periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

maps close to identity and universal maps in the newhouse domain given an n dimensional c r diffeomorphism g its renormalized iteration is an iteration of g restricted to a certain n dimensional ball and taken in some c r coordinates in which the ball acquires radius 1 we show that for any r 1 the renormalized iterations of c r close to identity maps of an n dimensional unit ball b n n 2 form a residual set among all orientation preserving c r diffeomorphisms b n r n in other words any generic n dimensional dynamical phenomenon can be obtained by iterations of c r close to identity maps with the same dimension of the phase space as an application we show that any c r generic two dimensional map that belongs to the newhouse domain i e it has a so called wild hyperbolic set so it is not uniformly hyperbolic nor uniformly partially hyperbolic and that neither contracts nor expands areas is c r universal in the sense that its iterations after an appropriate coordinate transformation c r approximate every orientation preserving two dimensional diffeomorphism arbitrarily well in particular every such universal map has an infinite set of coexisting hyperbolic attractors and repellers

symmetric galerkin boundary integral formulation for interface and multi zone problems domains containing an internal boundary such as a bi material interface arise in many applications e g composite materials and geophysical simulations this paper presents a symmetric galerkin boundary integral method for this important class of problems in this situation the physical quantities are known to satisfy continuity conditions across the interface but no boundary conditions are specied the algorithm described herein achieves a symmetric matrix of reduced size moreover the symmetry can also be invoked to lessen the numerical work involved in constructing the system of equations and thus the method is computationally very ecient a prototype numerical example with several variations in the boundary conditions and material properties is employed to validate the formulation and corresponding numerical procedure the boundary element results are compared with analytical solutions and with numerical results obtained with the nite element method 1997 by john wiley sons ltd

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

precorrected fft accelerated bem for large scale transient elastodynamic analysis using frequency domain approach summary a precorrected fast fourier transform pfft accelerated boundary element method bem for large scale transient elastodynamic analysis is developed and described in this paper the frequency domain approach is used to overcome the wrap around problem associated with the discrete fourier transform the exponential window method ewm is employed and incorporated in the frequency domain bem an improved implementation scheme of the pfft method based on polynomial interpolation technique is developed and applied to accelerate the elastodynamic bem this new scheme reduces the memory required to save the convolution matrix by a factor of 8 to further improve the efficiency of the code a newly developed linear system solver based on the induced dimension reduction method is employed its performance is investigated and compared with that of the well known gmres the accuracy and computational efficiency of the method are evaluated and demonstrated by three examples a classical benchmark a plate subject to an impact loading and a porous cube with nearly half million dofs subject to a step traction loading both analytical and experimental results are employed to validate the method it has been found that the ewm can effectively resolve the wrap around problem and accurate time responses for an arbitrarily chosen time period can be obtained copyright 2011 john wiley sons ltd

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

prevalent dynamics at the first bifurcation of henon like families we study the dynamics of strongly dissipative henon like maps around the first bifurcation parameter a at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set we prove that a is a full lebesgue density point of the set of parameters for which lebesgue almost every initial point diverges to infinity under positive iteration a key ingredient is that a corresponds to the non recurrence of every critical point reminiscent of misiurewicz parameters in one dimensional dynamics adapting on the one hand benedicks carleson s parameter exclusion argument we construct a set of good parameters having a as a full density point adapting benedicks viana s volume control argument on the other we analyze lebesgue typical dynamics corresponding to these good parameters

generic diffeomorphisms with superexponential growth of number of periodic orbits let m be a smooth compact manifold of dimension at least 2 and diff r m be the space of c r smooth diffeomorphisms of m associate to each diffeomorphism f 2 diff r m the sequence pn f of the number of isolated periodic points for f of period n in this paper we exhibit an open set n in the space of diffeomorphisms diff r m such for a baire generic diffeomorphism f 2 n the number of periodic points pnf grows with a period n faster than any following sequence of numbers fangn2zc along a subsequence i e pn f a ni for some ni 1 with i 1 in the cases of surface diffeomorphisms i e dim m d 2 an open set n with a supergrowth of the number of periodic points is a newhouse domain a proof of the man result is based on the gontchenko shilnikov turaev theorem gst a complete proof of that theorem is also presented

shil nikov configurations in any generic unfolding of the nilpotent singularity of codimension three on r3 abstract we are interested in the dynamics arising in generic unfoldings of the nilpotent singularity of codimension three since shil nikov type homoclinic orbits make up the simplest scenario where very complicated dynamics for three dimensional vector fields can be shown the aim of this paper is to prove analytically their existence in any of such unfoldings

modulation of estimators and confidence sets an unknown signal plus white noise is observed at n discretetime points within a large convex class of linear estimators of the signal we choose the one which minimizes estimated quadratic risk by construction the resulting estimator is nonlinear this estimation is done after orthogonal transformation of the data to a reasonable coordinate system the procedure adaptively tapers the coefficients of the transformed data if the class of candidate estimators satisfies a uniform entropy condition then our estimator is asymptotically minimax in pinsker s sense over certain ellipsoids in the parameter space and dominates the james stein estimatorasymptotically we describe computational algorithms for the modulation estimator and construct confidence sets for the unknown signal these confidence sets are centered at the estimator have correctasymptotic coverage probability and have relatively small risk asset valued estimators of the signal

dense heteroclinic tangencies near a bykov cycle abstract this article presents a mechanism for the coexistence of hyperbolic and non hyperbolic dynamics arising in a neighbourhood of a bykov cycle where trajectories turn in opposite directions near the two nodes we say that the nodes have different chirality we show that in the set of vector fields defined on a three dimensional manifold there is a class where tangencies of the invariant manifolds of two hyperbolic saddle foci occur densely the class is defined by the presence of the bykov cycle and by a condition on the parameters that determine the linear part of the vector field at the equilibria this has important consequences the global dynamics is persistently dominated by heteroclinic tangencies and by newhouse phenomena coexisting with hyperbolic dynamics arising from transversality the coexistence gives rise to linked suspensions of cantor sets with hyperbolic and non hyperbolic dynamics in contrast with the case where the nodes have the same chirality we illustrate our theory with an explicit example where tangencies arise in the unfolding of a symmetric vector field on the three dimensional sphere

transient analysis of the dynamic stress intensity factors using sgbem for frequency domain elastodynamics in this paper a two dimensional symmetric galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described the numerical implementation is carried out with quadratic elements allowing the use of an improved quarter point element for accurately determining frequency responses of the dynamic stress intensity factors dsifs to deal with singular and hypersingular integrals the formulation is decomposed into two parts the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities the treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary while the weakly singular integrals in the second part are handled by gauss quadrature time histories transient responses of the dsifs can be obtained in a post processing step by applying the standard fast fourier transform fft and algorithm to the frequency responses of these dsifs several test examples are presented for the calculation of the dsifs due to two types of impact loading heaviside step loading and blast loading the results suggest that the combination of the symmetric galerkin boundary element method and standard fft algorithms in determining transient responses of the dsifs is a robust and effective technique

toward a theory of rank one attractors introduction 1 statement of results part i preparation 2 relevant results from one dimension 3 tools for analyzing rank one maps part ii phase space dynamics 4 critical structure and orbits 5 properties of orbits controlled by critical set 6 identification of hyperbolic behavior formal inductive procedure 7 global geometry via monotone branches 8 completion of induction 9 construction of srb measures part iii parameter issues 10 dependence of dynamical structures on parameter 11 dynamics of curves of critical points 12 derivative growth via statistics 13 positive measure sets of good parameters appendices

react scatterplot smoothers superefficiency through basis economy abstract react estimators for the mean of a linear model involve three steps transforming the model to a canonical form that provides an economical representation of the unknown mean vector estimating the risks of a class of candidate linear shrinkage estimators and adaptively selecting the candidate estimator that minimizes estimated risk applied to one or higher way layouts the react method generates automatic scatterplot smoothers that compete well on standard datasets with the best fits obtained by alternative techniques historical precursors to react include nested model selection ridge regression and nested principal component selection for the linear model however react s insistence on working with an economical basis greatly increases its superefficiency relative to the least squares fit this reduction in risk and the possible economy of the discrete cosine basis of the orthogonal polynomial basis or of a smooth basis that generalizes the discrete cosine basis are illustrated by fitt

analytical integrations and sifs computation in 2d fracture mechanics analytical integrations in the framework of linear elastic problems modelled by means of boundary integral equations have been considered in a previous publication int j numer methods eng 2002 53 7 1695 1719 the present note aims at extending the subject to linear elastic fracture mechanics in such a context special shape functions have been recently proposed siam j appl math 1998 58 428 455 in order to increase accuracy in stress intensity factors approximation the closed form solution for integrals that arise from the boundary element method is a goal of the present work exploiting the analytical integrations asymptotical analysis around the crack tip are made possible with the purpose of formulating a coherent and accurate correlation between approximated stress intensity factors and crack opening displacements over the crack tip straight special elements copyright 2006 john wiley sons ltd

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

strange attractors in periodically kicked limit cycles and hopf bifurcations we prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with srb measures when certain simple dynamical systems are kicked at periodic time intervals the settings considered include limit cycles and stationary points undergoing hopf bifurcations

global bifurcations close to symmetry abstract heteroclinic cycles involving two saddle foci where the saddle foci share both invariant manifolds occur persistently in some symmetric differential equations on the 3 dimensional sphere we analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1 dimensional connection the saddle foci have the same chirality when part of the symmetry is broken the 2 dimensional invariant manifolds intersect transversely creating a heteroclinic network of bykov cycles we show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics there are n pulse heteroclinic tangencies trajectories that follow the original cycle n times around before they arrive at the other node each n pulse heteroclinic tangency is accumulated by a sequence of n 1 pulse ones this coexists with the suspension of horseshoes defined on an infinite set of disjoint strips where the first return map is hyperbolic we also show how as the system approaches full symmetry the suspended horseshoes are destroyed creating regions with infinitely many attracting periodic solutions

strange attractors with one direction of instability we give simple conditions that guarantee for strongly dissipative maps the existence of strange attractors with a single direction of instability and certain controlled behaviors only the d 2 case is treated in this paper although our approach is by no means limited to two phase dimensions we develop a dynamical picture for the attractors in this class proving they have many of the statistical properties associated with chaos positive lyapunov exponents existence of srb measures and exponential decay of correlations other results include the geometry of fractal critical sets nonuniform hyperbolic behavior symbolic coding of orbits and formulas for topological entropy

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

the collet eckmann condition for rational functions on the riemann sphere we show that the set of collet eckmann maps has positive lebesgue measure in the space of rational maps on the riemann sphere for any fixed degree d 2

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

global generic dynamics close to symmetry abstract our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in s 3 the cycle involves two saddle foci of different type and is structurally stable within the class of z 2 z 2 symmetric vector fields the cycle contains a two dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry breaking perturbations gradually breaking the symmetry in a two parameter family we get a wide range of dynamical behaviour an attracting periodic trajectory other heteroclinic trajectories homoclinic orbits n pulses suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of shilnikov type we also show that generically the coexistence of linked homoclinic orbits at the two saddle foci has codimension 2 and takes place arbitrarily close to the symmetric cycle

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

minimal attractors and bifurcations of random dynamical systems we consider attractors for certain types of random dynamical systems these are skew product systems whose base transformations preserve an ergodic invariant measure we discuss definitions of invariant sets attractors and invariant measures for deterministic and random dynamical systems under assumptions that include for example iterated function systems but that exclude stochastic differential equations we demonstrate how random attractors can be seen as examples of milnor attractors for a skew product system we discuss the minimality of these attractors and invariant measures supported by them as a further connection between random dynamical systems and deterministic dynamical systems we show how dynamical or d bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations and we relate the issue of branching at such d bifurcations to branching at blowout bifurcations

hopf saddle node bifurcation for fixed points of 3d diffeomorphisms analysis of a resonance bubble the dynamics near a hopf saddle node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study a two parameter model map g is constructed such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1 to investigate the effect of resonances the complex eigenvalues are selected to have a 1 5 resonance it is shown that near the origin of the parameter space the family g has two secondary hopf saddle node bifurcations of period five points a cone like structure exists in the neighbourhood formed by two surfaces of saddle node and a surface of hopf bifurcations quasi periodic bifurcations of an invariant circle forming a frayed boundary are numerically shown to occur in model g along such cantor like boundary an intricate bifurcation structure is detected near a 1 5 resonance gap subordinate quasi periodic bifurcations are found nearby suggesting the occurrence of a cascade of quasi periodic bifurcations c 2008 elsevier b v all rights reserved

quasi linear wavelet estimation abstract the main paradigm of the modern wavelet theory of spatial adaptation formulated by donoho and johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods on the other hand it is well known that linear adaptive estimates are the best whenever an estimated function is smooth is it possible to suggest a quasi linear wavelet estimate by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark sureshrink the answer is yes and we discuss quasi linear estimation both theoretically and via a monte carlo study in particular i show that asymptotically a quasi linear procedure not only matches properties of sureshrink over the besov scale but also allows

from invariant curves to strange attractors we prove that simple mechanical systems when subjected to external periodic forcing can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied in particular the existence of global strange attractors with fully stochastic properties is proved for a class of second order odes

high order boundary and finite elements for 3d fracture propagation in brittle materials abstract the quasi static propagation of fracture in brittle materials was studied in several recent publications a variational formulation salvadori 2008 salvadori and carini 2011 salvadori and fantoni 2013 led to three dimensional crack tracking strategies salvadori and fantoni 2014 2 3 salvadori and fantoni 2016 one of the complexities of this new class of algorithms is the evaluation of high order terms of the expansion of the crack opening and sliding in this paper new types of finite and boundary elements are formulated that capture the near crack front asymptotical displacement behavior up to the order 3 2 the use of these elements with the quasi static crack propagation algorithms of the above references is demonstrated for a simple crack configuration

dynamic elastic plastic analysis by a symmetric galerkin boundary element method with time independent kernels abstract boundary integral equations formulated by the combined use of single layer and double layer sources in such a way that the integral operator is symmetric with respect to a suitable bilinear form and their discretization by a galerkin weighted residual procedure are the basic features of the symmetric galerkin boundary element method sgbem this approach which has been applied to various mechanical problems in the last few years is proposed and developed herein for elastic plastic dynamic analysis using elastostatic kernels i e time independent green functions for unbounded spaces for displacements and tractions on the boundary and for stresses and displacements on the domain thus only double integrations are required instead of quadruple required by elastodynamic kernels at the price of domain integrals regularization procedures for singular integrals are developed assuming material behaviours as elastic plastic stable in drucker sense and suitable multifield modelling in space basic results for a computational theory are established extremum property and uniqueness of time step solution convergence of an iterative solution algorithm algorithmic stability along the step sequence finally the sgbem developed is corroborated by two numerical comparative tests

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

positive lyapunov exponents in families of one dimensional dynamical systems we give a generalization of jakobson s theorem ja and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures

on the dynamics of dominated splitting let f m m be a c 2 dieomorphism of a compact surface we give a complete description of the dynamics of any compact invariant set having dominated splitting in particular we prove a spectral decomposition theorem for the limit set l f under the assumption of dominated splitting moreover we describe all the bifurcations that these systems can exhibit and the dierent types of dynamics that could follow for small c r perturbations

periodic orbits for interval maps with sharp cusps abstract we study families of unimodal interval maps having a sharp cusp at a critical point we describe bifurcations and distribution of periodic orbits period adding for attracting periodic orbits is observed that is period k is followed by k 1 etc coexistence of attractors of successive periods as well as of an attracting fixed point and chaos occurs periodic orbits of various periods are grouped into bunches and occupy a relatively small part of the space

persistence of instanton connections in chemical reactions with time dependent rates the evolution of a system of chemical reactions can be studied in the eikonal approximation by means of a hamiltonian dynamical system the fixed points of this dynamical system represent the different states in which the chemical system can be found and the connections among them represent instantons or optimal paths linking these states we study the relation between the phase portrait of the hamiltonian system representing a set of chemical reactions with constant rates and the corresponding system when these rates vary in time we show that the topology of the phase space is robust for small time dependent perturbations in concrete examples and state general results when possible this robustness allows us to apply some of the conclusions on the qualitative behavior of the autonomous system to the time dependent situation

heteroclinic tangles in time periodic equations abstract in this paper we prove that when a heteroclinic loop is periodically perturbed three types of heteroclinic tangles are created and they compete in the space of where is a parameter representing the magnitude of the perturbations the three types are a transient heteroclinic tangles containing no gibbs measures b heteroclinic tangles dominated by sinks representing stable dynamical behavior and c heteroclinic tangles with strange attractors admitting srb measures representing chaos we also prove that as 0 the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections the theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical birkhoff melnikov smale method

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

entropy and rotation intervals for circle maps near saddle node bifurcations abstract consider a one parameter family of c r endomorphisms f lambda of the circle suppose that f 0 is on the boundary of an interval of phase locking and that f lambda generically unfolds f 0 so that f lambda exhibits nontrivial behavior for lambda 0 we show that the topological entropy and the width of the rotation interval of f lambda satisfy certain scaling laws as lambda searrow 0 and give estimates for each of these characteristics these scalings depend only on f 0 not on the particular family f lambda which unfolds it

large deviations for non uniformly expanding maps we obtain large deviation bounds for non uniformly expanding maps with non flat singularities or criticalities and for partially hyperbolic non uniformly expanding attracting sets that is given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map showing that the lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved as easy by products we deduce escape rates from subsets of the basins of physical measures for these types of maps the rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states

critical saddle node bifurcations and morse smale maps abstract we study the saddle node bifurcation in diffeomorphisms with a critical homoclinic orbit to the saddle node point in a typical family f of diffeomorphisms that undergoes a saddle node bifurcation at 0 the diffeomorphisms possibly after reparameterization for 0 have two periodic points which coalesce for 0 and then disappear for 0 if the saddle node points has a critical homoclinic orbit it is known that complicated dynamics can occur for 0 we show that there are families such that for 0 there are parameter values arbitrarily close to 0 for which the map is morse smale such parameter values are shown to have positive frequency at 0 in the process we show that the boundary of the set of morse smale diffeomorphisms possesses comb like structures we also show that there are other families unfolding a saddle node for which there are no morse smale maps for 0 these results rely heavily on projecting to circle endomorphisms we conclude with some numerical results from such maps

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

convolution quadrature method based symmetric galerkin boundary element method for 3 d elastodynamics the boundary integral equations in 3 d elastodynamics contain convolution integrals with respect to the time they can be performed analytically or with the convolution quadrature method the latter time stepping procedure s benefit is the usage of the laplace transformed fundamental solution therefore it is possible to apply this method also to problems where analytical time dependent fundamental solutions might not be known to obtain a symmetric formulation the second boundary integral equation has to be used which unfortunately requires special care in the numerical implementation since it involves hypersingular kernel functions therefore a regularization for closed surfaces of the laplace transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one supplementarily a weakly singular formulation of the laplace transformed elastodynamic double layer potential is presented this results in a time domain boundary element formulation involving at least only weakly singular integral kernels finally numerical studies validate this approach with respect to different spatial and time discretizations further a comparison with the wider used collocation method is presented it is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods copyright 2008 john wiley sons ltd

how often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency abstract here we study an amazing phenomenon discovered by newhouse s newhouse non density of axiom a a on s 2 in proc sympos pure math vol 14 amer math soc 1970 pp 191 202 s newhouse diffeomorphisms with infinitely many sinks topology 13 1974 9 18 s newhouse the abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms publ math inst hautes etudes sci 50 1979 101 151 it turns out that in the space of c r smooth diffeomorphisms diff r m of a compact surface m there is an open set u such that a baire generic diffeomorphism f u has infinitely many coexisting sinks in this paper we make a step towards understanding how often does a surface diffeomorphism have infinitely many sinks our main result roughly says that with probability one for any positive d a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by d or those whose period is relatively large compared to its cyclicity it verifies a particular case of palis conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are baire generic they have probability zero one of the key points of the proof is an application of newton interpolation polynomials to study the dynamics initiated in v kaloshin b hunt a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms i ann of math in press 92 pp v kaloshin a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms ii preprint 85 pp

strange attractors across the boundary of hyperbolic systems we study a bifurcation of axiom a hyperbolic vector fields in dimension three leading to robust strange attractors with singularities the axiom a vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift the attractors arising from this kind of bifurcation are not equivalent to the geometric lorenz attractors

suppression of chaos by cyclic parametric excitation in two dimensional maps we study the qualitative change of the dynamics of a generalized two dimensional quadratic map under the influence of parametric perturbations which operate in the chaotic parameter set it is shown that such perturbations can lead to the suppression of chaos and appearance of a regular periodic behaviour numerically we can argue that the suppression of chaos due to the parametric excitation is caused by a shift of the windows of periodic behaviour in the bifurcation diagram

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

singular heteroclinic cycles abstract we study the unfolding of heteroclinic cycles of vector fields in r n that possess a hyperbolic singularity and a saddle node the principal eigenvalues at the hyperbolic singularity are assumed to be real but the weak hyperbolic eigenvalues at the saddle node may be either real or complex conjugate we discuss the bifurcation diagrams of all codimension two such bifurcations in some of the occurring cases chaotic dynamics appears in the unfolding for the cases with only simple dynamics in the unfolding we obtain the complete bifurcation diagram we derive exponential expansions for the transition map near the saddle node

at the other side of a saddle node we describe phenomena occurring just before a saddle node bifurcation for one parameter families of interval maps in particular as a parameter approaches the bifurcation value attracting periodic orbits of periodsk k 1 k 2 k 3 can appear we make a detailed study of a family of cusp shaped maps where this phenomenon occurs in a pure form

solution of the basin problem for henon like attractors for a large class of non uniformly hyperbolic attractors of dissipative diffeomorphisms we prove that there are no holes in the basin of attraction stable manifolds of points in the attractor fill in a full lebesgue measure subset then lebesgue almost every point in the basin is generic for the srb sinai ruelle bowen measure of the attractor this solves a problem posed by sinai and by ruelle for this class of systems

bifurcation analysis of a system of coupled chemical oscillators bromate chlorite iodine abstract we examine the dynamical behavior of two chemical oscillators bromate iodide and chlorite iodide as well as the coupled system bromate chlorite iodide study of the period and amplitude of oscillation at the bifurcation points yields phase portraits for the uncoupled systems these are combined to give insights into the nature of the higher dimensional coupled system which exhibits a wide variety of dynamical phenomena including multiple stationary states birhythmicity and chaos analysis of the topology of the phase portraits and of the experimentally determined return map suggests that the coupled system may possess a lorenz type attractor

stable manifolds under very weak hyperbolicity conditions abstract we present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions

strange attractors and wandering domains near a homoclinic cycle to a bifocus abstract in this paper we explore the three dimensional chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence if the invariant manifolds of the bifocus satisfy a non degeneracy condition a sequence of hyperbolic suspended horseshoes arises near the cycle with one expanding and two contracting directions we extend previous results on the field and we show that the first return map to a given cross section may be approximated by a map exhibiting heteroclinic tangencies associated to two periodic orbits under an additional hypothesis about the coexistence of two heteroclinically related periodic points one without dominated splitting into one dimensional sub bundles the heteroclinic tangencies can be slightly modified in order to satisfy tatjer s conditions for a generalized tangency of codimension two this configuration may be seen as the organizing center by which one can obtain bogdanov takens bifurcations and therefore strange attractors infinitely many sinks and non trivial contracting wandering domains

ordered linear smoothers this paper deals with the following approach for estimating the mean mu of an n dimensional random vector y first a family s of n x n matrices is specified then an element s is an element of s is selected by mallows c l and mu s y the case is considered that s is an ordered linear smoother according to some easily interpretable qualitative conditions examples include linear smoothing procedures in nonparametric regression as e g smoothing splines minimax spline smoothers and kernel estimators stochastic probability bounds are given for the difference 1 n parallel to mu s y parallel to 2 2 1 n parallel to mu s mu y parallel to 2 2 where s mu denotes the minimizer of 1 n parallel to mu s y parallel to 2 2 for s is an element of s these probability bounds are generalized to the situation that s is the union of a moderate number of ordered linear smoothers the results complement work by li on the asymptotic optimality of c l implications for nonparametric regression are studied in detail it is shown that there exists a direct connection between james stein estimation and the use of smoothing procedures leading to a decision theoretic justification of the latter further conclusions concern the choice of the order of a smoothing spline or a minimax spline smoother and the rates of convergence of smoothing parameters

homoclinic tangencies and routes to chaos in a dripping faucet experiment we present the analysis of two attractor sequences from a dripping faucet experiment which were taken in different regions of the control parameter dripping rate we used symbolic dynamics analysis to enlighten the similarities between some experimental attractors and henon like attractors such similarities are clear evidences that the two transitions to chaos concern the breakup of a t2 torus due to a homoclinic bifurcation a homoclinic tangency of the manifolds of saddle node points embedded in the torus we also present a phenomenological model to explain how a dripping faucet experiment can show such homoclinic bifurcations quasiattractors and also long range anticorrelations behaviors similar to the ones presented by biological systems

a variational approach to boundary element elastodynamic analysis and extension to multidomain problems abstract time dependent discontinuities of both tractions and displacements are considered on the boundary of a homogeneous linear elastic solid conceived as embedded in the elastic space the analysis of the dynamic response of the solid to external actions is formulated in the time domain by means of suitable integral representations of boundary displacements and of boundary tractions using time dependent fundamental solutions the integral operator generated is proven to be symmetric in both space and time with respect to a time convolutive bilinear form as a consequence the history of boundary tractions and displacements along any assigned time interval 0 is shown to be characterized by a variational theorem the variational formulation developed is generalized to zonewise homogeneous systems and to elastically constrained bodies the linear algebraic equations generated by boundary element modelling in space and time turn out to exhibit symmetry of the coefficient matrix

symmetry breaking perturbations and strange attractors the asymmetrically forced damped duffing oscillator is introduced as a prototype model for analyzing the homoclinic tangle of symmetric dissipative systems with textit symmetry breaking disturbances even a slight fixed asymmetry in the perturbation may cause a substantial change in the asymptotic behavior of the system e g transitions from two sided to one sided strange attractors as the other parameters are varied moreover slight asymmetries may cause substantial asymmetries in the relative size of the basins of attraction of the unforced nearly symmetric attracting regions these changes seems to be associated with homoclinic bifurcations numerical evidence indicates that textit strange attractors appear near curves corresponding to specific secondary homoclinic bifurcations these curves are found using analytical perturbational tools

hyperbolicity and the creation of homoclinic orbits we consider one parameter families sp t e r of diffeomorphisms on surfaces which display a homoclinic tangency for yt 0 and are hyperbolic for yt 0 i e pf has a hyperbolic nonwandering set the tangency unfolds into transversal homoclinic orbits for yt positive for many of these families we prove that p is also hyperbolic for most small positive values of yt which implies much regularity of the dynamical structure a main assumption concerns the limit capacities of the basic set corresponding to the homoclinic tangency

hybrid shrinkage estimators using penalty bases for the ordinal one way layout this paper constructs improved estimators of the means in the gaussian saturated one way layout with an ordinal factor the least squares estimator for the mean vector in this saturated model is usually inadmissible the hybrid shrinkage estimators of this paper exploit the possibility of slow variation in the dependence of the means on the ordered factor levels but do not assume it and respond well to faster variation if present to motivate the development candidate penalized least squares pls estimators for the mean vector of a one way layout are represented as shrinkage estimators relative to the penalty basis for the regression space this canonical representation suggests further classes of candidate estimators for the unknown means monotone shrinkage ms estimators or soft thresholding st estimators or most generally hybrid shrinkage hs estimators that combine the preceding two strategies adaptation selects the estimator within a candidate class that minimizes estimated risk under the gaussian saturated one way layout model such adaptive estimators minimize risk asymptotically over the class of candidate estimators as the number of factor levels tends to infinity thereby adaptive hs estimators asymptotically dominate adaptive ms and adaptive st estimators as well as the least squares estimator local annihilators of polynomials among them difference operators generate penalty bases suitable for a range of numerical examples in case studies adaptive hs estimators recover high frequency details in the mean vector more reliably than pls or ms estimators and low frequency details more reliably than st estimators

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates

crack tip interpolation revisited it is well known that the near tip displacement eld on a crack surface can be represented in a power series in the variable p r where r is the distance to the tip it is shown herein that the coecients of the linear terms on the two sides of the crack are equal equivalently the linear term in the crack opening displacement vanishes the proof is a completely general argument valid for an arbitrary e g multiple nonplanar crack conguration and applied boundary conditions moreover the argument holds for other equations such as laplace a limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces evaluation of the nite surface stress and examination of potentially singular terms lead to the result inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress elds in the tip region and thus a more accurate evaluation of stress intensity factors