adaptive coupling and enhanced synchronization in coupled phase oscillators we study the dynamics of an adaptive coupled array of phase oscillators the adaptive law is designed in such a way that the coupling grows stronger for the pairs which have larger phase incoherence the proposed scheme enhances the synchronization and achieves a more reasonable coupling dynamics for the network of oscillators with different intrinsic frequencies the synchronization speed and the steady state phase difference can be adjusted by the parameters of the adaptive law besides global coupling nearest neighbor ring coupling is also considered to demonstrate the generality of the method

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

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

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

the geometric approach for constructing sinai ruelle bowen measures an important class of physically relevant measures for dynamical systems with hyperbolic behavior is given by sinai ruelle bowen srb measures we survey various techniques for constructing srb measures and studying their properties paying special attention to the geometric push forward approach after describing this approach in the uniformly hyperbolic setting we review recent work that extends it to non uniformly hyperbolic systems

rotational chaos and strange attractors on the 2 torus we construct a 2 torus homeomorphism h homotopic to the identity with an attracting r h bing s pseudocircle c such that the rotation set of h c is not a unique vector the only known examples of such an attractor were birkhoff s attractors arising for dissipative maps with a twist our construction relies heavily on the barge martin method for constructing attractors as inverse limits of graphs

comment on adaptive feedback control algorithm we point out a problem in the parametric estimation method based on the adaptive feedback algorithm proposed by d huang phys rev e 73 066204 2006 both illustratively and analytically furthermore under some hypotheses we prove that it is not the so called chaotic dynamical characteristic but the linear independence between the right hand functions and the estimated parameters of the intrinsic system in its attractor that counts in the availability of this algorithm in estimating the unknown parameters

attractors for graph critical rational maps we call a rational map f graph critical if any critical point either belongs to an invariant finite graph g or has minimal limit set or is non recurrent and has limit set disjoint from g we prove that for any conformal measure either for almost every point of the julia set j f its limit set coincides with j f or for almost every point of j f its limit set coincides with the limit set of a critical point of f

synchronization of spatiotemporal chaos with positive conditional lyapunov exponents computer simulations show that following the scalar driving approach synchronization of spatiotemporal chaos can also be achieved in driven subsystem with positive conditional lyapunov exponents this is a result of the so called extreme trap in the neighborhood of the maximum value the higher order term of difference causes a first order dissipative effect which is not reflected by the conditional lyapunov exponents hence the actual average divergence rate is smaller than the maximum conditional lyapunov number and thus causes the driven system to synchronize even when the conditional lyapunov exponents are positive

the method of upper and lower solutions and interval method for first order differential equations combining the method of upper and lower solutions and lower solutions and the interval method existence of solutions of initial and periodic value problems of first order differential equations is considered

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

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

a simple adaptive feedback scheme for identical synchronizing chaotic systems with uncertain parameters an adaptive feedback control scheme for identical synchronizing two linearly coupled chaotic systems and identifying uncertain parameters is proposed in comparison with the previous methods it inherited the present scheme is more flexible and accessible this is because that the prerequisites on the given chaotic systems are much looser and the adaptive feedback controllers involved are relatively independent moreover the present scheme is a thoroughly amendatory one by which the uncertain parameters can be identified three concrete chaotic systems are employed to show the effectiveness of the presented scheme in addition how the coefficient in the adaptive law affects the convergence rate and the final strength is studied numerically

chaos synchronization and parameter estimation of single degree of freedom oscillators via adaptive control this paper addresses the problem of synchronizing a class of single degree of freedom oscillators with uncertain parameters a modified adaptive control scheme is proposed to achieve globally asymptotic stable synchronization between the master and slave oscillators with arbitrary different initial conditions based on the barbalat s lemma one of the advantages of the method is that it requires only a scalar driving signal which may be easily designed in practical applications on the other hand the method can ensure that the unknown parameters in the slave oscillator would be completely estimated with an adaptive updating law numerical simulations are performed on two examples to verify the analytical results

a realistic minimal middle layer for neural networks abstract to demonstrate some dynamical consequences of making the formal characteristic of neuronal elements and their connectivities more realistic than those of modern neural computer algorithms we study two multiplicatively coupled logistic maps each with the quadratic response characteristics of neurons with autoreceptor mechanisms which iteratively exchange their outputs the dynamic phenomenology of this two neuron system though noninvertible demonstrates characteristics of both automorphisms of the real line and diffeomorphisms of the plane with smooth parametric transitions between them we portray a unified parameter space containing the negative positive and mixed coupled regimes categorize the phase space structures using characteristic exponent inequalities explore representative parametric paths demonstrate generic bifurcation sequences and report number theoretic ordering in parameter space we conjecture that most if not all middle layer like brain systems in contrast to primary sensory and motor information transport systems are dominated by intrinsic dynamics of the sort demonstrated here external input serves as a perturbation of these already ongoing complex systems the intrinsic instabilities of these middle layer dynamics return the unpredictability required by theoretical studies of real brain function to models of neural networks we demonstrate a variety of parameter dependent intrinsically ordered attractor sequences and number theoretic series which emerge from the global dynamics and exemplify the rich dynamical machine language potential for specific coding by brain like information transport and learning algorithms

periodicity versus chaos in one dimensional dynamics we survey recent results in one dimensional dynamics and as an application we derive rigorous basic dynamical facts for two standard models in population dynamics the ricker and the hassell families we also informally discuss the concept of chaos in the context of one dimensional discrete time models first we use the model case of the quadratic family for an informal exposition we then review precise generic results before turning to the population models our focus is on typical asymptotic behavior seen for most initial conditions and for large sets of maps parameter sets corresponding to different types of attractors are described in particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models natural measures asymptotic distributions and their stability properties are considered

remarks on nonlinear adaptive observer design abstract a solution to the problem of state estimation for systems with unknown parameters is to use some on line adaptation of the observer parameters on the basis of various existing results for such adaptive observer designs a unifying adaptive observer form is proposed in this paper this form indeed emphasizes properties allowing some asymptotic state estimation in spite of unknown parameters as well as additional properties which further allow parameter estimation as an example it is shown how an adaptive observer can be designed for a class of state affine systems

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

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

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

some phase transitions in coupled map lattices we study finite coupled map lattices of size d 2 with individual maps 0 1 0 1 and constant diffuse coupling for x 2x mod 1 we give sufficient conditions that the coupled system has a continuum of ergodic components in the case d 2 we determine the number of ergodic components for all coupling strengths if is a mixing tent map close to the transition from mixing to a periodic interval with period 2 the uncoupled system is mixing whereas numerical studies suggest that coupling with a suitable strength breaks up the phase space into domains which are interchanged with period 2 in case d 2 we prove this rigorously

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

failure of parameter identification based on adaptive synchronization techniques in this paper several examples as well as their numerical simulations are provided to show some possible failures of parameter identification based on the so called adaptive synchronization techniques these failures might arise not only when the synchronized orbit produced by the driving system is designed to be either some kind of equilibrium or to be some kind of periodic orbit but also when this orbit is deliberately designed to be chaotic the reason for emergence of these failures is theoretically analyzed in the paper and the boundedness of all trajectories generated by the coupled systems is rigorously proved moreover synchronization techniques are proposed to realize complete synchronization and unknown parameter identification in a class of systems where nonlinear terms are not globally lipschitz in addition unknown parameter identification is studied in coupled systems with time delays

metric attractors for smooth unimodal maps we classify the measure theoretic attractors of general c 3 unimodal maps with quadratic critical points the main ingredient is the decay of geometry

brain behaviour and mathematics are we using the right approaches mathematics are used in biological sciences mostly as a quantifying tool for it is the science of numbers after all there is a long standing interest in the application of mathematical methods and concepts to neuroscience in attempts to decipher brain activity while there has been a very wide use of mathematical physical methodologies less effort has been made to formulate a comprehensive and integrative theory of brain function this review concentrates on recent developments uses and abuses of mathematical formalisms and techniques that are being applied in brain research particularly the current trend of using dynamical system theory to unravel the global collective dynamics of brain activity it is worth emphasising that the theoretician neuroscientist eager to apply mathematical analysis to neuronal recordings has to consider carefully some crucial anatomo physiological assumptions that may not be as accurate as the specific methods require on the other hand the experimentalist neuro physicist with an inclination to implement mathematical thoughts in brain science has to make an effort to comprehend the bases of the theoretical concepts that can be used as frameworks or as analysis methods of brain electrophysiological recordings and to critically inspect the accuracy of the interpretations of the results based on the neurophysiological ground it is hoped that this brief overview of anatomical and physiological presumptions and their relation to theoretical paradigms will help clarify some particular points of interest in current trends in brain science and may provoke further reflections on how certain or uncertain it is to conceptualise brain function based on these theoretical frameworks if the physiological and experimental constraints are not as accurate as the models prescribe

basic control for the viscous moore greitzer partial differential equation the notions of basic controllability and basic control are defined a quadratic optimal control of the linearized viscous moore greitzer equation is presented and it is confirmed that stall is uncontrollable in this model a basic control is constructed for the nonlinear viscous moore greitzer equation which can control both surge and stall some extensions of this construction are discussed numerical simulations of the basic control are presented and its performance is compared to the performance of a backstepping control constructed by banaszuk hauksson and mezic siam j control optim 37 1999 pp 1503 1537 it is shown that the viscous moore greizer equation with throttle control is not basically controllable but under certain conditions adding air injection control will make the equation basically controllable

is every approximate trajectory of some process near an exact trajectory of a nearby process this paper deals with the problem can a noisy orbit be tracked by a real orbit in particular we will study the one parameter family of tent maps and the one parameter family of quadratic maps we writeg for eitherf orf withf x x forx 1 2 andf x 1 x forx 1 2 andf x x 1 x for a given we will say g permits increased parameter shadowing if for each x 0 there exists some 0 and some f 0 such that every f pseudog orbit starting in some invariant interval can be x shadowed by a realg orbit with we show thatg typically permits increased parameter shadowing

on the global structure of invariant regions of flows with asymptotically stable attractors several authors have pointed out the usefulness of shape theory which was introduced by k borsuk in 1968 as a tool in the study of dynamical systems in particular bogatyi and gutsu 4 garay 10 gunther and segal 11 hastings 12 robbin and salamon 16 rogers 17 and tezer 24 have obtained interesting results in dynamical systems using shape theoretical techniques one of the authors of the present article has given another connection between shape and topological dynamics 20 based on the approach to shape developed in 18 and 19 see also 21 for applications of shape theory to the study of uniform attractors most of the mentioned papers study properties of flows although 16 and 24 are devoted to discrete dynamical systems in particular a shape conley index is constructed in 16 for discrete dynamical systems and some shape theoretical ideas are used in 24 to study properties of topological conjugacy of shifts and homotopical shift equivalence for maps of polyhedra in the papers 4 10 11 12 20 21 some topological properties of the attractors of flows are studied for instance hastings established in 12 a higher dimensional poincare bendixson theorem proving the existence of attractors of certain flows in the interior of submanifolds of the euclidean space and comparing the shape of the attractor with that of the manifold other properties studied in the above papers are related to strong cellularity of the global attractors 10 and polyhedral shape of asymptotically stable compacta 4 11 20 a remarkable fact proved by

higher order chebyshev basis functions for two point boundary value problems cubic basis functions in one dimension for the solution of two point boundary value problems are constructed based on the zeros of chebyshev polynomials of the first kind a general formula is derived for the construction of polynomial basis functions of degree r where 1 r a galerkin finite element method using the constructed basis functions for the cases r 1 2 and 3 is successfully applied to three different types of problem including a singular perturbation problem

attractors synthesis for a lotka volterra like system the aim of this paper is to prove numerically via computer graphic simulations that the synthesis algorithmprovided by danca et al 1 can be utilized to synthesize any attractor of a dynamical system modeling a two predator one prey lotka volterra like system the algorithm switches in a periodic deterministic or a random way the control parameter inside a set of a chosen values the obtained attractor is the same with the attractor obtained for parameter value taken as averaged value of the switched control values this simple and effective algorithm relies on a convex property induced in the set of the attractors corresponding to the chosen switching parameters the algorithm was tested successfully on systems depending linearly on the control parameter like lorenz chen rossler networks and other systems

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

a computational study of finite element methods for second order linear two point boundary value problems a computational study of five finite element methods for the solution of a single second order linear ordinary differential equation subject to general linear separated boundary conditions is described in each method the approximate solution is a piecewise polynomial expressed in terms of a b spline basis and is determined by solving a system of linear algebraic equations with an almost block diagonal structure the aim of the investigation is twofold to determine if the theoretical orders of convergence of the methods are realized in practice and to compare the methods on the basis of cost for a given accuracy in this study three parametrized families of test problems containing problems of varying degrees of difficulty are used the conclusions drawn are rather straightforward collocation is the cheapest method for a given accuracy and the easiest to implement also for solving the linear algebraic equations the use of a special purpose solver which takes advantage of the structure of the equations is advisable 1 introduction finite element methods for two point boundary value problems for a single ordinary differential equation may take many forms the simplest type of method is probably collocation 3 where an approximate solution is sought in some finite space subject to the constraint that it satisfy the differential equation at certain specified points this type of method has been applied very successfully in the case of mixed order systems of boundary value problems see for example 1 other methods can be described based on the standard l2 galerkin approximation 9 or on a combination of this and the collocation approach 7 13 25 in addition different weak formulations of the boundary value problem lead to two other techniques the hx and galerkin methods 12 14 15 17 how one would apply these methods to a system of differential equations with the exception of the collocation case is not immediately clear in addition it is not apparent what the relative advantages are of these various methods even when applied to a single equation in this paper we restrict our attention to the case of a single linear differential equation our aim is to investigate numerically five finite element techniques for a second order linear two point boundary value problem with separated boundary conditions of a general nature in particular we consider i the organization of the methods ii the treatment of the boundary conditions

synchronization with positive conditional lyapunov exponents synchronization of chaotic systems may occur only when the largest conditional lyapunov exponent of the driven system is negative the synchronization with positive conditional lyapunov exponents reported in a recent paper phys rev e 56 2272 1997 is a combined result of the contracting region of the system and the finite precision in computer simulations

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

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

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

synchronization based estimation of all parameters of chaotic systems from time series by a simple combination of adaptive scheme and linear feedback with the updated feedback strength for a large class of chaotic systems it is proved rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically this approach supplies a systematic and analytical procedure for estimating parameters from time series and it is simple to implement in practice in addition this method is quite robust against the effect of noise and able to respond rapidly to changes in operating parameters of the experimental system lorenz and r ossler hyperchaos systems are used to illustrate the validity of this technique

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

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

state estimation for coupled uncertain stochastic networks with missing measurements and time varying delays the discrete time case this paper is concerned with the problem of state estimation for a class of discrete time coupled uncertain stochastic complex networks with missing measurements and time varying delay the parameter uncertainties are assumed to be norm bounded and enter into both the network state and the network output the stochastic brownian motions affect not only the coupling term of the network but also the overall network dynamics the nonlinear terms that satisfy the usual lipschitz conditions exist in both the state and measurement equations through available output measurements described by a binary switching sequence that obeys a conditional probability distribution we aim to design a state estimator to estimate the network states such that for all admissible parameter uncertainties and time varying delays the dynamics of the estimation error is guaranteed to be globally exponentially stable in the mean square by employing the lyapunov functional method combined with the stochastic analysis approach several delay dependent criteria are established that ensure the existence of the desired estimator gains and then the explicit expression of such estimator gains is characterized in terms of the solution to certain linear matrix inequalities lmis two numerical examples are exploited to illustrate the effectiveness of the proposed estimator design schemes

on typicality in nonequilibrium steady states from the statistical mechanical viewpoint relaxation of macroscopic systems and response theory rest on a notion of typicality according to which the behavior of single macroscopic objects is given by appropriate ensembles ensemble averages of observable quantities represent the measurements performed on single objects because almost all objects share the same fate in the case of non dissipative dynamics and relaxation toward equilibrium states almost all is referred to invariant probability distributions that are absolutely continuous with respect to the lebesgue measure in other words the collection of initial micro states single systems that do not follow the ensemble is supposed to constitute a set of vanishing phase space volume this approach is problematic in the case of dissipative dynamics and relaxation to nonequilibrium steady states because the relevant invariant distributions attribute probability 1 to sets of zero volume while evolution commonly begins in equilibrium states i e in sets of full phase space volume we consider the relaxation of classical thermostatted particle systems to nonequilibrium steady states we show that the dynamical condition known as omega t mixing is necessary and sufficient for relaxation of ensemble averages to steady state values moreover we find that the condition known as weak t mixing applied to smooth observables is sufficient for ensemble relaxation to be independent of the initial ensemble lastly we show that weak t mixing provides a notion of typicality for dissipative dynamics that is based on the non invariant lebesgue measure and that we call physical ergodicity

simple adaptive feedback controller for identical chaos synchronization based on the invariance principle of differential equations a simple adaptive feedback scheme is proposed to strictly synchronize almost all chaotic systems unlike the usual linear feedback the variable feedback strength is automatically adapted to completely synchronize two almost arbitrary identical chaotic systems so this scheme is analytical and simple to implement in practice moreover it is quite robust against the effect of noise the famous lorenz and rossler hyperchaos systems are used as illustrative examples

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

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

sensitivity of dendrite maps abstract suppose that x is a dendrite and f x x is a sensitive continuous map we show that a x f contains a bilaterally transitive subsystem with nonempty interior b the system x f satisfies only one of the following two conditions b1 x f contains a topologically transitive subsystem with nonempty interior b2 there exists an f invariant nowhere dense closed subset a of x such that the attraction basin basin a f contains a residual subset b of an open set and the strong attraction basin sbasin a f is dense in b c if x is completely regular then x f contains a relatively strongly mixing subsystem with nonempty interior dense periodic points and positive topological entropy unlike for interval maps we construct a sensitive dendrite map with zero topological entropy

attractors and recurrence for dendrite critical polynomials we call a rational map f dendrite critical if all its recurrent critical points either belong to an invariant dendrite d or have minimal limit sets we prove that if f is a dendrite critical polynomial then for any conformal measure either for almost every point its limit set coincides with the julia set of f or for almost every point its limit set coincides with the limit set of a critical point c of f moreover if is non atomic then c can be chosen to be recurrent a corollary is that for a dendrite critical polynomial and a non atomic conformal measure the limit set of almost every point contains a critical point

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

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

backward stability for polynomial maps with locally connected julia sets we study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability then we deduce from this that a polynomial f with a locally connected julia set is backward stable outside any neighborhood of its attracting and neutral cycles for a conformal measure this easily implies that one of the following holds 1 for a e x e j f x j f 2 for a e x j f x c x for a critical point c x depending on x

parameter identification of dynamical systems from time series in this paper synchronization based parameter identification of dynamical systems from time series is carefully revisited it is shown based on rigorous theoretical analysis and concrete counterexamples that some recent research reports on this issue are incomplete or even incorrect a linear independence condition is pointed out which is sufficient for such parameter identification of general dynamical systems

on hausdorff dimension of unimodal attractors there exists a universal constant 1 such that every attractor of every c 4 unimodal map with a non degenerate critical point is an analytic manifold or its hausdorff dimension is equal to or less than

reconstruction of complex networks with delays and noise perturbation based on generalized outer synchronization this paper proposes an approach to identify the topological structure and unknown system parameters of a weighted complex dynamical network with delay and noise perturbation based on the barbalat type invariance principle for stochastic differential equations an effective adaptive feedback technique with an updated law is developed to realize generalized outer synchronization the unknown topological structure and parameters are identified simultaneously through the established technique the weight configuration matrix was found to be unnecessarily symmetric numerical examples are examined to illustrate the effectiveness of the analytical results

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

applications of interval analysis to minimal and maximal solutions of differential equations the method of interval analysis and the method of upper and lower solutions will be combined to investigate differential equations both existence of extremal solutions and bounds of solutions will be obtained when the right hand side of differential equations is discontinuous

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

a simple universal adaptive feedback controller for chaos and hyperchaos control a simple universal adaptive feedback controller is proposed for chaos control in comparison with previous methods the proposed scheme which uses a single feedback gain and converges very fast is suitable for application to a larger class of chaotic hyperchaotic and nonhyperbolic chaotic systems a sufficient condition for selecting the least feedback terms is given and a numerical example using the lorenz system verifies the correctness and effectiveness of the proposed approach

analysis of spurious synchronization with positive conditional lyapunov exponents in computer simulations abstract synchronization of chaotic systems has been a field of great interest and potential applications the necessary condition for synchronization is the negativity of the largest conditional lyapunov exponent some researches have shown that negativity of the largest conditional lyapunov exponent is not a sufficient condition for high quality synchronization in the presence of small perturbations however it was reported that synchronization can be achieved with positive conditional lyapunov exponents based on numerical simulations in this paper we first analyze the behavior of synchronization with positive conditional lyapunov exponents in computer simulations of the synchronization of chaotic systems with various couplings and demonstrate that synchronization is an outcome of finite precision in numerical simulations it is shown that such a numerical artifact is quite common and easily occurs in numerical simulations thus can be confusing and misleading some behavior and properties of the synchronized system with slightly positive conditional lyapunov exponents can be understood based on the theory of on off intermittency we also study the effects of finite precision on numerical simulation of on off intermittency special care should be taken in numerical simulations of chaotic systems in order not to mistake numerical artifacts as physical phenomena

adaptive feedback control algorithm this paper is motivated by giving the detailed proofs and some interesting remarks on the results the author obtained in a series of papers phys rev lett 93 214101 2004 phys rev e 71 037203 2005 69 067201 2004 where an adaptive feedback algorithm was proposed to effectively stabilize and synchronize chaotic systems this note proves in detail the strictness of this algorithm from the viewpoint of mathematics and gives some interesting remarks for its potential applications to chaos control a synchronization in addition a significant comment on synchronization based parameter estimation is given which shows some techniques proposed in literature less strict and ineffective in some cases

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

stochastic adaptive synchronization for time varying complex delayed dynamical networks with heterogeneous nodes stochastic synchronization for a time varying complex delayed dynamical networks with heterogeneous nodes is studied in this paper using adaptive synchronization scheme for detailed analysis and discussion based on lyapunov stability theory the properties of weiner process brownian motions some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of the complex delayed dynamical networks model a sufficient condition is given to ensure that the proposed controlled networks model is globally asymptotically synchronized in mean square sense a simulation example is presented to show the effectiveness and applicability of the proposed results

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

adaptive complete synchronization of the noise perturbed two bi directionally coupled chaotic systems with time delay and unknown parametric mismatch in this paper we design an adaptive feedback controller to synchronize a class of noise perturbed two bi directionally coupled chaotic systems with time delay and unknown parametric mismatch based on invariance principle of stochastic time delay differential equations some sufficient conditions of adaptive complete synchronization are given comparing with other papers here we consider the effect of internal noise time delay and parametric mismatch in the synchronized process as the illustrative examples the famous lorenz system and rossler system are considered here in order to validate the proposed scheme numerical simulations are performed and the numerical results show that our scheme is very effective

the bandcount increment scenario iii deformed structures bifurcation structures in two dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely in a series of three papers we investigated the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of one dimensional canonical form for discontinuous maps in part i the basic structures in the chaotic region are explained by the bandcount increment scenario in part ii fine self similar substructures nested into the bandcount increment scenario are explained by the bandcount adding and doubling scenarios nested into each other ad infinitum hereby we fixed in both previous parts one of the parameters to a non generic value and studied the remaining two dimensional parameter subspace in this part iii finally we investigated the structures under variation of this third parameter remarkably this step is the most important with respect to practical applications since it cannot be expected that these operate exactly at the previously investigated specific value

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

adaptive classification of temporal signals in fixed weight recurrent neural networks an existence proof recurrent neural networks with fixed weights have been shown in practice to successfully classify adaptively signals that vary as a function of time in the presence of additive noise and parametric perturbations we address the question can this ability be explained theoretically we provide a mathematical proof that these networks have this ability even when parametric perturbations enter the signals nonlinearly the restrictions that we impose on the signals to be classified are that they satisfy an assumption of nondegeneracy and that noise amplitude is sufficiently small further we demonstrate that the recurrent neural networks may not only classify uncertain signals adaptively but also can recover the values of uncertain parameters of the signals up to their equivalence classes

attractors and characteristic exponents a natural definition of an attractor as an invariant measure is given based on the ergodic theory of axiom a diffeomorphisms and some results are proved which support this definition it is also proved that if an attractor has every characteristic exponent less than zero in a set of nonzero measure then the support set of the attractor is an asymptotic stable periodic orbit

synchronization based approach for parameters identification in delayed chaotic neural networks abstract in this paper an adaptive procedure to the problem of synchronization and parameters identification for chaotic neural networks with time varying delay is introduced by combining the adaptive control and linear feedback with appropriate update law based on the invariance principle of functional differential equations all the connection weight matrices can be efficiently estimated according to a simple rigorous and systematic technique this approach is also able to track the changes in the operating parameters of the experimental neural networks rapidly the speed of synchronization and parameters estimation can be adjusted under the adaptive gain properly chosen in addition the method is simple to implement in practice and it is quite robust against the effect of slight noise in the given time series and the estimated value of a parameter fluctuates around the correct value

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

adaptive synchronization of cohen grossberg neural network with mixed time varying delays and stochastic perturbation in this paper based on the lasalle invariant principle of stochastic differential delay equations and the stochastic analysis theory as well as the adaptive feedback technique several sufficient conditions ensuring the adaptive synchronization of cohen grossberg neural network with mixed time varying delays and stochastic perturbation are derived in particular the synchronization criterion considered globally is the almost surely asymptotic stability of the error dynamical system our synchronization criterion is easily verified and does not solve any linear matrix inequality these results generalized a few previous known results at last a numerical example and its simulations are provided to demonstrate the effectiveness and advantage of the theoretical results