater than wolf s algorithm the latter however appeared more accurate if reasonably large signal lengths are available and noise levels are sufficiently low due to its modularity the presented benchmark test can be used to evaluate and tune any estimation method to perform optimally for arbitrary experimental data

analysis adaptive control and circuit simulation of a novel nonlinear finance system in the last three decades a growing interest in developing nonlinear dynamical systems for economic models displaying chaotic behavior has been developed to this direction a novel 3 d nonlinear finance chaotic system consisting of two nonlinearities is presented the dynamical analysis of the proposed system confirms its complex dynamic behavior which is studied by using well known simulation tools of nonlinear theory such as the bifurcation diagram lyapunov exponents and phase portraits also some interesting phenomena related with nonlinear theory are observed such as route to chaos through a period doubling sequence antimonotonicity and crisis phenomena in addition an interesting scheme of adaptive control of finance system s behavior is presented furthermore the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package cadence orcad in order to confirm the feasibility of the theoretical model

localization of flexural waves in a disordered periodic piezoelectric beam abstract localization of bending waves in a disordered periodic piezoelectric beam is studied in this paper the equation of the wave motion for a piezoelectric beam is derived on the assumption of an euler bernoulli beam and the harmonic solution is presented the transfer matrix between two consecutive unit cells in the structures is obtained by using the continuity conditions the expression of the localization factor is given by wolf s algorithm numerical examples are presented and the effects of several disordered parameters on the localization factor are analyzed the results show that piezoelectricity has obvious effects on the passbands and stopbands of the periodic piezoelectric beam the behavior of wave propagation and localization in disordered periodic piezoelectric beams can be altered by tuning different structural parameters

a scaling exponent based detector of chaos in oscillatory circuits abstract in this paper a scaling exponent based approach is proposed to determine the state of chaotic circuits and the scaling exponent is calculated using detrended fluctuation analysis dfa the corresponding detector is designed using the fact that the scaling exponent changes for various states of chaotic circuits simulation examples in this paper are performed for chua s circuit and other chaotic systems and compared with the state of the art in the field the proposed detector outperforms existing techniques in ability to distinguish the chaotic and periodic states in the circuits for relatively high noise

a stable marriage of poisson and lebesgue let e be a discrete set in r d call the elements of e centers the well known voronoi tessellation partitions r d into polyhedral regions of varying sizes by allocating each site of r d to the closest center here we study fair allocations of r d to e in which the regions allocated to different centers have equal volumes we prove that if e is obtained from a translation invariant point process then there is a unique fair allocation which is stable in the sense of the galeshapley marriage problem i e sites and centers both prefer to be allocated as close as possible and an allocation is said to be unstable if some site and center both prefer each other over their current allocations we show that the region allocated to each center is a union of finitely many bounded connected sets however in the case of a poisson process an infinite volume of sites are allocated to centers further away than we prove power law lower bounds on the allocation distance of a typical site it is an open problem to prove any upper bound in d 1

non linear dynamical system theory and primary visual cortical processing abstract the vertebrate brain consists of a large number of neurons each with highly complex non linear dynamics these neurons communicate with each other with a complex nexus of connecting nerve axons the current work examines the dynamical properties of neurons located in cat primary visual cortex from the perspective of non linear dynamical theory the temporal patterns of activation of such neurons achieved with periodic stimuli suggest that the dynamics are relatively simple and may be modeled using a small set of coupled non linear equations predictions are made based as to the patterns of activation to be found in populations of neurons

non linear dynamics and chaos control for an electromagnetic system a non linear mathematical model has been obtained by applying a modified conventional identification technique based on the principle of harmonic balance in this study analytical work is carried out on this identified non linear model by applying the first harmonic approximation solution and the floquet theory the resulting criteria for bifurcations can be used to evaluate the operational range of a system employing such a non linear actuator we also employ the method of lyapunov exponents to show the occurrence of chaotic motion and to verify the above analyses finally various methods such as the state feedback control and injection of dither signal control are used to control chaos effectively

onset of spatiotemporal chaos in a nonlinear system we describe the onset of spatiotemporal chaos in a spatially extended nonlinear dynamical system as a result of the loss of transversal stability of an invariant manifold representing a spatially homogeneous and temporally chaotic state the onset of spatiotemporal chaos is characterized by the switching between spatially homogeneous and nonhomogeneous states with statistical properties of on off intermittency

statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability severe obstruction to shadowing of computer generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability that is when the dimension of the unstable eigenspace changes along a trajectory in the invariant set no true trajectory of reasonable length can be found to exist near any numerically generated trajectory an important quantity characterizing the shadowability of numerical trajectories is the shadowing time which measures for how long a trajectory remains valid this time depends sensitively on initial condition here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors an algebraic scaling for short times and an exponential scaling for long times the exponential behavior depends on system details but the small time algebraic behavior appears to be universal we describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors

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

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

the problem of spurious lyapunov exponents in time series analysis and its solution by covariant lyapunov vectors we briefly recall the methods for the determination of lyapunov exponents from time series data and their limitations one particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system leading to extra so called spurious lyapunov exponents several methods to identify the true ones have been proposed which do not give satisfactory results we show that the geometric information contained in covariant lyapunov vectors can be used to identify the true exponents we illustrate its use and its limitations by applying it to experimental nmr laser data this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

transition to chaos in complex dynamical networks abstract the transition from a non chaotic state to a chaotic state is a commonly concerned issue in the study of coupled dynamical networks in this work we consider a network consisting of nodes that are in non chaotic states with parameters in non chaotic regions before they are coupled together we show that if these non chaotic nodes are linked together through a suitable structural topology positive lyapunov exponents of the coupled network can be generated by choosing a certain uniform coupling strength and the threshold for this coupling strength is determined by the complexity of the network topology moreover we show that topological effects of scale free and random networks which are two basic types of complex network models can be visualized based on their topological sensitivity to random failures and intentional attacks our simulation results on a 1000 node scale free network and a 1000 node random network of the logistic maps have verified that during the transition from non chaotic to chaotic states if the topology is more heterogenous then the coupling strength required to achieve the transition can be decreased

new invariants of open symplectic and contact manifolds the first specifically symplectic invariant of open symplectic manifolds the width was defined by m gromov in gr gromov s width was generalized by h hofer i ekeland e zehnder and a floer see eh hz fh they defined numerous new invariants i e symplectic capacities all these invariants are monotonic with respect to the inclusion and serve as obstructions for symplectic embeddings for contact manifolds so far no invariants are known except overtwisting phenomenon in dimension three see bel ell it turns out that much more rich obstructions for symplectic see 2 2 and what is especially interesting for contact embeddings see 3 2 and 3 5 can be defined if we consider embeddings in a prescribed homotopy class this paper is inspired by a recent discovery by benci and sikorav see bc sil ls and 2 4 1 below recently j c sikorav see si2 independently developed some of the ideas discussed in this paper his interests are exclusively in the symplectic field while i am interested mainly in applications to the contact topology the most interesting result of this paper is theorem 3 5 1 which shows in particular the existence of nonisomorphic open solid tori in the standard contact s3

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

route to high dimensional chaos we present a route to high dimensional chaos that is chaos with more than one positive lyapunov exponent in this route as a system parameter changes a subsystem becomes chaotic through say a cascade of period doubling bifurcations after which the complementary subsystem becomes chaotic leading to an additional positive lyapunov exponent for the whole system a characteristic feature of this route as suggested by numerical evidence is that the second largest lyapunov exponent passes through zero continuously three examples are presented a discrete time map a continuous time flow and a population model for species dispersal in evolutionary ecology

symmetry increasing bifurcation as a predictor of a chaos hyperchaos transition in coupled systems in weakly coupled systems it is possible to observe the coexistence of the chaotic attractors which are located out of the invariant manifold and are not symmetrical in relation to this manifold when the control parameter is changed these attractors can undergo a chaos hyperchaos transition we give numerical evidence that before this transition the coexisting attractors merge together creating an attractor symmetrical with respect to the invariant manifold we argue that the attractors that are not located at the invariant manifold can exhibit dynamical behavior similar to bubbling and on off intermittency previously observed for the attractors located at the invariant manifold and we describe the mechanism of these phenomena

computation of lyapunov spectra effect of interactive noise and application to a chemical oscillator abstract a refined version of the sano sawada algorithm to compute lyapunov spectra has been implemented and tested with several models which include periodic chaotic and hyperchaotic attractors under conditions designed to mimic experimental situations it was found that reliable lyapunov exponents can be obtained for time series of moderate length and point density also in the presence of interactive noise as long as its amplitude does not exceed 1 of the oscillation amplitude the algorithm was then applied to experimental data from the co oxidation on pt 110 clearly showing the occurrence of chaos as well as hyperchaos

cross flow induced chaotic vibrations of heat exchanger tubes impacting on loose supports abstract the chaotic dynamics of heat exchanger tubes impacting on the generally loose baffle plates is studied using an analytic model that involves delay differential equations the critical flow velocity for the local instability of the flexible cylinder near the static equilibrium position is obtained by assuming a harmonic solution in the discretized linearized model and solving the resulting algebraic equations numerical solutions show that with increasing flow beyond the critical the amplitude of motion grows until impacting with the loose support occurs more complex motions then arise leading to chaos for a sufficiently high flow velocity a lyapunov exponent technique is developed for delay differential equations and is applied to the present model showing definitely and unequivocally for the first time for this system that chaotic motions do occur to understand the system behaviour better a one degree of freedom forced negatively damped impact oscillator is studied the response is quite complex and results in chaos in certain parameter regions the analysis is performed by finding periodic solutions and determining their stability and bifurcations with the poincare map technique

comments on nonlinear dynamics of a non ideal duffing rayleigh oscillator numerical and analytical approaches an analytical and numerical investigation into the dynamic interaction between a cantilever beam with nonlinear damping and stiffness behavior modeled by the duffing rayleigh equation and a non ideal motor that is connected to the end of the beam is presented non stationary and steady state responses in the resonance region as well as the passage through resonance behavior when the frequency of the excitation is varied are analyzed the influences of nonlinear stiffness nonlinear damping and the extent of the unbalance in the motor are examined it is found that in this situation so called sommerfeld effects may be observed the increase required by a source operating near the resonance results in a small change in the frequency but there is a large increase in the amplitude of the resultant vibration and the jump phenomenon occurs

tsallis like information measures and the analysis of complex signals abstract a tsallis like information measure that elaborates upon those of the tsallis type is shown to be a rather powerful tool for the analysis of complex non stationary signals as exemplified by the henon map and eegs the ensuing results could be of importance concerning the automatic detection of epileptic spikes

determining the complexity of fh ss sequence by approximate entropy high complexity of frequency hopping fh spreadspectrum ss sequence is of great importance to high security multiple access communication systems for it makes fh ss sequence difficult to be analyzed with the growing development in the design of fh ss sequence in much wider fields the wellknown complexity measuresiquestthe linear complexity lc the linear complexity profile lcp and the k error linear complexity k error lc iquestare widely used but not sufficient to evaluate the complexities of the sequences available such as the cryptographical sequence and the chaotic sequence families in this paper a new complexity metric to evaluate the unpredictability of fh ss sequence based on the approximate entropy apen is proposed in the view of the maximal randomness of the sequences with arbitrary length and the theoretical bounds of the apen are derived from a probabilistic point of view simulations and analysis results show that the proposed apen works effectively to discern the changing complexities of the fh ss sequences with small number of samples which provide superior performance over its candidates

route to hyperchaos in a system of coupled oscillators with multistability this work presents the results of a detailed experimental study into the transition between synchronized low dimensional and unsynchronized high dimensional dynamics using a system of coupled electronic chaotic oscillators novel data analysis techniques have been employed to reveal that a hyperchaotic attractor can arise from the amalgamation of two nonattracting sets these originate from initially multistable low dimensional attractors which experience a smooth transition from low to high dimensional chaotic behavior losing stability through a bubbling bifurcation numerical techniques were also employed to verify and expand on the experimental results giving evidence on the locally unstable invariant sets contained within the globally stable hyperchaotic attractor this particular route to hyperchaos also results in the possibility of phenomena such as unstable dimension variability that can be a major obstruction to shadowing and predictability in chaotic systems

small scale turbulence in the plankton low order deterministic chaos or high order stochasticity abstract almost all regions of the oceans are heavily influenced by the effects of physical processes such as turbulence which in turn influence the distribution and ecology of organisms that occupy these regions there is a real need for additional focus placed on the precise knowledge of both physical and biological processes which is often difficult using basic time series analysis in that way we applied nonlinear analysis techniques to high frequency time series of temperature salinity and phytoplankton concentrations recorded in different hydrodynamical regimes related to tidal forcing in a tidally mixed coastal ecosystem techniques devoted to the identification of low order deterministic chaos cannot find evidence of chaos while a lower dimensionality was identified in low hydrodynamic conditions the results rather suggest stochastic time series with many degrees of freedom no obvious attractor in phase space trajectory lle indistinguishable from zero absence of convergence of the correlation integral we then applied to these data specific multifractal analysis techniques and showed that these time series clearly exhibit high order stochasticity in addition the stochastic structure of purely passive scalars i e temperature and salinity remained invariant while the one of phytoplankton biomass must be regarded as highly structured in time by both hydrodynamic and advective processes

estimating the lyapunov exponent of a chaotic system with nonparametric regression abstract we discuss procedures based on nonparametric regression for estimating the dominant lyapunov exponent 1 from time series data generated by a nonlinear autoregressive system with additive noise for systems with bounded fluctuations 1 0 is the defining feature of chaos thus our procedures can be used to examine time series data for evidence of chaotic dynamics we show that a consistent estimator of the partial derivatives of the autoregression function can be used to obtain a consistent estimator of 1 the rate of convergence we establish is quite slow a better rate of convergence is derived heuristically and supported by simulations simulation results from several implementations one local thin plate splines and three global neural nets radial basis functions and projection pursuit are presented for two deterministic chaotic systems local splines and neural nets yield accurate estimates of the lyapunov exponent however the spline method is sensitive to the choice of the emb

the road to chaos by time asymmetric hebbian learning in recurrent neural networks this letter aims at studying the impact of iterative hebbian learning algorithms on the recurrent neural network s underlying dynamics first an iterative supervised learning algorithm is discussed an essential improvement of this algorithm consists of indexing the attractor information items by means of external stimuli rather than by using only initial conditions as hopfield originally proposed modifying the stimuli mainly results in a change of the entire internal dynamics leading to an enlargement of the set of attractors and potential memory bags the impact of the learning on the network s dynamics is the following the more information to be stored as limit cycle attractors of the neural network the more chaos prevails as the background dynamical regime of the network in fact the background chaos spreads widely and adopts a very unstructured shape similar to white noise next we introduce a new form of supervised learning that is more plausible from a biological point of view the network has to learn to react to an external stimulus by cycling through a sequence that is no longer specified a priori based on its spontaneous dynamics the network decides on its own the dynamical patterns to be associated with the stimuli compared with classical supervised learning huge enhancements in storing capacity and computational cost have been observed moreover this new form of supervised learning by being more respectful of the network intrinsic dynamics maintains much more structure in the obtained chaos it is still possible to observe the traces of the learned attractors in the chaotic regime this complex but still very informative regime is referred to as frustrated chaos

controlling chaos in systems of coupled oscillators abstract we study the dynamical behavior of a one dimensional network of phase oscillators displaying a variety of cluster or traveling wave solutions either stable or unstable according to the type of interaction function for some range of parameters these solutions coexist with a chaotic attractor we analyze a mechanism for controlling chaos and stabilizing the periodic solutions we also analyze the minimal number of controllers needed to stabilize the system and the relation between the location of the controllers and the symmetry of the reference state to be stabilized

tight contact structures on solid tori in this paper we study properties of tight contact structures on solid tori in particular we discuss ways of distinguishing two solicl tori with tight contact structures we also give examples of unusual tight contact structures on solid tori we prove the existence of a 2 valued and a llt 27re valued invaria nt of a closed solid torus we call them the self linking number and the rotation number respectively we then extend these definitions to the case of an open solid torus we show that these invariants exhibit certain monotonicity properties with respect to inclusion we also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic at the same time we discuss various ways of constructing a tight contact structure on a solid torus we then produce examples of solid tori with tight contact structures and calculate self linking and rotation numbers for these tori these examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori at the end we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite sheeted cover of that solid torus is no longer tight this answers negatively a question asked by eliashberg in 1990 we also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure another example o1 unexpected behavior

evaluation of the threshold of stability for the human spine the threshold of stability tos is introduced as a new tool for evaluating spinal stability current methods have evaluated the magnitude of spinal kinematic variability stability diffusion or lyapunov exponents this present method differs by modifying task difficulty to obtain a critical value delineating regions of stability and instability conceptually as task difficulty increases kinematic variability of the system increases and the basin of stability decreases when kinematic variability exceeds the basin of stability stability cannot be maintained over time this concept is first illustrated using a mathematical model from which an effective potential function was calculated to show the relationships among kinematic variability the basin of stability and task difficulty in addition an experiment was performed to evaluate the sensitivity of the tos to changes in postural control using visual feedback as a control variable the tos was found to be more sensitive than the lyapunov exponent to removal of visual feedback suggesting it may have use as a diagnostic indicator e g for low back pain furthermore this new method has an additional advantage in which minimal instrumentation is needed its simplicity sensitivity and low cost suggest that the tos has potential as a diagnostic or prognostic tool in a clinical setting

a new modified hyperchaotic lu system abstract this paper proposes a new four dimensional continuous autonomous hyperchaotic system which is constructed based on a modified lu system by introducing a nonlinear state feedback controller the detailed dynamical behaviors of this hyperchaotic system are further investigated including equlibria and stability various attractors bifurcation analysis and lyapunov exponents spectrum moreover a novel analog circuit diagram is then designed for verification of various attractors

orthosymplectic integration of linear hamiltonian systems the authors describe a continuous orthogonal and symplectic factorization procedure for integrating unstable linear hamiltonian systems the method relies on the development of an orthogonal symplectic change of variables to block triangular hamiltonian form integration is thus carried out within the class of linear hamiltonian systems use of an appropriate timestepping strategy ensures that the symplectic pairing of eigenvalues is automatically preserved for long term integrations as are needed in the calculation of lyapunov exponents the favorable qualitative properties of such a symplectic framework can be expected to yield improved estimates the method is illustrated and compared with other techniques in numerical experiments on the henon heiles and spatially discretized sine gordon equations

two boundary value problems for the ginzburg landau equation abstract two boundary value problems for the ginzburg landau equation are considered extensive numerical calculations have been performed in each case including bifurcation histories spectral analysis poincare sections and hausdorff dimension estimates the approach to the inviscid limit is given detailed treatment in this case universal behavior has been found to exist arguments are presented to account for this behavior

symbolic dynamics of noisy chaos abstract one model of randomness observed in physical systems is that low dimensional deterministic chaotic attractors underly the observations a phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer the amount of information available in observations and just how this information affects predictions of the system s future behavior in an effort to develop such a description we discuss the information theory of highly discretized observations of random behavior metric entropy and topological entropy are well defined invariant measures of such an attractor s level of chaos and are computable using symbolic dynamics real physical systems that display low dimensional dynamics are however inevitably coupled to high dimensional randomless e g thermal noise we investigate the effects of such fluctuations coupled to deterministic chaotic systems in particular the metric entropy s response to the fluctuations we find that the entropy increases with a power law in the noise level and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory we also argue that in addition to the metric entropy there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations i0 the maximum average information obtainable about the initial condition that produces a particular sequence of measurements or symbols

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

the necessity for a time local dimension in systems with time varying attractors abstract we show that a simple non linear system for ordinary differential equations may possess a time varying attractor dimension this indicates that it is infeasible to characterize eeg and meg time series with a single time global dimension we suggest another measure for the description of non stationary attractors

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow