subharmonic solutions in singular systems abstract we consider the problem of bifurcation from homoclinic towards periodic orbits for a periodic singular system of differential equations we use a functional analytic approach and the lyapunov schmidt method to obtain a bifurcation function for this problem which tends as e 0 to the well known melnikov function for the bifurcation towards transverse homoclinic orbits we also study the behavior of the bifurcation functions as the period of the periodic orbit tends to infinity

multiple internal layer solutions generated by spatially oscillatory perturbations abstract for a singularly perturbed system of reaction diffusion equations we study the bifurcation of internal layer solutions due to the addition of a spatially oscillatory term in the singular limit the existence and stability of internal layer solutions are determined by the intersection of a fast jump surface 1 and a slow switching curve c the case when the intersection is transverse was studied by x b lin construction and asymptotic stability of structurally stable internal layer solutions preprint in this paper we show that when 1 intersects with c tangentially saddle node or cusp type bifurcation may occur higher order expansions of internal layer solutions and eigenvalue eigenfunctions are also presented to find a true internal layer solution and true eigenvalue eigenfunctions we use a newton s method in functions spaces that is suitable for numerical computations

equilibrium phase transitions in coupled map lattices a pedestrian approach a class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed it can be solved analytically in terms of the statistical mechanics of spin lattices the corresponding hamiltonian is written down explicitly in terms of the parameters of the map the approach follows the line of recent mathematical investigations but the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail although the method works only for map lattices with repelling invariant sets some of the conclusions i e the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical systems

non trivial collective behavior in coupled map lattices a transfer operator perspective abstract we discuss the link between the presence of non trivial collective behavior in coupled map lattices and the spectral properties of their transfer or perron frobenius operator in particular it is argued that non trivial collective behavior corresponds to a perron frobenius operator possessing a cyclical spectral decomposition known as asymptotic periodicity we also discuss to what extent changes in the spectral properties of the perron frobenius operator are related to the phase transitions observed between two types of non trivial collective behavior

monte carlo simulation of classical spin models with chaotic billiards it has recently been shown that the computing abilities of boltzmann machines or ising spin glass models can be implemented by chaotic billiard dynamics without any use of random numbers in this paper we further numerically investigate the capabilities of the chaotic billiard dynamics as a deterministic alternative to random monte carlo methods by applying it to classical spin models in statistical physics first we verify that the billiard dynamics can yield samples that converge to the true distribution of the ising model on a small lattice and we show that it appears to have the same convergence rate as random monte carlo sampling second we apply the billiard dynamics to finite size scaling analysis of the critical behavior of the ising model and show that the phase transition point and the critical exponents are correctly obtained third we extend the billiard dynamics to spins that take more than two states and show that it can be applied successfully to the potts model we also discuss the possibility of extensions to continuous valued models such as the xy model

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

bifurcation from a heteroclinic solution in differential delay equations on etudie une classe d equations differentielles fonctionnelles x t af x t 1 a non linearite periodique f r r o f dans ao et f o dans o b f a f o f b o on demontre que pour une certaine valeur critique a a o il existe une solution heterocline allant de la solution d equilibre t a a l equilibre t b

exponential dichotomies and homoclinic orbits in functional differential equations abstract suppose an autonomous functional differential equation has an orbit which is homoclinic to a hyperbolic equilibrium point the purpose of this paper is to give a procedure for determining the behavior of the solutions near of a functional differential equation which is a nonautonomous periodic perturbation of the original one the procedure uses exponential dichotomies and the fredholm alternative it is also shown that any smooth function p t defined on the reals which approaches zero monotonically as t is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero examples illustrating the results are also given

subharmonic solutions of systems of difference equations with periodic perturbations part i existence we consider the existence of subharmonic solutions of systems of difference equations with periodic perturbations the theory of coincidence degree coupled with some detailed a priori estimates is applied to show that the perturbed system admits subharmonic solutions near a hyperbolic periodic orbit of the unperturbed system if the perturbation is sufficiently small

subharmonic solutions bifurcated from homoclinic orbits for weakly coupled singular systems abstract the problem of bifurcation from homoclinic solution towards periodic solution was considered for weekly coupled singular systems by using functional analytic approach based on the lyapunov schmidt reduction we obtained some functions h r d 1 r r d the simple roots of the equations h 0 correspond to the existence of subharmonic solutions and if the vector field is 2 period then for any integer m the weakly coupled singular system has 2 m period solution

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

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

phase ordering and persistence relative effects of space discretization chaos and anisotropy abstract the peculiar phase ordering properties of a lattice of coupled chaotic maps studied recently lemaitre chate phys rev lett 82 1999 1140 are revisited with the help of detailed investigations of interface motion it is shown that normal curvature driven like domain growth is recovered at larger scales than considered before and that the persistence exponent seems to be universal using generalized persistence spectra the properties of interface motion in this deterministic chaotic lattice system are found to interpolate between those of the two canonical reference systems the probabilistic ising model and the deterministic space continuous time dependent ginzburg landau equation

chaotic behavior in differential equations driven by a brownian motion abstract in this paper we investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed point under an unbounded random forcing driven by a brownian motion we prove that for almost all sample paths of the brownian motion in the classical wiener space the forced equation admits a topological horseshoe of infinitely many branches this result is then applied to the randomly forced duffing equation and the pendulum equation

bifurcation from homoclinic to periodic solutions in singular ordinary differential equations abstract problems of bifurcations from homoclinic to periodic orbits are considered for periodic singular differential equations a functional analytic approach together with the method of lyapunov schmidt is used to obtain bifurcation functions for these problems which yield periodic solutions of all integer multiples of the period of the vector field

steady flows of inviscid fluids under localized perturbations on essaie d obtenir tous les ecoulements stationnaires dans un canal avec un petit obstacle pour des nombres de froude proches de la valeur critique fo

codimension 3 bifurcations of a class of homoclinic loop with saddle point abstract in this work bifurcations in the class that the homoclinic orbit connects the strong stable and strong unstable manifolds of a saddle are investigated for four dimensional vector fields the existence numbers coexistence and incoexistence of 1 homoclinic orbit 1 periodic orbit 2 n homoclinic orbit and 2 n periodic orbit are obtained the approximate expressions of the corresponding bifurcation surfaces and the bifurcation diagrams are also presented

microextensive chaos of a spatially extended system by analyzing chaotic states of the one dimensional kuramoto sivashinsky equation for system sizes l in the range 79l ll 93 we show that the lyapunov fractal dimension d scales microextensively increasing linearly with l even for increments ensuremath delta l that are small compared to the average cell size of 9 and to various correlation lengths this suggests that a spatially homogeneous chaotic system does not have to increase its size by some characteristic amount to increase its dynamical complexity

multi hump pulses in systems with reflection and phase invariance abstract we investigate the existence and stability of standing and travelling multi hump waves in partial differential equations with reflection and phase symmetries we focus on 2 and 3 pulse solutions that arise near bi foci and apply our results to the complex cubic quintic ginzburg landau equation

heteroclinic chaotic behavior driven by a brownian motion abstract in this paper we study the chaotic behavior of a planar ordinary differential system with a heteroclinic loop driven by a brownian motion an unbounded random forcing unlike the case of homoclinic loops two random melnikov functions are needed in order to investigate the intersection of stable segments of one saddle and unstable segments of the other saddle we prove that for almost all paths of the brownian motion the forced system admits a topological horseshoe of infinitely many branches we apply this result to the josephson junction and the soft spring duffing oscillator

phase transitions in coupled map lattices and in associated probabilistic cellular automata analytical tools are applied to investigate piecewise linear coupled map lattices in terms of probabilistic cellular automata the so called disorder condition of probabilistic cellular automata is closely related with attracting sets in coupled map lattices the importance of this condition for the suppression of phase transitions is illustrated by spatially one dimensional systems invariant densities and temporal correlations are calculated explicitly ising type phase transitions are found for one dimensional coupled map lattices acting on repelling sets and for a spatially two dimensional miller huse like system with stable long time dynamics critical exponents are calculated within a finite size scaling approach the relevance of detailed balance of the resulting probabilistic cellular automaton for the critical behavior is pointed out

stabilization of causally and non causally coupled map lattices abstract two dimensional coupled map lattices have global stability properties that depend on the coupling between individual maps and their neighborhood the action of the neighborhood on individual maps can be implemented in terms of causal coupling to spatially distant past states or non causal coupling to spatially distant simultaneous states in this contribution we show that globally stable behavior of coupled map lattices is facilitated by causal coupling thus indicating a surprising relationship between stability and causality the influence of causal versus non causal coupling for synchronous and asynchronous updating as a function of coupling strength and for different neighborhoods is analyzed in detail

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

the sampling theorem dirichlet series and hankel transforms abstract some very surprising relations between fundamental theorems and formulas of signal analysis of analytic number theory and of applied analysis are presented it is shown that generalized forms of the classical whittaker kotelnikov shannon sampling theorem as well as of the brown butzer splettstoser approximate sampling expansion for non band limited signal functions can be deduced via the theory of dirichlet series with functional equations from a new summation formula for hankel transforms this counterpart to poisson s summation formula is shown to be essentially equivalent to the famous functional equation of riemann s zeta function to the modular relation of the theta function to the nielsen doetsch summation formula for bessel functions and to the partial fraction expansion of the periodic hilbert kernel

critical exponents for extended dynamical systems with simultaneous updating the case of the ising model the behavior of a simultaneously updated metropolis simulation of the two dimensional 2d ising model is explored the configurations generated by such a simulation are not constructed to be equilibrium configurations and may therefore display a different critical behavior assuming a phase transition is present this is relevant to the recently posed question of how the updating scheme affects the critical exponents for chaotic extended systems that show continuous phase transitions it is found that the simultaneously updated algorithm drives the lattice for all non zero temperatures into blinking maximum energy configurations making it of little use a small modification given by a reduction in the acceptance probabilities for spin flips restores the ability of the algorithm to generate ordered and disordered states with a well defined phase transition between them the critical temperature becomes a function of the acceptance probability for energy lowering spin flips the critical exponents still fall into the 2d ising universality class this happens even though the simultaneously updated metropolis algorithm does not really simulate the ising model in equilibrium

percolation transition at growing spatiotemporal fractal patterns in models of mesoscopic neural networks spike packet propagation is modeled in mesoscopic scale networks composed of locally and recurrently coupled neural pools and embedded in a two dimensional lattice site dynamics is governed by three key parameters pool connectedness probability synaptic strength following the steady state distribution of some realizations of spike timing dependent plasticity learning rule and the neuron refractoriness formation of spatiotemporal patterns in our model synfire chains exhibits critical behavior with the emerging percolation phase transition controlled by the probability for nonzero synaptic strength value applying the finite size scaling method we infer the critical probability dependence on synaptic strength and refractoriness and determine the effects of connection topology on the pertaining percolation clusters fractal dimensions we find that the directed percolation and the pair contact process with diffusion constitute the relevant universality classes of phase transitions observed in a class of mesoscopic scale network models which may be related to recently reported data on in vitro cultures

chaos in singular impulsive o d e it is well known l that the existence of a transversal homoclinic orbit of a diffeomorphism implies chaos i e the diffeomorphism possesses a compact domain which is invariant for some of its iterates and such that the dynamics of such an iterate on it is homeomorphic to the bernoulli shift on a finite number of symbols however the location of transversal homoclinic points is generally difficult one of the most common approaches is a perturbation method based on the derivation of the so called melnikov functions 2 i e we look for a transversal homoclinic point of a perturbed diffeomorphism when the unperturbed diffeomorphism has a known dynamics analytical methods based on the lyapunov schmidt procedure are presented in 3 51 for periodically regularly perturbed autonomous ordinary differential equations o d e and in 6 81 for diffeomorphisms regularly perturbed impulsive o d e are studied in 8 singular perturbation problems are studied analytically in 9 lo for o d e and in l 11 for impulsive o d e in all these papers it is assumed that the so called reduced o d e has a homoclinic orbit in particular the following impulsive o d e is studied in ll

bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite dimensional systems sous certaines conditions generiques on montre comment une orbite periodique stable unique peut bifurquer a partir d une orbite homocline pour des equations paraboliques semi lineaires et des equations differentielles fonctionnelles a retard

superlinear indefinite equations on the real line and chaotic dynamics abstract in this paper we are concerned with a differential equation of the form x c x q t g x 0 t a b where a b q has infinitely many zeros in a b and g is superlinear we prove the existence of solutions with prescribed nodal properties in the intervals of negativity and positivity of q when c 0 and q is periodic we show that the equation under consideration exhibits chaotic like dynamics

toom s model with glauber rates an exact solution using elementary methods a simple spatially two dimensional stochastic cellular automaton with asymmetric coupling and synchronous updating according to glauber rates is considered while detailed balance is violated it is still possible to compute analytically the stationary probability distribution by elementary means the stationary distribution can be written as a canonical equilibrium distribution of a spin system on a triangular lattice with nearest neighbour coupling thus the cellular automaton shows a nonequilibrium phase transition with ising critical behaviour

analytic and algebraic conditions for bifurcations of homoclinic orbits i saddle equilibria abstract we study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four dimensional systems which may be hamiltonian or not only one parameter is enough to treat these types of bifurcations in hamiltonian systems but two parameters are needed in general systems we apply a version of melnikov s method due to gruendler to obtain saddle node and pitchfork types of bifurcation results for homoclinic orbits furthermore we prove that if these bifurcations occur then the variational equations around the homoclinic orbits are integrable in the meaning of differential galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds we illustrate our theories with an example which arises as stationary states of coupled real ginzburg landau partial differential equations and demonstrate the theoretical results by numerical ones

rigorous chaos verification in discrete dynamical systems abstract in this paper we show how interval analysis can be used to calculate rigorously valid enclosures of transversal homoclinic points in discrete dynamical systems the existence of such points guarantees that a smale horseshoe is embedded in the mapping which therefore is chaotic we take special interest in the 2 dimensional case and describe a method to obtain a computer assisted proof of chaos for these systems numerical results are presented for the standard map