the geometry of mixing in time periodic chaotic flows i asymptotic directionality in physically realizable flows and global invariant properties abstract this paper demonstrates that the geometry and topology of material lines in 2d time periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality this property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the jacobian matrix of the n th iterative of the poincare map associated with the flow asymptotic directionality also determines the geometry of the invariant unstable manifolds which are everywhere tangent to the field of asymptotic eigendirections this fact is used to derive simple non perturbative methods for reconstructing the global invariant manifolds to any desired level of detail the geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a poincare map of the flow defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation the connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the poincare map in a sense that will be made clear in the article the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures two physically realizable systems are analyzed closed cavity flow and flow between eccentric cylinders asymptotic directionality provides evidence of a global self organizing structure characterizing chaotic flow which is analogous to that of anosov diffeomorphisms which turns out to be the basic prototype of mixing systems in this framework we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure different from the ergodic measure associated with the dynamical system generated by the field of asymptotic unstable eigenvectors

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

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

derivatives and constraints in chaotic flows asymptotic behaviour and a numerical method in a smooth flow the leading order response of trajectories to infinitesimal perturbations in their initial conditions is described by the finite time lyapunov exponents and associated characteristic directions of stretching we give a description of the second order response to perturbations in terms of lagrangian derivatives of the exponents and characteristic directions these derivatives are related to generalised lyapunov exponents which describe deformations of phase space elements beyond ellipsoidal when the flow is chaotic care must be taken in evaluating the derivatives because of the exponential discrepancy in scale along the different characteristic directions two matrix decomposition methods are used to isolate the directions of stretching the first appropriate in finding the asymptotic behaviour of the derivatives analytically the second better suited to numerical evaluation the derivatives are shown to satisfy differential constraints that are realised with exponential accuracy in time with a suitable reinterpretation the results of the paper are shown to apply to the eulerian framework as well

computation of orthonormal factors for fundamental solution matrices in this work we introduce and analyze two new techniques for obtaining the q factor in the qr factorization of some or all columns of a fundamental solution matrix y of a linear differential system these techniques are based on elementary householder and givens transformations we implement and compare these new techniques with existing approaches on some examples

distortions of reconstruction for chaotic attractors abstract we consider the problem of quality evaluation for delay reconstruction of chaotic attractors from scalar time series the proposed approach is based upon local linear analysis of the reconstruction mapping it is shown that some linear distortions in principle could be corrected but usually nonlinear distortions make it dificult to efficiently apply this technique the new algorithms of estimating the scale of nonlinear distortion and irrelevance time upper limit for the reconstruction window from a time series are proposed both of them do not require pricise knowledge of minimal embedding dimension for the attractor studied

a comparative study of computation of lyapunov spectra with different algorithms abstract in this paper we make a detailed numerical comparison between three algorithms for the computation of the full lyapunov spectrum as well as the associated eigenvectors of general dynamical systems they are a the standard method b a differential formulation of the standard method and c a new algorithm which does not require rescaling and reorthogonalization we also bring out the relations among these methods moreover we give a simplified formulation of the new algorithm when the dimensionality of the system is 4 we find that there is reasonable agreement among the lyapunov spectra obtained using the three algorithms in most cases however the standard method seems to be the most efficient followed by the new method and the differential version of the standard method in that order as far as the cpu time for the computation of the lyapunov spectra is concerned the new method is hardly suitable for finding the eigenvectors whereas the other procedures give nearly identical numerical results

discrete time minimal control synthesis adaptive algorithm this article proposes a discrete time minimal control synthesis mcs algorithm for a class of single input single output discrete time systems written in controllable canonical form as it happens with the continuous time mcs strategy the algorithm arises from the family of hyperstability based discrete time model reference adaptive controllers introduced in landau y 1979 adaptive control the model reference approach new york marcel dekker inc and is able to ensure tracking of the states of a given reference model with minimal knowledge about the plant the control design shows robustness to parameter uncertainties slow parameter variation and matched disturbances furthermore it is proved that the proposed discrete time mcs algorithm can be used to control discretised continuous time plants with the same performance features contrary to previous discrete time implementations of the continuous time mcs algorithm here a formal proof of asymptotic stability is given for generic n dimension

fluctuations of finite time stability exponents in the standard map and the detection of small islands some statistical properties of finite time stability exponents in the standard map can be estimated analytically the mean exponent averaged over the entire phase space behaves quite differently from all the other cumulants whereas the mean carries information about the strength of the interaction and only indirect information about dynamical correlations the higher cumulants carry information about dynamical correlations and essentially no information about the interaction strength in particular the variance and higher cumulants of the exponent are very sensitive to dynamical correlations and easily detect the presence of very small islands of regular motion via their anomalous time scalings the average of the stability matrix inverse trace is even more sensitive to the presence of small islands and has a seemingly fractal behavior in the standard map parameter the usual accelerator modes and the small islands created through double saddle node bifurcations which come halfway between the positions in interaction strength of the usual accelerator modes are clearly visible in the variance whose time scaling is capable of detecting the presence of islands as small as 0 01 of the phase space we study these quantities with a local approximation to the trace of the stability matrix which significantly simplifies the numerical calculations as well as allows for generalization of these methods to higher dimensions we also discuss the nature of this local approximation in some detail

finite time lyapunov exponent and advection diffusion equation the diffusive transport of a scalar in a flowing fluid is determined by the advection diffusion equation if the flow is chaotic the properties of the transport are determined by the spatial and temporal dependence of the finite time lyapunov exponent the rapid diffusive transport occurs only along the field line of the vector s which defines the stable direction in which neighboring points asymptotically converge the geometry of the s line affects the diffusive transport through the finite time lyapunov exponent which varies smoothly along an s line for example the finite time lyapunov exponent of a 2d conservative system can always be written as t t with s 0 0 and the infinite time lyapunov exponent the spatial dependence of the finite time lyapunov exponent is related to the geometry of the s line through by s 0 0 s 0 diffusion is empeded if the finite time lyapunov exponent becomes small along the field line of the vector it is shown that the finite time lyapunov exponent has a sharp dip where the line encounters a sharp bend hence the barriers for diffusion are located at these sharp bends of the lines which are generic for chaptic flows due to nonhyperbolicity this new class of diffusive transport barrier persists even in systems that are far from integrable

the calculation of lyapunov spectra abstract linearization around the chaotic orbits of dynamical systems offers good prospects for understanding complex systems the naturally available information that can be obtained from such studies are the magnitudes and directions in phase space of stretching and shrinking of an initial small sphere of realizations this information is a property of a time displacement matrix that relates the initial bundle of realizations to its later configuration in this paper we extract information from this matrix through its singular values and the associated orthogonal matrices this given an algebraic foundation to the geometric methods that have been used previously the equations of the orthogonal principal stretching and contraction directions show that asymptotically after a long time they depend only on the location in phase space and not on the orbit as an example we show that these directions sketch out the attractor of the lorenz model

hydrodynamic lyapunov modes in coupled map lattices in this paper numerical and analytical results are presented which indicate that hydrodynamic lyapunov modes hlms also exist for coupled map lattices cmls the dispersion relations for the hlms of cmls are found to fall into two different universality classes it is characterized by lambda approximately k for coupled standard maps and lambda approximately k2 for coupled circle maps the conditions under which hlms can be observed are discussed the role of the hamiltonian structure conservation laws translational invariance and damping is elaborated our results are as follows 1 the hamiltonian structure is not a necessary condition for the existence of hlms 2 conservation laws or the translational invariance alone cannot guarantee the existence of hlms 3 including a damping term in the system of coupled hamiltonian maps does not destroy the hlms the lambda k dispersion relation of hlms however changes to the universality class with lambda k2 under damping in contrast no hlms survives in the system of coupled circle maps under damping 4 an on site potential destroys the hlms 5 the study of zero value lyapunov exponents les and associated lyapunov vectors lvs shows that translational invariance and conservation laws play different roles in the tangent space dynamics 6 the dynamics of the coordinate and momentum parts of lvs in hamiltonian systems are related but different furthermore numerical results for a two dimensional system show that the appearance of hlms in cmls is not restricted to the one dimensional case

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

direct and indirect model reference adaptive control for multivariable piecewise affine systems this article proposes direct and indirect model reference adaptive control strategies for multivariable piecewise affine systems which constitute a popular tool to model hybrid systems and approximate nonlinear systems a chosen reference model which can be linear or also piecewise affine describes the desired closed loop system behavior that is to be achieved by the adaptive controllers for unknown system dynamics each subsystem acquires its own set of control gains which is tuned under careful consideration of the switching behavior in the indirect approach the use of dynamic gain adjustment avoids singularities in the certainty equivalence principle it is shown for both algorithms that the state of the reference model is tracked asymptotically given a common lyapunov function for the switched reference model is available furthermore parameter convergence in both the direct and indirect approach is proven for sufficiently rich reference signals finally both algorithms are evaluated in numerical simulations and their advantages and disadvantages are discussed

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

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

looking at the equilibrium measures in dynamical systems the authors discuss how finite time fluctuations of the chaoticity degree permit one to observe a family of equilibrium measures defined in the thermodynamical formalism for expanding one dimensional maps and for axion a systems by means of generalised lyapunov exponents one can thus calculate the kolmogorov entropies the lyapunov exponents and the hausdorff dimensions for this set of measures they perform such a calculation for the lozi map an almost everywhere hyperbolic diffeomorphism of the plane they stress the heuristic power of their approach which can be extended to more generic non hyperbolic systems in this case they suggest that phase transition phenomena might appear as a consequence of the existence of laminar like regular periods during chaotic evolutions

bifurcations in a model of magnetoconvection abstract bifurcations from oscillatory solutions are studied in a truncated model of two dimensional boussinesq magnetoconvection the fifth order system of nonlinear differential equations is integrated numerically and in certain parameter regimes there is a bifurcation from symmetrical to asymmetrical oscillations followed by a period doubling cascade after the accumulation point there is a semiperiodic cascade leading to chaotic behaviour then the semiperiodic cascade is repeated in reverse followed by a period halving cascade and a bifurcation back to symmetry finally the branch of oscillatory solutions terminates with a symmetrical heteroclinic orbit that connects two saddle foci the interval with aperiodic solutions contains many pairs of narrow windows with asymmetrical or symmetrical periodic solutions each with its own cascade this pattern of behaviour is likely to occur when a periodic orbit approaches a symmetrical pair of saddle foci with eigenvalues that satisfy shil nikov s inequality

a comparative study for the pair creation contact process using series expansions a comparative study between two distinct perturbative series expansions for the pair creation contact process is presented in contrast to the ordinary contact process whose supercritical series expansions provide accurate estimates for its critical behaviour the supercritical approach does not work properly when applied to the pair creation process to circumvent this problem a procedure is introduced in which one site creation is added to the pair creation an alternative method is the generation of subcritical series expansions which works even for the case of the pure pair creation process in contrast to the supercritical case the subcritical series yields estimates that are compatible with numerical simulations

characterization of chaos in random maps abstract we discuss the characterization of chaotic behaviors in random maps both in terms of the lyapunov exponent and of the spectral properties of the perron frobenius operator in particular we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the perron frobenius operator

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

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

turbulence of polymer solutions we investigate high reynolds number turbulence in dilute polymer solutions we show the existence of a critical value of the reynolds number which separates two different regimes in the first regime below the transition the influence of the polymer molecules on the flow is negligible so they can be regarded as passively embedded in the flow this case admits a detailed investigation of the statistics of the polymer elongations the second state is realized when the reynolds number is larger than the critical value this regime is characterized by the strong back reaction of polymers on the flow we establish some properties of the statistics of the stress and velocity in this regime and discuss its relation to the drag reduction phenomenon

phase transition in conservative diffusive contact processes we determine the phase diagrams of conservative diffusive contact processes by means of numerical simulations these models are versions of the ordinary diffusive single creation pair creation and triplet creation contact processes in which the particle number is conserved the transition between the frozen and active states was determined by studying the system in the subcritical regime and the nature of the transition whether continuous or first order was determined by looking at the fractal dimension of the critical cluster for the single creation model the transition remains continuous for any diffusion rate for pair and triplet creation models however the transition becomes first order for high enough diffusion rate our results indicate that in the limit of infinite diffusion rate the jump in density equals 2 3 for the pair creation model and 5 6 for the triplet creation model

stretching and curvature of material lines in chaotic flows as a streak of dye is advected by a chaotic flow it stretches and folds and becomes indistinguishable from a one dimensional idealized material line the variation along a material line of the total stretching experienced by fluid elements is examined and it is found that it can be decomposed into an overall time dependent factor constant along the line and a smooth time independent deviation the stretching is related by a power law to the curvature of the line near sharp bends this is confirmed numerically and motivated by a simple model a conservation law for lyapunov exponents explains deviations from a power law

dynamic stability analysis of viscoelastic plates by lyapunov exponents abstract the dynamic stability of uniform homogeneous viscoelastic plates subjected to periodic in plane loads is investigated the viscoelastic behavior of the plate is given in terms of the boltzmann superposition principle allowing any linear viscoelastic character the proposed method is based on the evaluation of the lyapunov exponents by which the stability or instability can be determined the numerical procedure is straightforward and easy for computer implementation

canonical forms of generic piecewise linear continuous systems this technical note reports a simple but effective strategy to derive a single coordinate transformation of a generic n dimensional piecewise linear continuous vector field that puts both of its subsystems into the same observer or control canonical form the only hypothesis required is that both subsystems are either observable or controllable to derive the result the structure of the system is used in order to prove that it suffices to consider the same transformation for both subsystems the theoretical results are illustrated by means of the colpitts oscillator as a practical case of study

computing lyapunov exponents on a stiefel manifold the problem of numerical computation of a few lyapunov exponents les of finite dimensional dynamical systems is considered from the viewpoint of the differential geometry of stiefel manifolds whether one computes one many or all les of a continuous dynamical system by time integration discrete or continuous orthonormalization is essential for stable numerical integration a differential geometric view of continuous orthogonalization suggests that one restricts the linearized vectorfield to a stiefel manifold however the stiefel manifold is not in general an attracting submanifold of the ambient euclidean space it is a constraint manifold with a weak numerical invariant new numerical algorithms for this problem are then designed which use the fiber bundle characterization of these manifolds this framework leads to a new class of systems for continuous orthogonalization which have strong numerical invariance properties and the strong skew symmetry property numerical integration of these new systems with geometric integrators leads to a new class of numerical methods for computing a few les which preserve orthonormality to machine accuracy this idea is also taken a step further by making the stiefel manifold an attracting invariant manifold in which case standard explicit runge kutta algorithms can be used this leads to an algorithm which requires only marginally more computation than a standard explicit integration without orthogonalization these class of methods should be particularly effective for computing a few les for large dimension dynamical systems the new schemes are straightforward to implement a test case is presented for illustration and an example from dynamical systems is presented where a few les are computed for an array of coupled oscillators

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

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

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

minimal control synthesis adaptive control of continuous bimodal piecewise affine systems this paper is concerned with the derivation of a model reference adaptive control mrac scheme for bimodal piecewise affine pwa continuous systems the algorithm is based on an extension of the minimal control synthesis algorithm originally developed as an mrac for smooth systems the resulting adaptive algorithm is a switched feedback controller able to cope with uncertain continuous pwa systems the proof of stability based on the newly developed passivity theory for hybrid systems is provided and the effectiveness of the new proposed control is shown on a representative example

adaptive control of piecewise linear systems the state tracking case nonlinear controlled systems at multiple operating points are modeled as piecewise linear systems where changes in operating points are modeled as switches between constituent linearized systems this note studies the adaptive state feedback for state tracking control problem for such systems piecewise linear reference model systems are used for generating desired state trajectories and their stability properties are studied adaptive state feedback control schemes are developed and their stability and tracking performance are analyzed and evaluated by simulation examples it is shown that exponential tracking performance can be achieved if the reference input is sufficiently rich and the switches are sufficiently slow

measure theoretical properties of the unstable foliation of two dimensional differentiable area preserving systems this article analyzes in detail the statistical and measure theoretical properties of the nonuniform stationary measure referred to as the w invariant measure associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two dimensional differentiable area preserving systems the analysis is developed starting from a sequence of analytical approximations for the associated density these approximations are related to the properties of the jacobian matrix of the n mathrm th iteration of a poincar e map the w invariant measure plays a fundamental role in the study of transport phenomena in laminar chaotic fluid mixing systems for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids the w invariant measure turns out to be singular and exhibits multifractal features its associated density displays local self similarity in an ensuremath varepsilon neighborhood of hyperbolic periodic points the cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed the only case for which the w invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism the connections with the statistical properties and in particular with the stretching dynamics are addressed in detail

first order transitions in a two dimensional nonequilibrium replicator model abstract a single component two dimensional lattice model of self replicating entities replicators which exhibits a nonequilibrium first order phase transition into a unique absorbing state is introduced and studied through extensive numerical simulations the basic dynamic processes are auto catalytic replication in which groups of at least k 1 adjacent replicators are required to create a new replicator in a neighboring empty cell and spontaneous decay in contrast to the one dimensional versions known as pair creation k 1 and trimer creation k 2 models which present only continuous transitions into the vacuum we find a discontinuous transition for k 2 provided the replication probability is less than some threshold value beyond which the transition becomes continuous for the k 0 and k 1 models the transitions are continuous and belong to the class of universality of directed percolation regardless of the value of the replication probability using the conserved contact process we show that in the discontinuous case the density of replicators exhibits a non monotonic dependence on the decay probability this steady state analysis is complemented by the spreading analysis which allows the calculation of the dynamic critical exponents associated to the asymptotic behavior of the survival probability of a small colony of replicators in an infinite lattice

hybrid model reference adaptive control of piecewise affine systems this paper is concerned with the derivation of a model reference adaptive control mrac scheme for multimodal piecewise affine pwa and piecewise linear systems the control allows the plant to track asymptotically the states of a multimodal piecewise affine or smooth reference model the reference model can be characterized by a number and geometry of phase space regions that can be entirely different from those of the plant numerical simulations on a set of representative examples confirm the theoretical derivation and proof of stability

the number dependence of the maximum lyapunov exponent we examine the number dependence of the largest lyapunov exponent for nonlinear dynamical systems in one two and three cartesian dimensions our results suggest that the largest lyapunov exponent diverges logarithmically with system size independently of the number of cartesian dimensions and the interaction potential

parametric stability of non linearly elastic composite plates by lyapunov exponents abstract the dynamic stability of non linearly elastic composite plates subjected to periodic in plane loading is investigated infinitely wide plates made of resin matrix composite are considered the non linearly elastic behavior of the resin matrix is modelled by the generalized ramberg osgood representation the effect of the matrix non linearity on the overall response of the composite is predicted by the micromechanical method of cells the dynamic stability analysis is performed by evaluating the largest lyapunov exponent the sign of which indicates whether the system is stable or not it is shown that this approach forms a convenient tool for predicting parametric stability of non linear composite structures

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

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

the n site approximation for the triplet creation model the nonequilibrium phase transition of the one dimensional triplet creation model is investigated using the n site approximation scheme we find that the phase diagram in the space of parameters d where is the particle decay probability and d is the diffusion probability exhibits a tricritical point for n 4 however the fitting of the tricritical coordinates t dt using data for 4 n 13 predicts that t becomes negative for n 26 indicating thus that the phase transition is always continuous in the limit n however the large discrepancies between the critical parameters obtained in this limit and those obtained by monte carlo simulations as well as a puzzling non monotonic dependence of these parameters on the order of the approximation n argue for the inadequacy of the n site approximation to study the triplet creation model for computationally feasible values of n