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

asymptotic behavior of global solutions for a chemical reaction model this paper deals with a system of reaction diffusion equations modeling a chemical reaction process involving three substances the mathematical system consists of three semilinear partial differential equations of parabolic type with homogenous neumann boundary conditions the goal is to determine whether each substance has a positive asymptotic limit by the method of upper and lower solutions

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

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

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

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

parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter abstract in this work we consider an optimal accurate numerical approximation for the boundary layer adaptive solution of a system of reaction diffusion problems involving a small perturbation parameter to generate the layer adaptive mesh we use the equidistribution of a positive monitor function this technique enables the mesh movement to follow the rapidly changing behavior of the solution it is observed that the discrete solution is second order convergent on the layer adaptive equidistributed nonuniform mesh irrespective of the value of the perturbation parameter this convergence rate is optimal with respect to the discretization of the continuous problem we illustrate the performance of the present method based on a moving mesh algorithm by numerical experiments

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

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

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

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

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

multiple blow up for a porous medium equation with reaction the present paper is concerned with the cauchy problem left begin array ll partial t u delta u m u p quad rm in mathbb r n times 0 infty u x 0 u 0 x geq 0 quad rm in mathbb r n end array right with p m 1 a solution u with bounded initial data is said to blow up at a finite time t if lim rm sup t nearrow t u t l infty mathbb r n infty for n 3 we obtain in a certain range of values of p weak solutions which blow up at several times and become bounded in intervals between these blow up times we also prove a result of a more technical nature proper solutions are weak solutions up to the complete blow up time

on nonlinear reaction diffusion systems abstract this paper presents a qualitative analysis for a coupled system of two reaction diffusion equations under various boundary conditions which arises from a number of physical problems the nonlinear reaction functions are classified into three basic types according to their relative quasi monotone property for each type of reaction functions an existence comparison theorem in terms of upper and lower solutions is established for the time dependent system as well as some boundary value problems three concrete physical systems arising from epidemics biochemistry and engineering are taken as representatives of the basic types of reacting problems through suitable construction of upper and lower solutions various qualitative properties of the solution for each system are obtained these include the existence and bounds of time dependent solutions asymptotic behavior of the solution stability and instability of nontrivial steady state solutions estimates of stability regions and finally the blowing up property of the solution special attention is given to the homogeneous neumann boundary condition

optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction diffusion boundary value problems in this article we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction diffusion problems having diffusion parameters with different magnitudes the central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle an a priori monitor function is obtained from the error estimate a suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second order parameter uniform convergence we present the results of numerical experiments for linear and semilinear reaction diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

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 relationships between equilibria and positive solutions of certain nonlinear elliptic systems abstract letm u v 0 n u v 0 define two distinct phase curves 1 2in the u v phase plane this paper presents results on the relationships among the positive equilibria the phase curves and the existence of positive solutions to the pde system u um u v 0 v vn u v 0 in r n under dirichlet boundary conditions where is a bounded domain andm nare monotone functions

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

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

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

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

simplicity of zeros in scalar parabolic equations soit le systeme u t u xx f x u x 0 1 u t 0 u t 1 0 f x 0 0 u 0 x u 0 x soit t 0 u t a seulement des zeros simples soit z u t le nombre de zeros de u 0 on suppose z u 0 fini alors est ouvert et dense dans l intervalle de temps maximal d existence de u t

local behavior near blow up points for semilinear parabolic equations abstract this paper is concerned with solutions of u t u xx u p p 1 defined in a cylinder qr x t r 0 which blowup at t 1 more precisely we are interested in the asymptotic behaviour of such solutions near a blow up point as t 1 we show that whenever the blow up set can be confined to the interior of the interval r r asymptotics are the same as those corresponding to solutions defined in the whole line which assume bounded initial values however if the blow up set reaches the boundary of the interval r r a different behaviour may appear as indicated by theorem 2 in the introduction

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

asymptotic behavior and positive solutions of a chemical reaction diffusion system abstract this paper is concerned with some qualitative analysis for a coupled system of three reaction diffusion equations which arises from certain chemical reactions first discovered by belousov and zhabotinskii the analysis includes the existence of a bounded global time dependent solutions the stability and instability of the zero solution and the existence and nonexistence of a positive steady state solution including a global attractor of the system the global existence and stability problem is determined by the method of upper and lower solutions and the existence of a positive steady state solution is based on the fixed point index and bifurcation theory this analysis leads to a necessary and sufficient condition for the existence and nonexistence of a positive steady state solution in relation to the various physical parameters of the system

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0