the transversal homoclinic solutions and chaos for stochastic ordinary differential equations abstract we consider the persistence of a transversal homoclinic solution and chaotic motion for ordinary differential equations with a homoclinic solution to a hyperbolic equilibrium under an unbounded random forcing driven by a brownian force by lyapunov schmidt reduction the persistence of transversal homoclinic solution is reduced to find the zeros of some bifurcation functions defined between two finite spaces it is shown that for almost all sample paths of the brownian motion the perturbed system exhibits chaos

non conservative perturbations of homoclinic snaking scenarios abstract homoclinic snaking refers to the continuation of homoclinic orbits to an equilibrium e near a heteroclinic cycle connecting e and a periodic orbit p typically homoclinic snaking appears in one parameter families of reversible conservative systems here we discuss perturbations of this scenario which are both non reversible and non conservative we treat this problem analytically in the spirit of the work 3 the continuation of homoclinic orbits happens with respect to both the original continuation parameter and the perturbation parameter the continuation curves are parametrised by the dwelling time l of the homoclinic orbit near p it turns out that l tends to zero while the vs l diagram displays isolas or criss cross snaking curves in a neighbourhood of the original snakes and ladder structure in the course of our studies we adapt both fenichel coordinates near p and the analysis of shilnikov problems near p to the present situation

a melnikov method for homoclinic orbits with many pulses we present an extension of the melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near integrable systems the melnikov function in this situation is the sum of the usual melnikov functions evaluated with some appropriate phase delays we show that a nonfolding condition which involves the logarithmic derivative of the melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist

chaotic motions near homoclinic manifolds and resonant tori in quasiperiodic perturbations of planar hamiltonian systems abstract we study chaotic dynamics of nonlinear oscillators with the form of a two frequency quasiperiodic perturbation of a planar hamiltonian system possessing a homoclinic orbit whose interior contains a one parameter family of periodic orbits in the extended phase space the unperturbed system has a three dimensional homoclinic manifold and a one parameter family of invariant 3 tori using melnikov s technique and the second order averaging method we show that chaotic motions may exist near the unperturbed homoclinic manifold and the unperturbed resonant tori these chaotic motions result from transverse intersection between the stable and unstable manifolds of normally hyperbolic invariant tori and are characterized by a generalization of the bernoulli shift we also give an example for the quasiperiodically forced duffing oscillator and demonstrate the existence chaotic motions by numerical simulation

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

heteroclinic bifurcation and singularly perturbed boundary value problems abstract we study a singularly perturbed boundary value problem in r m n x f x y e e y g x y e b 1 x 0 y 0 e 0 b 2 x 0 y 0 e 0 given a candidate for the 0th order approximation which exhibits both boundary layers and interior layers we present a complete procedure to compute higher order expansions and a procedure to compute the real solution near a truncated asymptotic expansion assuming the hyperbolicity of the regular layers and some generic assumptions similar results concerning the existence of periodic solutions relaxation oscillations are also presented several ideas from dynamical systems theory are employed e g exponential dichotomies fredholm alternatives and heteroclinic bifurcations

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

genericity of the multibump dynamics for almost periodic duffing like systems in this paper we consider slowly oscillating perturbations of almost periodic du ng like systems i e systems of the form u u a t t w0 u t r u rn where w c2n rn r is superquadratic and a and are positive and almost periodic by variational methods we prove that if 0 is small enough then the system admits a multibump dynamics as a corollary we get that the system u u a t w0 u t r u rn admits multibump solutions whenever a a where a is an open dense subset of a c r r a is almost periodic and a t 0 t r

dynamics of 4d symplectic maps near a double resonance abstract we study the dynamics of a family of 4 d symplectic mappings near a doubly resonant elliptic fixed point we derive and discuss algebraic properties of the resonances required for the analysis of a takens type normal form in particular we propose a classification of the double resonances adapted to this problem including cases of both strong and weak resonances around a weak double resonance a junction of two resonances of two different orders both being larger than 4 the dynamics can be described in terms of a simple in general non integrable hamiltonian model the non integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder we use a 4 d generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 d symplectic map we evaluate numerically the volume of a 4 d parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively in good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 d analog different numerical studies point out that double resonances play a key role to understand arnold diffusion this paper has to be seen also as a first step in this direction

melnikov functions and heteroclinic orbits in delay differential equations let c c r 01 r e 0 co be the banach space of continuous functions 4 r 0 r with the sup norm for any continuous function z r tr and per z e c is defined by means of z 0 z t e r 8 0 the main aim of this paper is to give sufficient conditions assuring the existence of bounded solutions to perturbed delay differential equations d d e of the following type

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

horseshoes in two degree of freedom hamiltonian systems with saddle centers we consider a class of two degree of freedom hamiltonian systems which have saddle centers with homoclinic orbits and do not take the form of small perturbations of integrable systems using a melnikov type global perturbation technique we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle centers we apply the theory to systems with potentials and give concrete examples in particular we can detect the presence of horseshoes in a class of systems to which previous results did not apply

use of simple shooting to obtain chaos abstract we discuss an elementary method of proving that certain smooth differential equations exhibit a kind of chaos in some cases such as a forced pendulum the proof is simpler than an approach through poincare maps and melnikov theory in other cases new results are obtained even for the pendulum there is a generalization of previous results because the forcing need not be periodic or almost periodic

transversality of double pulse homoclinic orbits in the inviscid lorenz krishnamurthy system abstract we study a 5 dimensional system originally introduced by lorenz 1986 apart from its atmospheric interest this system also exhibits interesting dynamical behavior we introduce a general technique based on the fenichel coordinates introduced by jones and kopell 1994 which systematically tracks the evolution of the tangent space of the unstable manifold along a trajectory that spends a long time o log i where is a small parameter inside a neighborhood of a two dimensional normally hyperbolic invariant manifold the estimates for the components of tangent vectors are nontrivial precisely because of this long time more specifically we establish the transversality of the double pulse homoclinic orbits constructed by camassa 1993 transversality gives rise to chaotic behavior near the double pulse homoclinic orbits in particular there is a countable infinity of homoclinic orbits in the system with even number of pulses

on the detection and dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations we develop a global perturbation technique similar to that of melnikov 1963 for detecting the presence of orbits homoclinic to hyperbolic periodic orbits and normally hyperbolic invariant tori in a class of ordinary differential equations our techniques are more general in that they apply to systems undergoing large amplitude excitation at low frequencies and to systems undergoing quasiperiodic excitation we discuss the dynamical consequences of orbits homoclinic to hyperbolic periodic orbits and the recent results concerning orbits homoclinic to tori obtained by wiggins 1986 and meyer and sell 1986 we illustrate the method with an example of a parametrically excited pendulum additionally we show how our results are complementary to a result of arnold 1962 concerning the infinite time behavior of adiabatic invariants in hamiltonian systems we illustrate this by using our method to prove the existence of smale horseshoes for a pendulum whose length undergoes a slow periodic variation

chaotic dynamics in a vertically driven planar pendulum as a first application we prove the presence of chaotic dynamics for a vertically driven planar pendulum the theory of topological horseshoes and linked twist maps developed so far along with phase plane analysis are used to show the presence of chaotic dynamics we propose in the general setting of topological spaces a definition of two dimensional oriented cell and consider maps which possess a property of stretching along the paths with respect to oriented cells for these maps we prove some theorems on the existence of fixed points periodic points and sequences of iterates which are chaotic in a suitable manner our results motivated by the study of the poincare map associated to some nonlinear equations extend and improve some recent work

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

the existence of localized vegetation patterns in a systematically reduced model for dryland vegetation abstract in this paper we consider the 2 component reaction diffusion model that was recently obtained by a systematic reduction of the 3 component gilad et al model for dryland ecosystem dynamics gilad et al 2004 the nonlinear structure of this model is more involved than other more conceptual models such as the extended klausmeier model and the analysis a priori is more complicated however the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations moreover we find that the model exhibits a richness of analytically tractable patterns that exceeds that of klausmeier type models our study focuses on the 4 dimensional dynamical system associated with the reaction diffusion model by considering traveling waves in 1 spatial dimension we use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic homoclinic periodic orbits that jump between normally hyperbolic slow manifolds representing various kinds of localized vegetation patterns the basic 1 front invasion patterns and 2 front spot gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model by exploiting the rich nonlinear structure of the model we construct many multi front patterns that are novel both from the ecological and the mathematical point of view in fact we argue that these orbits patterns are not specific for the model considered here but will also occur in a much more general singularly perturbed reaction diffusion setting we conclude with a discussion of the ecological and mathematical implications of our findings

on the geometry of an atmospheric slow manifold abstract we examine the hyperbolic structure and the invariant manifolds of a model proposed by lorenz to introduce the concept of an atmospheric slow manifold within the framework of dynamical system theory we address the question of the long time asymptotic behaviour of the system using the global geometric point of view it is shown that the model can be reduced to the classical example of a pendulum coupled to a harmonic oscillator the dynamical regimes of interest for the slow manifold hypothesis correspond to regions of phase space near the saddle center fixed point of this model which were not previously explored these phase space regions are analysed using a combination of melnikov type methods and ideas from singular perturbation theory by using the reversible symmetries of the model an extension of the melnikov theory is derived this extension allows us to find homoclinic orbits and determine their approximation by simply computing the zeros of a certain function which is constructed in terms of the usual melnikov function countable infinities of global homoclinic bifurcations and existence of chaotic dynamics can be shown to exist by using the new tool

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

counting coloured planar maps differential equations we address the enumeration of q coloured planar maps counted by the number of edges and the number of monochromatic edges we prove that the associated generating function is differentially algebraic that is satisfies a non trivial polynomial differential equation with respect to the edge variable we give explicitly a differential system that characterizes this series we then prove a similar result for planar triangulations thus generalizing a result of tutte dealing with their proper q colourings in statistical physics terms we solve the q state potts model on random planar lattices this work follows a first paper by the same authors where the generating function was proved to be algebraic for certain values of q including q 1 2 and 3 it is known to be transcendental in general in contrast our differential system holds for an indeterminate q for certain special cases of combinatorial interest four colours proper q colourings maps equipped with a spanning forest we derive from this system in the case of triangulations an explicit differential equation of order 2 defining the generating function for general planar maps we also obtain a differential equation of order 3 for the four colour case and for the self dual potts model

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

multi dimensional homoclinic jumping and the discretized nls equation we consider a class of dynamical systems that arise frequently in multi mode truncations and discretizations of partial differential equations including the perturbed nls we develop a general method to detect the existence of multi pulse solutions that are doubly asymptotic to an invariant manifold with two different time scales we use our method together with some recent results of li and mclaughlin to show the existence of several families of multi pulse orbits for the ablowitz ladik discretization of the perturbed nls these orbits include n pulse heteroclinic orbits and n pulse silnikov type orbits for arbitrarily large n

orbits homoclinic to resonances the hamiltonian case abstract in this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits hyperbolic fixed points or combinations of hyperbolic fixed points and or periodic orbits in a class of two degree of freedom integrable hamiltonian systems subject to arbitrary hamiltonian perturbations our methods differ from previous methods in that the invariant sets periodic orbits fixed points are created and become hyperbolic as a result of the interaction of the perturbation with a resonance in the unperturbed system this results in a very degenerate situation that requires a combination of geometric singular perturbation theory higher dimensional melnikov type methods and transversality theory we establish a simple energy phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance we apply our methods to a two mode truncation of the driven nonlinear schrodinger equation first studied by bishop et al in this example we show that as the energy is increased at resonance orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle node bifurcation of homoclinic orbits

the dilute potts model on random surfaces we present a new solution of the asymmetric two matrix model in the large n limit which only involves a saddle point analysis the model can be interpreted as ising in the presence of a magnetic field on random dynamical lattices with the topology of the sphere resp the disk for closed resp open surfaces we elaborate on the resulting phase diagram the method can be equally well applied to a more general q 1 matrix model which represents the dilute potts model on random dynamical lattices we discuss in particular duality of boundary conditions for open random surfaces

critical exponents on fortuin kasteleyn weighted planar maps in this paper we consider random planar maps weighted by the self dual fortuin kastelyn model with parameter q2 0 4 using a bijection due to sheeld and a connection to planar brownian motion in a cone we obtain rigorously the value of critical exponents associated with the length of cluster interfaces which is shown to be 4 arccos p 2 p q 2

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

construction and asymptotic stability of structurally stable internal layer solutions we introduce a geometric asymptotic method to treat structurally stable internal layer solutions we consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability proofs of the existence of exact solutions and eigenvalue eigenfunctions are outlined multi layered solutions are constructed by a new shooting method through a sequence of pseudo poincare mappings that do not require the transversality of the flow to cross sections the critical eigenvalues are determined by a coupling matrix that generates the slep matrix the transversality of the shooting method is related to the nonzeroness of the critical eigenvalues an equivalent approach is given to mono layer solutions they can be determined by the intersection of a fast jump surface and a slow switching curve which reduces fenichel s transversality condition to the slow manifold the critical eigenvalue is determined by the angle of the intersection we present three examples the first treats the critical eigenvalues of the system studied by angenent mallet paret peletier the second shows that a key lemma in the slep method may not hold the third is a perturbed activator inhibitor system that can have any number of mono layer solutions some of the solutions can only be found with the new shooting method

local convergence of large random triangulations coupled with an ising model we prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size n decorated by an ising configuration with a weight proportional to the energy of this configuration to do so we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition in particular we show that these partition functions all have the same phase transition at the same critical temperature some properties of the limiting object called the infinite ising planar triangulation are derived including the recurrence of the simple random walk at the critical temperature

dynamical systems analysis of the maasch saltzman model for glacial cycles abstract this article is concerned with the internal dynamics of a conceptual model proposed by maasch and saltzman 1990 to explain central features of the glacial cycles observed in the climate record of the pleistocene epoch it is shown that in most parameter regimes the long term system dynamics occur on certain intrinsic two dimensional invariant manifolds in the three dimensional state space these invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the volume of the north atlantic deep water are well separated and they are center manifolds when these characteristic time scales are comparable in both cases the reduced dynamics on these manifolds are governed by bogdanov takens singularities and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles in addition knowledge of the reduced systems and their bifurcations is useful for understanding the effects of slowly varying parameters which cause passage through hopf bifurcations and of orbital milankovitch forcing both are central to the mechanism proposed by maasch and saltzman for the mid pleistocene transition in their model

multi bump orbits homoclinic to resonance bands we establish the existence of several classes of multi bump orbits homoclinic to resonance bands for completely integrable hamiltonian systems subject to small amplitude hamiltonian or dissipative perturbations each bump is a fast excursion away from the resonance band and the bumps are interspersed with slow segments near the resonance band the homoclinic orbits which include multi bump silnikov orbits connect equilibria and periodic orbits in the resonance band the main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

phase change between separatrix crossings abstract the change of the crossing parameter essentially the phase between separatrix crossings is calculated for hamiltonian systems with one degree of freedom and slow time dependence this completes the calculation of the map for an arbitrary sequence of separatrix crossings the change of the crossing parameter is used to calculate the retrapping probability it is found that even in the limit of infinitely slow time dependence correlations persist between the separatrix crossings and corrections to the usual zero order adiabatic approximation must be included comparison with numerical experiments show that the result derived here is accurate for quite large values of the slowness parameter

using the extended melnikov method to study the multi pulse global bifurcations and chaos of a cantilever beam the aim of this paper is to investigate the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended melnikov method in the resonant case first the extended melnikov method for studying the shilnikov type multi pulse homoclinic orbits and chaos in high dimensional nonlinear systems is briefly introduced in the theoretical frame then this method is utilized to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the cantilever beam how to employ this method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications is demonstrated through this example finally the results of numerical simulation are given and also show that the shilnikov type multi pulse chaotic motions can occur for the nonlinear non planar oscillations of the cantilever beam which verifies the analytical prediction

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

smooth linearization near a fixed point in this paper we extend a theorem of sternberg and bi leckii we study a vector field or a diffeomorphism in the vicinity of a hyperbolic fixed point we assume that the eigenvalues of the linear part a at the fixed point satisfy qth order algebraic inequalities where q 2 2 then there is ck linearization in the vicinity of the fixed point where k is the q smoothness of a we give an explicit and simple algorithm for com puting k an illustrative example from celestial mechanics is included which shows that our main theorem is an improvement over previously known theories we also show that if the vector field or diffeomorphism depends smoothly on a parameter then the linearizing conjugation de pends smoothly on the parameter

subharmonic and homoclinic bifurcations in the driven and damped sine gordon system abstract chaotic responses induced by an applied biharmonic driven signal on the sine gordon sg system influenced by a constant dc driven and the damping fields are investigated using a collective coordinate approach for the motion of the breather in the system for this biharmonic signal one term has a large amplitude at low frequency thus the classical melnikov method does not apply to such a system however we use the modified version of the melnikov method to homoclinic bifurcations of the perturbed sg system additionally resonant breathers are studied using the modified subharmonic melnikov theory this dynamic behavior is illustrated by some numerical computations

bounded solutions to singularly perturbed systems of o d e abstract we consider existence and stability properties of bounded solutions to singular systems of o d e the method we suggest considers the concept of singular maps between scales of banach spaces and their properties this theory is then applied to obtain existence and transversality of bounded solutions to singular systems of o d e

multiple transverse homoclinic solutions near a degenerate homoclinic orbit consider an autonomous ordinary differential equation in r n that has a homoclinic solution asymptotic to a hyperbolic equilibrium the homoclinic solution is degenerate in the sense that the linear variational equation has 2 bounded linearly independent solutions we study bifurcation of the homoclinic solution under periodic perturbations using exponential dichotomies and lyapunov schmidt reduction we obtain general conditions under which the perturbed system can have transverse homoclinic solutions and nearby periodic or chaotic solutions published by elsevier inc

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

drift of slow variables in slow fast hamiltonian systems abstract we study the drift of slow variables in a slow fast hamiltonian system with several fast and slow degrees of freedom keeping the slow variables frozen for any periodic trajectory of the fast subsystem we define an action for a family of periodic orbits the action is a scalar function of the slow variables and can be considered as a hamiltonian function which generates some slow dynamics these dynamics depend on the family of periodic orbits assuming that for the frozen slow variables the fast system has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories we prove that for any path composed of a finite sequence of slow trajectories generated by action hamiltonians there is a trajectory of the full system whose slow component shadows the path

smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems abstract in this paper we study linear differential systems 1 x a t x where a is an n n matrix valued function defined on the k torus t k and t t is a given irrational twist flow on t k first we show that if a c n t k where n 0 1 2 then the spectral subbundles are of class c n on t k next we assume that a is sufficiently smooth on t k and satisfies a suitable small divisors inequality we show that if 1 satisfies the full spectrum assumption then there is a quasi periodic linear change of variables x p t y that transforms 1 to a constant coefficient system y by finally we study the case where the matrix a t in 1 is the jacobian matrix of a nonlinear vector field x evaluated along a quasi periodic solution x t of 2 x x we give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system z defined in the vicinity of h the hull of so that the linearized system 1 can be represented in the form z dz where d is a constant matrix our results represent substantial improvements over known methods because we do not require that a be close to a constant coefficient system

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

scaling limits of random planar maps with large faces we discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha in 1 2 when the number n of vertices of the map tends to infinity the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process which can be constructed from a centered stable process with no negative jumps and index alpha in particular the profile of distances in the map rescaled by the factor n 1 2 alpha converges to a random measure defined in terms of the distance process with the same rescaling of distances the vertex set viewed as a metric space converges in distribution as n to infty at least along suitable subsequences towards a limiting random compact metric space whose hausdorff dimension is equal to 2 alpha

more on the o n model on random maps via nested loops loops with bending energy we continue our investigation of the nested loop approach to the o n model on random maps by extending it to the case where loops may visit faces of arbitrary degree this allows us to express the partition function of the o n loop model as a specialization of the multivariate generating function of maps with controlled face degrees where the face weights are determined by a fixed point condition we deduce a functional equation for the resolvent of the model involving some ring generating function describing the immediate vicinity of the loops when the ring generating function has a single pole the model is amenable to a full solution physically such a situation is realized upon considering loops visiting triangles only and further weighting these loops by some local bending energy our model interpolates between the two previously solved cases of triangulations without bending energy and quadrangulations with rigid loops we analyze the phase diagram of our model in details and derive in particular the location of its non generic critical points which are in the universality classes of the dense and dilute o n model coupled to 2d quantum gravity similar techniques are also used to solve a twisting loop model on quadrangulations where loops are forced to make turns within each visited square along the way we revisit the problem of maps with controlled possibly unbounded face degrees and give combinatorial derivations of the one cut lemma and of the functional equation for the resolvent

multi pulse chaotic dynamics in non planar motion of parametrically excited viscoelastic moving belt abstract this paper investigates the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the parametrically excited viscoelastic moving belt using an extended melnikov method in the resonant case using the kelvin type viscoelastic constitutive law and hamilton s principle the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation applying the method of multiple scales and galerkin s approach to the partial differential governing equation the four dimensional averaged equation is obtained for the case of 1 1 internal resonance and primary parametric resonance from the averaged equations obtained the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues based on the explicit expressions of normal form the extended melnikov method is used for the first time to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics the paper demonstrates how to employ the extended melnikov method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications numerical simulations show that for the nonlinear non planar oscillations of the viscoelastic moving belt the shilnikov type multi pulse chaotic motions can occur overall both theoretical and numerical studies suggest that the chaos for the smale horseshoe sense in motion exists

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

homoclinic motions and chaos in the quasiperiodically forced van der pol duffing oscillator with single well potential we study a two frequency quasiperiodically forced oscillator with single well po tential which reduces to the van der pol and duffing oscillators in certain special cases the unperturbed system without damping and forcing terms has a one parameter family of periodic orbits we concentrate on the dynamics near the unperturbed resonant periodic orbits using the second order averaging method and a version of melnikov s method we show that when double resonance occurs the stable and unstable manifolds of normally hyperbolic invariant tori intersect transversely i e transverse homoclinic motions exist near the unperturbed reso nant periodic orbits in certain parameter regions such homoclinic motions yield chaotic dynamics characterized by a generalization of the bernoulli shift numer ical simulation results are also given to demonstrate the theoretical results

critical behaviour of the fully packed loop model on the square lattice we investigate the critical behaviour of the fully packed o n loop model on the square lattice in which each vertex is visited once by a loop a transfer matrix analysis shows that this model can be interpreted as a superposition of a low temperature o n model and an solid on solid sos model as for the fully packed model on the honeycomb lattice however not all of the critical exponents are the same for both lattices in contrast the fully packed model on the triangular lattice appears to behave as a pure low temperature o n model

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

higher order melnikov theory for adiabatic systems in this paper we study adiabatic hamiltonian systems including those subject to small amplitude forcing and damping it is known that simple zeroes of the adiabatic poincare arnold melnikov function imply the existence of primary intersection points of the stable and unstable manifolds of hyperbolic orbits here we present an nth order melnikov function whose simple zeroes correspond to nth order transverse intersection points and hence to n pulse homoclinic orbits using this function it can be shown that n pulse homoclinic orbits arise in a plethora of adiabatic models including systems with slowly varying potentials the theory is illustrated on a damped hamiltonian system with a slowly varying cubic potential in addition the nth order adiabatic melnikov function is useful for showing the existence of multi pulse resonant periodic orbits in the special class of slow time periodic systems

simple maps hurwitz numbers and topological recursion we introduce the notion of fully simple maps which are maps with non self intersecting disjoint boundaries in contrast maps where such a restriction is not imposed are called ordinary we study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder we show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks we also obtain an elegant formula for cylinders these relations reproduce the relation between moments and higher order free cumulants established by collins et al 22 and implement the symplectic transformation x leftrightarrow y on the spectral curve in the context of topological recursion we conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y we propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1 hermitian matrix model which is believed to be true but has not been proved yet our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion our argument relies on an unconditional matrix model interpretation of fully simple maps via the formal hermitian matrix model with external field we also deduce a universal relation between generating series of fully simple maps and of ordinary maps which involves double monotone hurwitz numbers in particular ordinary maps without internal faces which are generated by the gaussian unitary ensemble and with boundary perimeters lambda 1 ldots lambda n are strictly monotone double hurwitz numbers with ramifications lambda above infty and 2 ldots 2 above 0 combining with a recent result of dubrovin et al 24 this implies an elsv like formula for these hurwitz numbers

loop models on random maps via nested loops the case of domain symmetry breaking and application to the potts model we use the nested loop approach to investigate loop models on random planar maps where the domains delimited by the loops are given two alternating colors which can be assigned different local weights hence allowing for an explicit z2 domain symmetry breaking each loop receives a non local weight n as well as a local bending energy which controls loop turns by a standard cluster construction that we review the q n2 potts model on general random maps is mapped to a particular instance of this problem with domain non symmetric weights we derive in full generality a set of coupled functional relations for a pair of generating series which encode the enumeration of loop configurations on maps with a boundary of a given color and solve it by extending well known complex analytic techniques in the case where loops are fully packed we analyze in detail the phase diagram of the model and derive exact equations for the position of its non generic critical points in particular we underline that the critical potts model on general random maps is not self dual whenever q 1 in a model with domain symmetric weights we also show the possibility of a spontaneous domain symmetry breaking driven by the bending energy this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

a recursive approach to the o n model on random maps via nested loops we consider the o n loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops this follows from a combinatorial decomposition that consists in cutting the o n model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees in the induced statistics these maps are drawn according to a boltzmann distribution whose parameters the face weights are determined by a fixed point condition in particular we show that the dense and dilute critical points of the o n model correspond to bipartite maps with large faces i e whose degree distribution has a fat tail the re expression of the fixed point condition in terms of linear integral equations allows us to explore the phase diagram of the model in particular we determine this phase diagram exactly for the simplest version of the model where the loops are rigid several generalizations of the model are discussed

n pulse homoclinic orbits in perturbations of resonant hamiltonian systems in this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable two degree of freedom resonant hamiltonian systems our energy phase method applies to both hamiltonian and dissipative perturbations and reveals families of multi pulse solutions which are not amenable to melnikov type methods as an example we study a two mode approximation of the nonlinear nonplanar oscillations of a parametrically forced inextensional beam in this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically

limits of the boundary of random planar maps we discuss asymptotics for the boundary of critical boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter alpha in 1 2 first in the dense phase corresponding to alpha in 1 3 2 we prove that the scaling limit of the boundary is the random stable looptree with parameter 1 alpha 1 2 second we show the existence of a phase transition through local limits of the boundary in the dense phase the boundary is tree like while in the dilute phase corresponding to alpha in 3 2 2 it has a component homeomorphic to the half plane as an application we identify the limits of loops conditioned to be large in the rigid o n loop model on quadrangulations proving thereby a conjecture of curien and kortchemski

combinatorics of bicubic maps with hard particles we present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles this is done by the use of a bijection with a particular class of blossom trees with particles obtained by an appropriate cutting of the maps although these trees have no simple local characterization we prove that their enumeration may be performed upon introducing a larger class of admissible trees with possibly doubly occupied edges and summing them with appropriate signed weights the proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non sectile edges the admissible trees are characterized by simple local rules allowing eventually for an exact enumeration of planar bicubic maps with hard particles we also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re derive the enumeration of quartic maps with ising spins in the present framework of admissible trees we finally comment on a possible interpretation in terms of branching processes

martingales in self similar growth fragmentations and their connections with random planar maps the purpose of the present work is twofold first we develop the theory of general self similar growth fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework in particular we establish many to one formulas for growth fragmentations and define the notion of intrinsic area of a growth fragmentation second we identify a distinguished family of growth fragmentations closely related to stable levy processes which are then shown to arise as the scaling limit of the perimeter process in markovian explorations of certain random planar maps with large degrees which are roughly speaking the dual maps of the stable maps of le gall and miermont in ann probab 39 1 69 2011 as a consequence of this result we are able to identify the law of the intrinsic area of these distinguished growth fragmentations this generalizes a geometric connection between large boltzmann triangulations and a certain growth fragmentation process which was established in bertoin et al ann probab accepted

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

exponential dichotomies heteroclinic orbits and melnikov functions the interest in the behaviour of the solutions to perturbed autonomous systems with bounded perturbative term like

the gaussian map for rational ruled surfaces in this paper the gaussian map 2 h 0 c k h 0 c 3k of a smooth curve c lying on a minimal rational ruled surface is computed it is shown that the corank of is determined for almost all such curves by the rational surface in which it lies hence expect for some special cases a curve cannot lie on two nonisomorphic minimal rational ruled surfaces

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere

nonintersecting string model and graphical approach equivalence with a potts model using a graphical method we establish the exact equivalence of the partition function of aq state nonintersecting string nis model on an arbitrary planar even valenced lattice with that of a q2 state potts model on a related lattice the nis model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types and the resulting potts model generally has multispin interactions for the square and kagome lattices this leads to the equivalence of a staggered nis model with potts models with anisotropic pair interactions indicating that these nis models have a first order transition forq 2 for the triangular lattice the nis model turns out to be the five vertex model of wu and lin and it relates to a potts model with two and three site interactions the most general model we discuss is an oriented nis model which contains the six vertex model and the nis models of stroganov and schultz as special cases