Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales

In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation x△=A（t）x on time scales. Moreover, for the nonlinear perturbed equation x△= A（t）x ＋ f（t, x） we give the instability of the zero solution when f is sufficiently small.

Pseudo Almost Automorphic Solutions for Non-autonomous Stochastic Differential Equations with Exponential Dichotomy

In this paper, we consider the existence and uniqueness of the solutions which are pseudo almost automorphic in distribution for a class of non-autonomous stochastic differential equations in a Hilbert space. In conclusion, we use the Banach contraction mapping principle and exponential dichotomy property to obtain our main results.

A GENERALIZATION OF A PROPOSITION ON EXPONENTIAL DICHOTOMY

It is well known that if x' = A(t)x admits an exponential dichotomy and f(t, x) is a bounded function and has a small Lipschitz constant, then system X' = A(x)x + f(t, x) has a unique bounded solution. In this paper, we shall generalize this theorem to the case when f(t, x) is an unbounded function.

EXPONENTIAL DICHOTOMIES,HOMOCLINIC ORBITS AND METHODS OF MELNIKOV

In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).

MELNIKOV VECTOR AND HOMOCLINICBIFURCATIONS IN A DEGENERATE CASE

The inch purpose of this paper is to study homocliaic bifurcation in a degenerate case of differential equations of form i = f(x, ε) and i = g(x) + h(t, x. ε). By using theory of exponential dichotomies and Liapunov-Schmidt method, the authors obtain the Melnikov-like vectors by which the existence of homoclinic orbits can be detected.

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

Based on the theory of generalized exponeatial dichotomies, some typical nonautonomouslinear systems with time-translation parameters or with time-scale parameters are discussed.Conditions of parameters are given to determine by spectral gaps the generalized exponentialdichotomies of those systems and their local L'-limit systexns. These conclusions are applied tosrvcalled SVC systems and HFO systems for their stability.

THE EXISTENCE OF ALMOST PERIODIC SOLUTION TO NONLINEAR DIFFERENTIAL EQUATIONS

By applying Horn fixed point theorem and the method of exponential dichotomy,we investigate the existence of almost periodic solution to nonlinear ordinarydifferential equation.

ON THE EXISTENCE OF PERIODIC SOLUTION OF HIGHER ORDER PERIODIC SYSTEM

In this paper, we deal with the problem on the existence of periodic solution of higher periodic system. Using the exponential dichotomy and the Schauder's fixed point theorem,we establish the sufficient conditions which guarantee the existence and uniqueness and stability of periodic solution.

Problems in Homoclinic Bifurcation with Higher Dimensions

In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.

Orbits Heteroclinic to Invariant Manifolds

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3-5].

Existence of local stable manifolds for some nondensely defined nonautonomous partial functional differential equations

In this paper,we establish the existence of local stable manifolds for a semi-linear differential equation,where the linear part is a Hille-Yosida operator on a Banach space and the nonlinear forcing term f satisfies the ψ-Lipschitz conditions,where ψ belongs to certain classes of admissible function spaces.The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.

Almost Periodic Solutions by the Harmonic Balance Method

We consider non-autonomous ordinary differential equations in two cases.One is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponential dichotomy.For differential equations of this kind,we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.

BIFURCATION OF DEGENERATE HOMOCLINIC ORBITS

The main purpose of this paper is to investigate existence of transversal homoclinic orbits of nonlinear system of two small parameters in a degenerate case. By making use ofthe theory of exponential dichotomy and Liapunov-Schmidt method, we construct Melnikov-like functions, and show that if these functions satisfy some conditions thentransversal homoclinic orbits of nonlinear system exists.

PERSISTENCE AND TRANSVERSALITY OF HOMOCLINIC ORBITS FOR DIFFEOMORPHISMS

A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeomorphisms.

THE EXISTENCE OF ALMOST PERIODIC SOLUTION TO A CLASS OF ALMOST PERIODIC DIFFERENTIAL SYSTEM

By means of the fixed point theorem and exponential dichotomy, in this paper we investigate the existence of almost periodic solutions to a class of ndimensional almost periodic systems, which is more general than the systems in [1-2]. We generalize and improve some results in [3].

ON THE EXISTENCE OF ALMOST PERIODIC SOLUTIONS OF SOME DELAY DIFFERENTIAL EQUATIONS

By means of exponential dichotomy, this paper investigates the existence of almost periodic solutions of some delay differential equations.

MELNIKOV VECTOR WITH HIGHER ORDER

By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of application are also given.

ALMOST PERIODIC SOLUTIONS TO STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

By applying the properties of almost periodic function and exponential dichotomy of linear system as well as Banach fixed point theorem,we establish the conditions for the existence and uniqueness of square-mean almost periodic solution to some stochastic functional differential equations.

PSEUDO-ALMOST PERIODIC SOLUTIONS TO SOME FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper is concerned with the pseudo-almost periodic solutions to some functional differential equations. By the exponential dichotomy theory and Schauder’s fixed-point theorem, some results on the existence and uniqueness of pseudo-almost periodic solutions to the system are obtained.

ASYMPTOTICALLY MEAN ALMOST PERIODIC SOLUTIONS TO RANDOM DIFFERENCE EQUATIONS

In this paper,we first investigate some basic properties of asymptotically mean almost periodic random sequences on Z + and then show some properties of asymptotically mean almost periodic solutions to random difference equations.