high frequency solutions for the singularly perturbed one dimensional nonlinear schrodinger equation this article is devoted to the nonlinear schrodinger equation when the parameter e approaches zero all possible asymptotic behaviors of bounded solutions can be described by means of envelopes or alternatively by adiabatic profiles we prove that for every envelope there exists a family of solutions reaching that asymptotic behavior in the case of bounded intervals we use a combination of the nehari finite dimensional reduction together with degree theory our main contribution is to compute the degree of each cluster which is a key piece of information in order to glue them

solitons and the electromagnetic field in a recent paper 4 it has been introduced a lorentz invariant equation in three space dimensions having soliton like solutions we recall that roughly speaking a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations see 6 15 13 10 the equation introduced in 4 is the euler lagrange equation of an action functional

stochastic hybrid control the objective of this paper is to study the stochastic version of a previous paper of the authors in which hybrid control for deterministic systems was considered the modelling is quite similar to the deterministic case we have a system whose state is composed of a continuous part and a discrete part they are affected by a continuous type control and an impulse control the dynamics is moreover perturbed by noise also a continuous and a discrete noise process the markovian character of the sate process is preserved we develop the model and show how the dynamic programming approach leads to some involved quasi variational inequality

shooting methods and topological transversality we show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds the interest of such results is twofold first these orbits persist under perturbations which destroy the structure allowing the shooting method and second topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics we shall focus on heteroclinic solutions in the extended fisher kolmogorov equation

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

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

positive solutions for some non autonomous schrodinger poisson systems abstract in this paper we study the schrodinger poisson system sp u u k x x u a x u p 1 u x r 3 k x u 2 x r 3 with p 3 5 assuming that a r 3 r and k r 3 r are nonnegative functions such that lim x a x a 0 lim x k x 0 and satisfying suitable assumptions but not requiring any symmetry property on them we prove the existence of positive solutions

numerical methods for controlled regime switching diffusions and regime switching jump diffusions this work is concerned with numerical methods for controlled regime switching diffusions and regime switching jump diffusions numerical procedures based on markov chain approximation techniques are developed convergence of the algorithms is derived by means of weak convergence methods in addition examples are also provided for demonstration purpose

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

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

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

existence of infinitely many large solutions for the nonlinear schrodinger maxwell equations abstract in this paper we study the nonlinear stationary schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 using the variant fountain theorem introduced by zou w zou variant fountain theorems and their applications manuscripta math 104 2001 343 358 under certain assumptions on v and f we get infinitely many large solutions for

infinitely many solutions for a class of sublinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many solutions for a class of sublinear schrodinger maxwell equations the proof is based on the variant fountain theorem established by zou recent results from the literature are extended

multiple solutions for a nonlinear schrodinger poisson system with sign changing potential this paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear schrodinger poisson system under some suitable conditions some criteria on the existence of nontrivial solutions including a ground state solution two solutions and a sequence of nontrivial solutions u k with u k 0 as k is established by applying the morse theory and variational methods some known results in the literatures are extended and improved

high energy solutions for the superlinear schrodinger maxwell equations abstract in this paper we study the existence of infinitely many large energy solutions for the superlinear schrodinger maxwell equations u v x u u f x u in r 3 u 2 in r 3 via the fountain theorem in critical point theory

positive solutions of asymptotically linear schrodinger poisson systems with a radial potential vanishing at infinity in this paper we study the schrodinger poisson system sp u v x u x u k x f u inr3 u2 u 0 inr3 and prove the existence of positive solutions for system sp when the nonlinearity f has growth at most linear for small allowing the potential v x to vanish at infinity in addition also we obtain the nonexistence of a nontrivial positive solution for 14

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

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

multi bump solutions to carrier s problem abstract we study the existence of well known singularly perturbed bvp problem e2y 1 y2 2b 1 x2 y y 1 y 1 0 introduced by g f carrier in particular we show that there exist multi spike solutions and the locations of interior spikes are clustered near x 0 and are separated by an amount of o e lne while only single spikes are allowed near the boundaries x 1

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

stochastic near optimal controls necessary and sufficient conditions for near optimality near optimization is as sensible and important as optimization for both theory and applications this paper concerns dynamic near optimization or near optimal controls for systems governed by the ito stochastic differential equations sdes where both the drift and diffusion terms are allowed to depend on controls and the systems are allowed to be degenerate necessary and sufficient conditions for a control to be near optimal are studied it is shown that any near optimal control nearly maximizes the hh function which is a generalization of the usual hamiltonian and is quadratic with respect to the diffusion coefficients in some integral sense and vice versa if certain additional concavity conditions are imposed error estimates for both the near optimality of the controls and the near maximum of the hh function are obtained based on some delicate estimates of the adjoint processes examples are presented to demonstrate the results

layers and spikes in non homogeneous bistable reaction diffusion equations we study e 2 u f u x au 1 u u where a a u x 0 x 0 1 and e 0 is sufficiently small on an interval 0 l with boundary conditions u 0 at x 0 l all solutions with an e independent number of oscillations are analyzed existence of complicated patterns of layers and spikes is proved and their morse index is determined it is observed that the results extend to f a u x u u u with x x x and also to an infinite interval

multiplicity of small negative energy solutions for a class of nonlinear schrodinger poisson systems this paper deals with the following nonlinear schrodinger poisson email protected v x u k x f x u h x f x u inr 3 email protected f k x u 2 inr 3 where v x k x and h x are nonnegative continuous functions under appropriate assumptions on v x k x h x and f x u we prove the existence of infinitely many small negative energy solutions by using the variant fountain theorem established by zou recent results from the literature are extended

oscillating solutions of the falkner skan equation for positive β on demontre l existence de solutions pour le probleme f ff 1 f 2 0 f o f o 0 f 1

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

technical communique near optimal control for a class of stochastic hybrid systems to address a computationally intractable optimal control problem for a class of stochastic hybrid systems this paper proposes a near optimal state feedback control scheme which is constructed by using a statistical prediction method based on approximate numerical solution that samples over the entire state space a numerical example illustrates the potential of the approach

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

multipulse like orbits for a singularly perturbed nearly integrable system abstract we use topological shooting arguments to study the existence of multipulse like orbits for a singularly perturbed 3 dimensional dynamical system established as a model problem for the 1 dimensional complex ginzburg landau equations our main result improves an earlier result obtained by doelman and hek

multi layered stationary solutions for a spatially inhomogeneous allen cahn equation abstract we consider stationary solutions of a spatially inhomogeneous allen cahn type nonlinear diffusion equation in one space dimension the equation involves a small parameter e and its nonlinearity has the form h x 2 f u where h x represents the spatial inhomogeneity and f u is derived from a double well potential with equal well depth when e is very small stationary solutions develop transition layers we first show that those transition layers can appear only near the local minimum and local maximum points of the coefficient h x and that at most a single layer can appear near each local minimum point of h x we then discuss the stability of layered stationary solutions and prove that the morse index of a solution coincides with the total number of its layers that appear near the local maximum points of h x we also show the existence of stationary solutions having clustering layers at the local maximum points of h x

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

optimal control of switching diffusions with application to flexible manufacturing systems a controlled switching diffusion model is developed to study the hierarchical control of flexible manufacturing systems the existence of a homogeneous markov nonrandomized optimal policy is established by a convex analytic method using the existence of such a policy the existence of a unique solution in a certain class to the associated hamilton jacobi bellman equations is established and the optimal policy is characterized as a minimizing selector of an appropriate hamiltonian

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

the falkner skan equation ii dynamics and the bifurcations of p and q orbits the falkner skan equation is a reversible three dimensional system of ordinary differential equations with two distinguished straight line trajectories which form a heteroclinic loop between fixed points at infinity we showed in the previous paper 1995 j differential equations119 336 394 that at positive integer values of the parameter there are bifurcations creating large sets of periodic and other interesting trajectories here we show that all but two of these trajectories are destroyed in another sequence of bifurcations as and by considering topological invariants and orderings on certain manifolds we obtain unusually detailed information about the sequences of bifurcations which can occur