In this note we sketch some results concerning a geometric method for periodic solutions of non-autonomous time-periodic differential equations. We give the definition of isolating chain and we provide a theorem on the existence of periodic solutions inside isolating chains. We recall some results on existence of chaotic dynamics which can be proved by the theorem. We provide several examples o...

In this paper, the existence and uniqueness of pullback attractors for the modified Swift-Hohenberg equation defined on R driven by both deterministic non-autonomous forcing and additive white noise are established. We first define a continuous cocycle for the equation in L(R), and we prove the existence of pullback absorbing sets and the pullback asymptotic compactness of solutions when the eq...

The aim of this paper is to present a numerical technique for the computation of connections between periodic orbits in non{autonomous and autonomous systems of ordinary diierential equations. First the existence and computation of connecting orbits between xed points in discrete dy-namical systems is discussed; then it is shown that the problem of nding connections between equilibria and perio...

We consider the non-autonomous Hamiltonian system Ju̇+∇H (t; u) = 0; (t; u)∈ ST × R ; where H ∈C1(R×R2N ;R) is T -periodic in t-variable, and J= ( 0 −IN IN 0 ) denotes the standard symplectic matrix and ∇ denotes the gradient with respect to the u-variable. We are interested in the existence of T -periodic solutions of (1). For the autonomous case, i.e. H is independent of t, in his pioneer work...

The energy equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. A suitable extension of this approach to the exi...

With the rise of importance of sport as a social phenomenon in recent decades, more public attention has been paid to the issue of the legal nature of the rules governing social relations in sports. The issue raised above are only a part of a much broader topic that involves questions of relations of a classical Westphalian state and changes in the international community as well as issues of t...

here we consider some non-autonomous ordinary differential equations of order n and present some results and theorems on the existence of periodic solutions for them, which are sufficient conditions, section 1. also we include generalizations of these results to vector differential equations and examinations of some practical examples by numerical simulation, section 2. for some special cases t...

In this paper, we theoretically prove the existence of periodic solutions for a nonautonomous discrete-time neural networks by using the topological degree theory. Sufficient conditions are also obtained for the existence of an asymptotically stable periodic solution. As a special case, we obtain the existence of a fixed point to the corresponding autonomous discrete-time neural networks which ...

In this paper, we study the cubic three-parameter autonomous planar system ẋ1 = k1 + k2x1 − x1 − x2, ẋ2 = k3x1 − x2, where k2, k3 > 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be a...

This article studies the existence of homoclinic solutions for the second-order non-autonomous Hamiltonian system q̈ − L(t)q + Wq(t, q) = 0, where L ∈ C(R, Rn ) is a symmetric and positive definite matrix for all t ∈ R. The function W ∈ C1(R × Rn, R) is not assumed to satisfy the global Ambrosetti-Rabinowitz condition. Assuming reasonable conditions on L and W , we prove the existence of at leas...