on the lebesgue measure of li yorke pairs for interval maps we investigate the prevalence of li yorke pairs for c2 and c3 multimodal maps f with non flat critical points we show that every measurable scrambled set has zero lebesgue measure and that all strongly wandering sets have zero lebesgue measure as does the set of pairs of asymptotic but not asymptotically periodic points

elliptic boundary value problems with λ dependent boundary conditions abstract in this paper second order elliptic boundary value problems on bounded domains r n with boundary conditions on depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework for a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure in the special case of rational nevanlinna or riesz herglotz functions on the boundary the solution operator is obtained in an explicit form in the product hilbert space l 2 l 2 m which is a natural generalization of known results on linear elliptic boundary value problems and rational boundary value problems for ordinary second order differential equations

invariant measures exist under a summability condition for unimodal maps for unimodal maps with negative schwarzian derivative a sufficient condition for the existence of an invariant measure absolutely continuous with respect to lebesgue measure is given namely the derivatives of the iterations of the map in the unique critical value must be so large that the sum of some root of the inverses is finite

on spurious detection of linear response and misuse of the fluctuation dissipation theorem in finite time series abstract using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems a goodness of fit statistics is developed for a linear statistical model of the observations based on results for central limit theorems for deterministic dynamical systems and used to detect linear response breakdown we apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series the magnitude of the perturbation and on the choice of the observable we find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed even for simple systems such as the logistic map one needs of the order of 10 6 observations to reliably detect the breakdown with a confidence level of 95 if less observations are available one may be falsely led to conclude that linear response theory is valid the amount of data required is larger the smaller the applied perturbation for judiciously chosen observables the necessary amount of data can be drastically reduced but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems furthermore we explore the use of the fluctuation dissipation theorem fdt in cases with limited data length or coarse graining of observations the fdt if applied naively to a system without linear response is shown to be very sensitive to the details of the sampling method resulting in erroneous predictions of the response

scrambled sets of continuous maps of 1 dimensional polyhedra let k be a 1 dimensional simplicial complex in r3 without isolated vertexes x iki be the polyhedron of k with the metric dk induced by k and f x x be a continuous map in this paper we prove that if k is finite then the interior of every scrambled set of f in x is empty we also show that if k is an infinite complex then there exist continuous maps from x to itself having scrambled sets with nonempty interiors and if x r or r then there exist c maps of x with the whole space x being a scrambled set

projection methods for nonlinear boundary value problems where l d l c x y is a linear map n x y is nonlinear and x and y are banach spaces recently cesari and mckenna s proved that if the underlying spaces possess a schauder basis or more generally they have what the authors call property l it is theoretically possible to establish a finite dimensional alternative problem for 1 3 their paper unities previous results 12 131 that established conditions under which 1 3 could be reduced to finite 282 0022 0396 85 3 00

galerkin s method for ordinary differential equations subject to generalized nonlinear boundary conditions where xk belongs to tc a finite dimensional subspace of cc l 11 r il ii and i qj is a projection mapping cc l 11 r i i x r onto a subspace of c 1 11 r 11 ii having the same dimension as rck there is a vast collection of literature concerning the application of galerkin s method to differential equations 1 3 12 16231 we make a special reference to the papers of cesari 4 5 and urabe 22 23 our results provide an extension and a unification of previous results of urabe that were conderned with the determination of periodic solutions to systems of ordinary differential equations 22 and with the solvability of 112 0022 0396192 3 00

nonlinear discrete sturm liouville problems at resonance abstract in this paper we study nonlinear bounded value problems of the form p t 1 y t 1 q t y t k y t f t y t h t a 11 y a a 12 y a 0 a 21 y b 1 a 22 y b 1 0 where k is an eigenvalue of the associated linear problem f is subject to the sublinear growth condition f t s a s b t a 1 b 1 s r for some 0 1 and a b 0 we prove the existence and multiplicity of solutions by using the connectivity properties of solution sets of parameterized families of compact vector fields

linear orderings and the full periodicity kernel for the n star abstract we show that baldwin s characterization of the set of periods of continuous self maps of the n star can be expressed in terms of a finite number of linear orderings additionally we study the minimal sets of periods which force a continuous self map of the n star to have periodic points of all periods

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

solvability of a nonlinear fourth order discrete problem at resonance let t be an integer with t 5 and let t 2 2 3 t we consider the nonlinear discrete boundary value problem d 4u t 2 l 1u t f t u t t f t h t t t 2 u 1 u t 1 d 2u 0 d 2u t 0 where l 1 is the first eigenvalue of the associated linear eigenvalue problem f is the corresponding eigenfunction f t 2xr r is continuous and f t s r with s 2 th t f t 0 we show the existence of solutions of the above problem our approaches are based on the krein rutman theorem connectivity properties of solution sets of parameterized families of compact vector fields

weakly nonlinear discrete multipoint boundary value problems abstract in this paper we study nonlinear discrete multipoint boundary value problems of the form x t 1 a t x t f t x t subject to b 0 x 0 b 1 x 1 b n x n 0 we provide sufficient conditions for the existence of solutions and we present a qualitative analysis of the way the solutions depend on the parameter

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

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

nonlinear discrete sturm liouville problems in this paper we study nonlinear boundary value problems of the form p t 1 y t 1 q t y t y t f y t t a 1 b 1 subject to a11y a a12 y a 0anda21y b 1 a22 y b 1 0 the parameter is an eigenvalue of the associated linear problem that is there is a nontrivial function u that satisfies the boundary conditions and also p t 1 u t 1 q t u t u t 0 for t in a 1 a 2 b 1 we establish sufficient conditions for the solvability of such problems in addition in those cases where the nonlinearity is small we provide a qualitative analysis of the relation between solutions of the nonlinear problem and eigenfunctions of the linear one

multipoint boundary value problems for nonlinear ordinary differential equations abstract in this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations we consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2 when this solution space is trivial we establish existence results via the schauder fixed point theorem in the resonance case we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem we accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

nonlinear discrete systems with global boundary conditions abstract this paper is devoted to the study of boundary value problems on infinite time intervals for nonlinear discrete time systems for problems of the form x k 1 f k x k h k k 0 1 2 subject to constraints or nonlocal boundary conditions of the form k 0 g k x k y we analyze how perturbations in h and y affect the existence of l solutions of this problem in this setting h is an element of l and y belongs to r p we also study the existence and behavior of bounded solutions to problems of the form x k 1 f k x k k 0 1 2 subject to k 0 g k x k 0 we place particular importance on the behavior of the solutions as a function of the parameter

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

bifurcation from infinity and multiple solutions for some discrete sturm liouville problems in this paper we study nonlinear discrete boundary value problems of the form d p t 1 dy t 1 q t y t l email protected y t f t y t h t email protected t a 1 1y a a 1 email protected a 0 a 2 1y b 1 a 2 email protected b 1 0 where l k is an eigenvalue of the associated linear problem l is a parameter and f satisfies the sublinear growth condition f t s a s a b email protected a 1 b 1 email protected r for some email protected email protected 1 and a email protected 0 we study the existence and multiplicity of solutions for the above mentioned problems by establishing some a priori bounds together with leray schauder continuation and bifurcation arguments

unbounded perturbations of nonlinear discrete periodic problem at resonance abstract in this paper we study the existence of solutions of nonlinear discrete boundary value problems 2 u t 1 k u t g t u t h t u 0 u t u 1 u t 1 where t 1 t h t r k is the k th eigenvalue of the linear problem 2 u t 1 u t 0 u 0 u t u 1 u t 1 g n r r satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u r

existence of solutions to functional boundary value problem of second order nonlinear differential equation abstract in this paper a functional boundary value problem is studied based on mawhin s coincidence degree theory some existence theorems are obtained in the case of non resonance and the cases of dim ker l 1 and dim ker l 2 at resonance the results not only generalize and improve some known results of multi point and integral boundary value problems but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x

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

quadratic maps without asymptotic measure an interval map is said to have an asymptotic measure if the time averages of the iterates of lebesgue measure converge weakly we construct quadratic maps which have no asymptotic measure we also find examples of quadratic maps which have an asymptotic measure with very unexpected properties e g a map with the point mass on an unstable fix point as asymptotic measure the key to our construction is a new characterization of kneading sequences

coupling induced statistical cycling in two diffusively coupled maps abstract this paper discusses the effects of diffusively coupling two identical one dimensional maps attention is focused on situations where the local isolated maps are statistically stable but where the coupled system is not a biologically motivated map and the quadratic map are numerically shown to display this behavior the piecewise linear tent map is then investigated analytically and we give a phase diagram of this system which displays the location of nonequilibrium phase transitions it is conjectured that the diffusive coupling of two chaotic but statistically stable maps i e with asymptotically stable perron frobenius operators can yield a two dimensional system which is not statistically stable whose associated perron frobenius operator is asymptotically periodic

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

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

on the solvability of nonlinear sturm liouville problems abstract the focus of this paper is the study of nonlinear differential equations subject to general non local boundary conditions to establish sufficient conditions for the existence of solutions we use properties of the nonlinearities and their relationship with the eigenvalues of an associated linear sturm liouville problem

statistical properties of unimodal maps the quadratic family we prove that almost every nonregular real quadratic map is colleteckmann and has polynomial recurrence of the critical orbit proving a conjecture by sinai it follows that typical quadratic maps have excellent ergodic properties as exponential decay of correlations keller and nowicki young and stochastic stability in the strong sense baladi and viana this is an important step in achieving the same results for more general families of unimodal maps

binary tree approach to scaling in unimodal maps ge rusjan and zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one dimensional unimodal maps we consider the scaling behavior in a modified tree which takes into account the self similarity of the window structure a nonuniversal geometric convergence of the associated superstable parameter values towards a misiurewicz point is observed for almost all binary sequences with periodic tails for these sequences the window period grows arithmetically down the binary tree there are an infinite number of exceptional sequences however for which the growth of the window period is faster numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences

solvability conditions for non local boundary value problems for two dimensional half linear differential systems abstract in this paper we consider two non local boundary value problems for two dimensional half linear differential systems we prove general fredholm type theorems which allow one to derive new efficient solvability criteria for the problems studied

continuous measures in one dimension families of unimodal maps satisfying 1 t 1 1 1 1 witht 1 1 and t 1 1 2 t x isc2 inx2 and and symmetric inx 3 t0 0 0 t1 0 1 with frac d d lambda t 0 0 are considered the results of guckenheimer 1982 are extended to show that there is a positive measure of for whicht has a finite absolutely continuous invariant measure

scalar discrete nonlinear multipoint boundary value problems abstract in this paper we provide sufficient conditions for the existence of solutions to scalar discrete nonlinear multipoint boundary value problems by allowing more general boundary conditions and by imposing less restrictions on the nonlinearities we obtain results that extend previous work in the area of discrete boundary value problems debra l etheridge jesus rodriguez periodic solutions of nonlinear discrete time systems appl anal 62 1996 119 137 debra l etheridge jesus rodriguez scalar discrete nonlinear two point boundary value problems j difference equ appl 4 1998 127 144

invariant measures exist without a growth condition given a non flat s unimodal interval map f we show that there exists c which only depends on the order of the critical point c such that if dfn f c c for all n sufficiently large then f admits an absolutely continuous invariant probability measure acip as part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small then f admits an acip as a special case any s unimodal map with critical order l 2 having no central returns possesses an acip these results imply that the summability assumptions in the theorems of nowicki van strien 21 and martens nowicki 17 can be weakened considerably

topological conditions for the existence of absorbing cantor sets this paper deals with strange attractors of s unimodal rnaps f it generalizes earlier results in the sense that very general topological conditions are given that either i guarantee the existence of an absorbing cantor set provided t he critical point of f is sufficiently degenerate or ii prohibit the existence of an absorbing cantor set altogether as a by product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for f

nonlocal boundary value problems for discrete systems abstract our goal in this paper is to provide sufficient conditions for the existence of solutions to discrete nonlinear systems subject to multipoint boundary conditions the criteria we present depends on the size of the nonlinearity and the set of solutions to the corresponding linear homogeneous boundary value problems our analysis is based on the lyapunov schmidt procedure and brouwer s fixed point theorem the results presented extend the previous work of d etheridge and j rodriguez 1996 1998 5 6 and j rodriguez and p taylor 2007 18 19

singular measures without restrictive intervals for the transformationt 0 1 0 1 defined byt x x 1 x with 0 4 a is shown to exist for whicht has no restrictive intervals hence is sensitive to initial conditions but for which no finite absolutely continuous invariant measure exists fort

generic family with robustly infinitely many sinks we show for every r d ge 0 or r d ge 2 the existence of a baire generic set of c d families of c r maps f a a in mathbb r k of a manifold m of dimension ge 2 so that for every a small the map f a has infinitely many sinks when the dimension of the manifold is ge 3 the generic set is formed by families of diffeomorphisms when m is the annulus this generic set is formed by local diffeomorphisms this is a counter example to a conjecture of pugh and shub

an alternative method for boundary value problems with large nonlinearities where g r r is continuously differentiable and b and d are nonsingular 12 x n matrices historically lyapunov schmidt and the alternative method are able to reduce the problem to a finite dimensional one provided the norm of the nonlinear term is sufficiently small in the region of interest that is the weakly nonlinear case recently the finite dimensional reduction for ordinary differential equations with periodic boundary conditions was accomplished 161 in this paper a constructive technique is developed that allows us to reduce 1 1 to a finite dimensional problem

regularity of the metric entropy for expanding maps the main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the perron frobenius operator for expanding mappings as a consequence we are able to show that the metric entropy of an expanding map has modulus of continuity x log 1 x on the space of c2 expandings we also give an explicit estimate of the rate of mixing for cl functions in terms of natural constants it seems that the method we present can be generalized to other classes of dynamical systems which have a distinguished invariant measure like axiom a diffeomorphisms it also can be adopted to show that the entropy of the quadratic family f x 1 ux2 computed with respect to the absolutely continuous invariant measure found in jakobson s theorem varies continuously the last result is going to appear somewhere else

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

on a resonance problem with the discrete p laplacian on finite graphs abstract in this paper we discuss the existence of a solution for a resonance problem involving the discrete p laplacian using asymptotic behaviors of g at infinity a nontrivial solution is established moreover we also find a necessary condition for asymptotic behavior of g at zero to guarantee the existence of at least two nontrivial solutions

on nonlinear discrete boundary value problems mo x 1 x n 0 1 4 where n is a fixed positive integer 8 represent n by n constant matrices a t is an n by n matrix for each te 0 l h t belongs to ir for each nonnegative integer t u is in r 1 is a real parameter f and g are continuously differentiable f r i ir and g r i r for systems 1 1 1 2 we prove that if the linear system x t l a l x t h t 1 5 subject to

onset of stochasticity for some one dimensional systems we study the appearance of stochastic dynamics in two situations 1 i for one parameter families of maps of the interval f undergoing an infinite sequence of period doubling bifurcations we consider 0 0 e beyond feigenbaum s parameter value 0 2 ii for a two parameter family of circle maps fb x x b2pisin2 x we consider b 1 1 e 0 1 in both cases we estimate the relative measure of parameter values corresponding to stochastic behavior

linear orderings and the sets of periods for star maps we show that baldwin s characterization of the set of periods of continuous self maps of the n star can be expressed in terms of a finite number of linear orderings