We prove the existence of periodic orbits with minimal period greater than any prescribed number for a natural Lagrangian autonomous system in several variables that is analytic and periodic in each variable and whose potential is nonconstant. Mathematics Subject Classification: 34C25, 37J45

An algorithm for obtaining rigorous results concerning the existence of chaotic invariant sets of dynamical systems generated by non-autonomous, time periodic differential equations with strong expansion is presented. The result is based on a new theoretical approach to the computation of the homology of the Poincaré map. A concrete numerical example concerning a time-periodic differential equa...

In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem d dt u(t) = A(t)u(t) + f(t), 0 ≤ s ≤ t ≤ T, L(t)u(t) = Φ(t)u(t) + g(t), 0 ≤ s ≤ t ≤ T, u(s) = x. The solution to this problem is obtained by a variation of constants formula.

A method for constructing a one-dimensional discrete mapping describing certain periodic process in general system of ordinary autonomous differential equations is proposed. The resulting then used to prove the existence chaos original equations.

A new implementation of an autonomous hysteresis chaotic circuit using only two primitive cellular neural networks (CNN) cells is presented. Both computer simulations and laboratory measurements have confirmed the chaotic behavior and show the existence of the double scroll attractor. Key-Words: CNN, chaotic circuits, chaos, attractor, bifurcation, oscillator.

Important concepts concerning tubes in dynamical systems are defined in details. In the cases of a homoclinic tube and a heteroclinically tubular cycle in autonomous systems, existence of tubular chaos is established. The main goal of this article is to stress the importance of tubes in high dimensional dynamical systems. 2006 Elsevier Ltd. All rights reserved.