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 structure of symmetric attractors we consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems we show for example that the symmetry of an attractor cannot in general be an arbitrary subgroup of the group of symmetries in addition there are group theoretic restrictions on the symmetry of connected components of a symmetric attractor the symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time average of a chaotic dynamical system

symmetry and bifurcations of a two degree of freedom vibro impact system abstract a two degree of freedom system with impact is considered the symmetry of the system and its poincare map is described the symmetric period n 2 motion corresponding to the symmetric fixed point of the poincare map is obtained if the jacobian matrix of the poincare map at the fixed point has a real eigenvalue crossing the unit circle at 1 the symmetric fixed point will bifurcate into two antisymmetric fixed points which have the same stability via pitchfork bifurcation the numerical simulation shows that the symmetric fixed points may have pitchfork bifurcations and hopf bifurcations while the control parameter changes continuously the two antisymmetric fixed points will give birth to two synchronous bifurcation sequences

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

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

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

an interior point method for large scale l 1 regularized logistic regression logistic regression with l1 regularization has been proposed as a promising method for feature selection in classification problems in this paper we describe an efficient interior point method for solving large scale l1 regularized logistic regression problems small problems with up to a thousand or so features and examples can be solved in seconds on a pc medium sized problems with tens of thousands of features and examples can be solved in tens of seconds assuming some sparsity in the data a variation on the basic method that uses a preconditioned conjugate gradient method to compute the search step can solve very large problems with a million features and examples e g the 20 newsgroups data set in a few minutes on a pc using warm start techniques a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently

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

observing the symmetry of attractors abstract we show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space equivariant projections have long been used but they can give misleading results if used improperly and have been considered untrustworthy we find conditions under which an equivariant projection generically shows the correct symmetry of the attractor

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

a bernstein problem for special lagrangian equations where is are the eigenvalues of the hessian d2u namely any global convex solution to 1 1 in rn must be a quadratic polynomial recall the classical result any global convex solution in rn to the laplace equation u 1 n c or the monge ampere equation log det d2u log 1 log n c must be quadratic equation 1 1 originates from special lagrangian geometry hl the lagrangian graph x u x rn rn is called special when the argument of the complex number 1 1 1 1 1 n is constant c or u satisfies 1 1 and it is special if and only if x u x is a minimal surface in rn rn hl theorem 2 3 proposition 2 17 in terms of minimal surface our result is the following

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

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

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

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

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

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

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

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

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

homogeneous solutions of fully nonlinear elliptic equations in four dimensions we prove that there is no nontrivial homogeneous order 2 analytic outside zero solutions of fully nonlinear uniformly elliptic equations in 2013 wiley periodicals inc

run and inspect method for nonconvex optimization and global optimality bounds for r local minimizers many optimization algorithms converge to stationary points when the underlying problem is nonconvex they may get trapped at local minimizers or stagnate near saddle points we propose the run and inspect method which adds an inspect phase to existing algorithms that helps escape from non global stationary points it samples a set of points in a radius r around the current point when a sample point yields a sufficient decrease in the objective we resume an existing algorithm from that point if no sufficient decrease is found the current point is called an approximate r local minimizer we show that an r local minimizer is globally optimal up to a specific error depending on r if the objective function can be implicitly decomposed into a smooth convex function plus a restricted function that is possibly nonconvex nonsmooth therefore for such nonconvex objective functions verifying global optimality is fundamentally easier for high dimensional problems we introduce blockwise inspections to overcome the curse of dimensionality while still maintaining optimality bounds up to a factor equal to the number of blocks we also present the sample complexities of these methods when we apply our method to the existing algorithms on a set of artificial and realistic nonconvex problems we find significantly improved chances of obtaining global minima

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

rigidity theorems for special lagrangian cones in this paper we derive that every n dimensional 2 le n le 7 special lagrangian cones in mathbb r n times mathbb r n with flat normal bundle must be flat and we also derive a rigidity result under a certain restriction on gauss map

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

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

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

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

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

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

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

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

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

nonconvex sparse logistic regression with weakly convex regularization in this paper we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem the idea is based on the finding that a weakly convex function as an approximation of the l 0 pseudo norm is able to better induce sparsity than the commonly used l 1 norm for a class of weakly convex sparsity inducing functions we prove the nonconvexity of the corresponding problem and study its local optimality conditions and the choice of the regularization parameter despite the nonconvexity a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression and its convergence behavior is studied theoretically then the general framework is applied to a specific weakly convex function and a local optimality condition and a bound on the logistic loss at a local optimum are provided the solution method is instantiated in this case as an iterative firm shrinkage algorithm and a nesterov acceleration is used with a convergence guarantee its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets

robust low rank tensor recovery via nonconvex singular value minimization tensor robust principal component analysis via tensor nuclear norm tnn minimization has been recently proposed to recover the low rank tensor corrupted with sparse noise outliers tnn is demonstrated to be a convex surrogate of rank however it tends to over penalize large singular values and thus usually results in biased solutions to handle this issue we propose a new definition of tensor logarithmic norm tln as the nonconvex surrogate of rank which can decrease the penalization on larger singular values and increase that on smaller ones simultaneously to preserve the low rank structure of a tensor then the strategy of tensor factorization is combined into the minimization of tln to improve computational performance to handle impulsive scenarios we propose a nonconvex ell p ball projection scheme with 0 instead of the conventional convex scheme with p 1 which enhances the robustness against outliers by incorporating the tln minimization and the ell p ball projection we finally propose two low rank recovery algorithms whose resulting optimization problems are efficiently solved by the alternating direction method of multipliers admm with convergence guarantees the proposed algorithms are applied to the synthetic data recovery and image and video restorations in real world experimental results demonstrate the superior performance of the proposed methods over several state of the art algorithms in terms of tensor recovery accuracy and computational efficiency

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

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

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