the pim simplex method an extension of the pim triple method to saddles with an arbitrary number of expanding directions abstract the growing interest in non attracting chaotic sets of high dimensional dynamical systems requires the development of numerical techniques for their study the pim triple method h e nusse j a yorke physica d 36 1989 137 is a very good method to obtain trajectories on saddles with one positive lyapunov exponent in this paper we combine the same ideas with an algorithm for finding local extrema of multi variable functions to develop an extension of the method the pim simplex method that is suitable for the study of sets with an arbitrary number of expanding directions

asymptotically periodic behaviour in the dynamics of chaotic mappings we will prove that for a chaotic mapping f belonging to a suitable class of c 1 alpha functions the set a infty f has lebesgue measure zero with a infty f a nonempty set consisting of points whose orbits do not converge to an asymptotically stable periodic orbit of f or to the absorbing boundary moreover almost every point is asymptotically periodic with period p for some positive integer p further we will show that the same conclusions hold for maps with nonpositive schwarzian derivative under some additional assumptions

remarks on differential harnack inequalities abstract in this paper we give a generalization of global and local differential harnack inequalities for heat equations obtained by li and xu j f li x j xu differential harnack inequalities on riemannian manifolds i linear heat equation adv math 226 5 2011 4456 4491 and baudoin and garofalo f baudoin n garofalo perelman s entropy and doubling property on riemannian manifolds j geom anal 21 2011 1119 1131 from this we can derive new harnack inequalities and new lower bounds for the associated heat kernel also we provide some new entropy formulas with monotonicity

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

unstable recurrent patterns in kuramoto sivashinsky dynamics we undertake an exploration of recurrent patterns in the antisymmetric subspace of the one dimensional kuramoto sivashinsky system for a small but already rather turbulent system the long time dynamics takes place on a low dimensional invariant manifold a set of equilibria offers a coarse geometrical partition of this manifold the newton descent method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions the attracting set appears surprisingly thin its backbone consists of several smale horseshoe repellers well approximated by intrinsic local one dimensional return maps each with an approximate symbolic dynamics the dynamics appears decomposable into chaotic dynamics within such local repellers interspersed by rapid jumps between them

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

on brauer stars this paper contains some results on how representations in the principal p block of a finite group with a cyclic sylow p subgroup yield information on the structure of the group throughout the paper g denotes a finite group p a fixed odd prime dividing 1 gi p a sylow p subgroup of g with p p and b p the principal p block of g if p is cyclic then the ordinary and modular irreducible characters in any p block of g can be described in terms of a graph called the bruuer tree belonging to the block as in 5 vii 61 a tree is called a star if there are no paths with more than two edges i e

chaos and crises in more than two dimensions instituto de astronomi ayfi sica del espacio conicet casilla de correo 67 socursal 28 1428 buenos aires argentina received 23 december 1996 noisy chaotic trajectories with nite time lyapunov exponents that uctuate about zero are basicallyunshadowable s dawson c grebogi t sauer and j a yorke phys rev lett 73 1927 1994 this canoccur when periodic orbits with different numbers of unstable directions coexist inside the attractor thepresence of a he non type chaotic saddle i e a nonattracting chaotic set with a structure similar to that of thehe non attractor guarantees such coexistence in a persistent manner s p dawson phys rev lett 76 4348 1996 in this paper we describe how these sets appear naturally in maps of more than two dimensions howthey can be found and what crises they produce s1063 651x 97 13005 7 pacs number s 05 45 1b 05 40 1ji introduction

monte carlo sampling in fractal landscapes we design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems we show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest lyapunov exponent of the chaotic system we propose a generalization of the wang landau algorithm which constructs not only the density of states transient time distribution but also the correct step length as a result we obtain a flat histogram monte carlo method which samples fractal landscapes in polynomial time a dramatic improvement over the exponential scaling of traditional uniform sampling methods our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions

bi instability and the global role of unstable resonant orbits in a driven laser driven class b lasers are devices which possess quadratic nonlinearities and are known to exhibit chaotic behavior we describe the onset of global heteroclinic connections which give rise to chaotic saddles these form the precursor topology which creates both localized homoclinic chaos as well as global mixed mode heteroclinic chaos to locate the relevant periodic orbits creating the precursor topology approximate maps are derived using matched asymptotic expansions and subharmonic melnikov theory locating the relevant unstable fixed points of the maps provides an organizing framework to understand the global dynamics and chaos exhibited by the laser 2000 published by elsevier science b v

devil staircase behavior of dynamical invariants in chaotic scattering a crisis in chaotic scattering is characterized by the merging of two or more nonattracting chaotic saddles the fractal dimension of the resulting chaotic saddle increases through the crisis we present a rigorous analysis for the behavior of dynamical invariants associated with chaotic scattering by utilizing a representative model system that captures the essential dynamical features of crisis our analysis indicates that the fractal dimension and other dynamical invariants are a devil staircase type of function of the system parameter our results can also provide insight for similar devil staircase behaviors observed in the parametric evolution of chaotic saddles of general dissipative dynamical systems and in communicating with chaos 2000 elsevier science b v all rights reserved

fractal boundaries in open hydrodynamical flows signatures of chaotic saddles we introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently it is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors the fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set we point out that these boundaries coincide with streak lines passing through appropriately chosen points as an illustrative numerical experiment we consider a model of the von k arm an vortex street a time periodic two dimensional flow of a viscous fluid around a cylinder

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

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

unpredictable behavior in the duffing oscillator wada basins this paper describes some numerical experiments giving evidence of wada basin boundaries for the duffing oscillator we suggest some mechanisms by which this fractal property of the boundaries appears and discuss the difficulties that the wada property presents for predicting the behavior of dynamical systems

the local structure of a chaotic attractor in four dimensions abstract a simple system of four ordinary differential equations exhibiting chaotic behavior is introduced the lyapunov exponents are approximately 3 34 0 00 1 79 and 5 55 points on the attractor where two variables simultaneously assume prechosen values are located by a numerical integration procedure in which suitable antecedent conditions are obtained by successive approximations the points are organized into curves which appear more numerous under higher resolution a backward integration procedure indicates that a line transverse to the curves intersects each curve rather than passing through gaps while the set of intersections forms a cantor set thus the attractor appears to be locally the product of three continua and one cantor set the continua are found to extend in the expanding neutral and less rapidly contracting local directions while the cantor set extends in the more rapidly contracting direction

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

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

persistent homoclinic tangencies in the he non family abstract homoclinic tangencies in the henon family f x y x 2 by x for the parameter values b 0 3 and 1 270 1 420 are investigated our main observation is that there exist three intervals comprising 93 percent of the values of the parameter such that for a dense set of parameter values in these intervals the henon family possesses a homoclinic tangency therefore one should expect long parameter intervals where the henon family is not structurally stable strong numerical support for this observation is provided

constructing constrained invariant sets in multiscale continuum systems we present a method that we name the constrained invariant manifold method a visualization tool to construct stable and unstable invariant sets of a map or flow where the invariant sets are constrained to lie on a slow invariant manifold the construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural mechanical system consisting of a pendulum coupled to a viscoelastic rod additionally we extend the step and stagger method d sweet h nusse and j yorke phys rev lett 86 2261 2001 to calculate a delta pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the lyapunov exponents of the saddle

differential harnack inequalities and perelman type entropy formulae for subelliptic operators abstract in this paper under the generalized curvature dimension inequality recently introduced by f baudoin and n garofalo we obtain differential harnack inequalities theorem 2 1 for the positive solutions to the schrodinger equation associated to subelliptic operator with potential as applications of the differential harnack inequality we derive the corresponding parabolic harnack inequality theorem 4 1 also we define the perelman type entropy associated to subelliptic operators and derive its monotonicity theorem 5 3

a procedure for finding numerical trajectories on chaotic saddles abstract examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors if almost every trajectory eventually leaves some region but the region contains a chaotic set then typical trajectories will behave chaotically for a while and then will leave the region and so we will observe chaotic transients the main objective that will be addressed is the dynamic restraint problem given a region that contains a chaotic set but does not contain an attractor find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time systems with horseshoes have such regions as do systems with fractal basin boundaries as does the henon map for suitably chosen parameters we present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory the method is called the pim triple procedure since it is based on so called pim triples a pim proper interior maximum triple is three points a c b in a straight line segment such that the interior point c i e c is between a and b has the maximum escape time that is its escape time from the region is greater than the escape time of both a and b proper means the segment from a to b is smaller than a previously obtained segment we show rigorously that the pim triple procedure works in ideal situations we find it works well even in less than ideal cases this procedure can also be used for the computation of lyapunov exponents furthermore the accessible pim triple procedure a refined pim triple procedure for finding accessible trajectories on the chaotic saddle will also be discussed

a sub riemannian curvature dimension inequality volume doubling property and the poincare inequality let mathbb m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l satisfying l1 0 and which is symmetric with respect to mu we show that if l satisfies with a non negative curvature parameter the generalized curvature inequality introduced by the first and third named authors in http arxiv org abs 1101 3590 then the following properties hold the volume doubling property the poincare inequality the parabolic harnack inequality the key ingredient is the study of dimension dependent reverse log sobolev inequalities for the heat semigroup and corresponding non linear reverse harnack type inequalities our results apply in particular to all sasakian manifolds whose horizontal webster tanaka ricci curvature is nonnegative all carnot groups of step two and to wide subclasses of principal bundles over riemannian manifolds whose ricci curvature is nonnegative

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

differential harnack inequalities on riemannian manifolds i linear heat equation abstract in the first part of this paper we get new li yau type gradient estimates for positive solutions of heat equation on riemannian manifolds with ricci m k k r as applications several parabolic harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with ricci curvature bounded from below in the second part we establish a perelman type li yau hamilton differential harnack inequality for heat kernels on manifolds with ricci m k which generalizes a result of l ni 2004 2006 20 21 as applications we obtain new harnack inequalities and heat kernel estimates on general manifolds we also obtain various entropy monotonicity formulas for all compact riemannian manifolds

shadowability of statistical averages in chaotic systems we ask whether statistical averages in chaotic systems can be computed or measured reliably under the influence of noise situations are identified where the invariance of such averages breaks down as the noise amplitude increases through a critical level an algebraic scaling law is obtained which relates the change of the averages to the noise variation this breakdown of shadowability of statistical averages as characterized by the algebraic scaling law can be expected in both low and high dimensional chaotic systems

enhancement of noise induced escape through the existence of a chaotic saddle we study the noise induced escape process in a prototype dissipative nonequilibrium system the ikeda map in the presence of a chaotic saddle embedded in the basin of attraction of the metastable state we find the novel phenomenon of a strong enhancement of noise induced escape this result is established by employing the theory of quasipotentials our finding is of general validity and should be experimentally observable

reverse poincare inequalities isoperimetry and riesz transforms in carnot groups abstract we prove an optimal reverse poincare inequality for the heat semigroup generated by the sub laplacian on a carnot group of any step as an application we give new proofs of the isoperimetric inequality and of the boundedness of the riesz transform in carnot groups

volume and distance comparison theorems for sub riemannian manifolds abstract in this paper we study global distance estimates and uniform local volume estimates in a large class of sub riemannian manifolds our main device is the generalized curvature dimension inequality introduced by the first and the third author in 3 and its use to obtain sharp inequalities for solutions of the sub riemannian heat equation as a consequence we obtain a gromov type precompactness theorem for the class of sub riemannian manifolds whose generalized ricci curvature is bounded from below in the sense of 3

controlling transient chaos the method of ott grebogi and yorke 1990 is extended to control transient chaos the controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor the time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime the number of controlled trajectories however decreases the maximum perturbation according to a power law applicability to experimental situations and comparison with permanent chaotic cases are discussed

approximating chaotic saddles for delay differential equations chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos they play a key role in mediating transport processes involving scattering and chaotic transients here we present evidence long chaotic transients and fractal basins of attraction of transient chaos in a logistic delay differential equation we adapt an existing method stagger and step to numerically construct the chaotic saddle for this system this is the first such analysis of transient chaos in an infinite dimensional dynamical system and in delay differential equations in particular using poincar e section techniques we illustrate approaches to visualizing the saddle set and confirm that the saddle has the cantor like fractal structure consistent with a chaotic saddle generated by horseshoe type dynamics

perelman s entropy formula for the witten laplacian on riemannian manifolds via bakry emery ricci curvature in this paper we study perelman s mathcal w entropy formula for the heat equation associated with the witten laplacian on complete riemannian manifolds via the bakry emery ricci curvature under the assumption that the m dimensional bakry emery ricci curvature is bounded from below we prove an analogue of perelman s and ni s entropy formula for the mathcal w entropy of the heat kernel of the witten laplacian on complete riemannian manifolds with some natural geometric conditions in particular we prove a monotonicity theorem and a rigidity theorem for the mathcal w entropy on complete riemannian manifolds with non negative m dimensional bakry emery ricci curvature moreover we give a probabilistic interpretation of the mathcal w entropy for the heat equation of the witten laplacian on complete riemannian manifolds and for the ricci flow on compact riemannian manifolds

analysis of chaotic saddles in low dimensional dynamical systems the derivative nonlinear schrodinger equation abstract in this paper we present a computational study of nonattracting chaotic sets known as chaotic saddles in a low dimensional dynamical system describing stationary solutions of the derivative nonlinear schrodinger equation a driven dissipative model for alfven waves these chaotic saddles have gaps which are filled at chaotic transitions such as a saddle node bifurcation and an interior crisis we give a detailed explanation of how to numerically determine the chaotic saddles and describe how a chaotic attractor after an interior crisis point can be decomposed into two chaotic saddles dynamically connected by a set of coupling unstable periodic orbits created by a gap filling explosion after the crisis this coupling between two chaotic saddles is responsible for the intermittent dynamics displayed by the chaotic system after the interior crisis

explosions of chaotic sets abstract large scale invariant sets such as chaotic attractors undergo bifurcations as a parameter is varied these bifurcations include sudden changes in the size and or type of the set an explosion is a bifurcation in which new recurrent points suddenly appear at a non zero distance from any pre existing recurrent points we discuss the following in a generic one parameter family of dissipative invertible maps of the plane there are only four known mechanisms through which an explosion can occur 1 a saddle node bifurcation isolated from other recurrent points 2 a saddle node bifurcation embedded in the set of recurrent points 3 outer homoclinic tangencies and 4 outer heteroclinic tangencies the term outer tangency refers to a particular configuration of the stable and unstable manifolds at tangency in particular we examine different types of tangencies of stable and unstable manifolds from orbits of pre existing invariant sets this leads to a general theory that unites phenomena such as crises basin boundary metamorphoses explosions of chaotic saddles etc we illustrate this theory with numerical examples

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

characterization of a high dimensional interior crisis in a nonlinear reactive diffusion equation abstract we report an investigation of interior crisis in extended spatiotemporal systems exemplified by the kuramoto sivashinsky equation we show that unstable periodic orbits and their associated invariant stable and unstable manifolds in the poincare hyperplane can effectively characterize the global bifurcation dynamics of high dimensional systems in particular we introduce a new technique to characterize the high dimensional homoclinic tangency responsible for an interior crisis using the stable manifolds of a chaotic saddle

on linear groups of degrees at most p 1 abstract in this paper we determine the structure of complex linear groups g of degree at most p 1 where p is a t i sylow p subgroup of g

the equality of fractal dimension and uncertainty dimension for certain dynamical systems mgoy introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system in mgoy and p it was conjectured that the box counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems in this paper our main result is that the box counting dimension the uncertainty dimension and the hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems which are uniformly hyperbolic on their basin boundary when the box counting dimension of the basin boundary is large that is near the dimension of the phase space this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in

moving finite size particles in a flow a physical example of pitchfork bifurcations of tori the motion of small spherical particles of finite size in fluid flows at low reynolds numbers is described by the strongly nonlinear maxey riley equations due to the stokes drag the particle motion is dissipative giving rise to the possibility of attractors in phase space we investigate the case of an infinite cellular flow field with time periodic forcing the dynamics of this system are studied in a part of the parameter space we focus particularly on the size of the particles whose variations are most important in active physical processes for example for aggregation and fragmentation of particles depending on their size the particles will settle on different attractors in phase space in the long term limit corresponding to periodic quasiperiodic or chaotic motion one of the invariant sets that can be observed in a large part of this parameter region is a quasiperiodic motion in the form of a torus we identify some of the bifurcations that these tori undergo as particle size and mass ratio relative to the fluid are varied in this way we provide a physical example for sub and supercritical pitchfork bifurcations of tori

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

symmetry restoration in a class of forced oscillators abstract this paper is concerned with a class of symmetric forced oscillators modeled by nonlinear ordinary differential equations the forced duffing oscillator and the forced pendulum belong to this class of oscillators another example is found in an electric power system and is associated with a phenomenon known as ferroresonance solutions for this class of systems can be symmetric or non symmetric sometimes as a physical parameter is varied solutions lose or gain the symmetry at a bifurcation point it will be shown that bifurcations which might otherwise be called symmetry breaking symmetry creation via collision and symmetry creation via explosion are all the result of a collision between conjugate attractors i e attractors that relate to each other by the symmetry and a symmetric limit set the same mechanism seems plausible for at least some bifurcations involving non attracting sets this point is illustrated with some examples

noise induced escape through a chaotic saddle lowering of the activation energy the noise induced escape mechanism is investigated for a prototype time discrete nonequilibrium system describing a laser in a ring cavity with optical feedback the ikeda map for certain parameter values a chaotic saddle is embedded in the basin of attraction of the metastable state this results in a lowering of the escape threshold for the activation energy which is established by employing the theory of quasipotentials our findings are explained by the computation of the most probable exit path mpep which lies partly on the chaotic saddle the enhancement of escape turns out to be maximal if the mpep lies entirely on the chaotic saddle

remarks on li yau inequality on graphs abstract in this paper we study li yau gradient estimates for the solutions u to the heat equation t u u on graphs under the curvature condition c d n k which is introduced by bauer et al in 5 as applications we derive harnack inequalities and heat kernel estimates on graphs also we present a type of hamilton gradient estimates

li yau gradient estimate and entropy formulae for the cr heat equation in a closed pseudohermitian 3 manifold in this paper we derive two subgradient estimates of the cr heat equation in a closed pseudohermitian 3 manifold which are served as the cr version of the li yau gradient estimate with its applications we first get a subgradient estimate of the logarithm of the positive solution of the cr heat equation secondly we have the harnack inequality and upper bound estimate for the heat kernel finally we obtain perelman type entropy formulae for the cr heat equation

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases

persistence of order and structure in chaos abstract in this paper some results are presented concerning one dimensional chaotic maps with arbitrarily many critical points let f be a chaotic map belonging to some suitable class of c 1 maps from a nontrivial interval x into itself assuming that f is of class c 1 for some 0 we have that the set of aperiodic points for f has lebesgue measure zero further if f x is bounded then there exists a positive integer p such that almost every point in the interval is asymptotically periodic with period p moreover it will turn out that this asymptotically periodic behaviour in the complicated dynamics of f is persistent under small smooth perturbations the topological structure of the nonwandering set of f will be described and this structure is invariant under small c 1 perturbations of the map f assuming that f is of class c 2 the map f is c 2 structurally stable provided that f satisfies some suitable conditions finally it will turn out that maps with a negative schwarzian derivative belong to the suitable class of maps mentioned above

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

log sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality let m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l which is symmetric with respect to mu we assume that l satisfies a generalized curvature dimension inequality as introduced by baudoin garofalo cite bg1 our goal is to discuss functional inequalities for mu like the poincare inequality the log sobolev inequality or the gaussian logarithmic isoperimetric inequality

ensemble data assimilation for hyperbolic systems ensemble based methods are now widely used in applications such as weather prediction but there are few rigorous results regarding their application the broad goal of this paper is to provide some theoretical evidence of their applicability in the computational study of dynamical systems in some idealized yet interesting setting the specific goal of this paper is to investigate a data assimilation procedure dap an ensemble kalman filter ekf in the context of hyperbolic systems we show that with appropriate assumptions on observations for every trajectory on an attractor the predictions produced by the dap remain close to the truth for all time provided the ensemble is properly initialized making the dap reliable we deal with the case of a one dimensional unstable direction first and later extend to higher dimensional unstable spaces a feature of this approach is that no model linearizations are involved making it efficient and potentially of interest for applications in high dimensional systems lyapunov exponents are also investigated