Barnabas M. Garay

Uniform persistence and chain recurrence

Soit (#7B-E, d) un espace metrique localement compact et E un sous-ensemble clos de #7B-E. Soit π un systeme dynamique sur E. On reconsidere la notion de persistance uniforme dans le cadre de la theorie de Conley des ensembles invariants

OPTIMIZATION AND THE MIRANDA APPROACH IN DETECTING HORSESHOE-TYPE CHAOS BY COMPUTER

We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form , and thus check certain sufficient conditions for chaotic behavior of dynamical systems in a rigorous way. Our proof of the underlying abstract theorem avoids referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem. The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach. Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Henon mapping.

A Verified Optimization Technique to Locate Chaotic Regions of Hénon Systems

We present a new verified optimization method to find regions for Hénon systems where the conditions of chaotic behaviour hold. The present paper provides a methodology to verify chaos for certain mappings and regions. We discuss first how to check the set theoretical conditions of a respective theorem in a reliable way by computer programs. Then we introduce optimization problems that provide a model to locate chaotic regions. We prove the correctness of the underlying checking algorithms and the optimization model. We have verified an earlier published chaotic region, and we also give new chaotic places located by the new technique.

On Cj-closeness between the solution flow and its numerical approximation

Ordinary differential equations of the form and their p-th order one-step discretization approximations of class are considered. On intervals of length T, the time-h-map of the induced solution flow is compared to the discrete-time dynamical system obtained via discretization. For stepsize h sufficiently small, their Cj - distance is proved (see Theorem 1.1) to be less than const1(T). where, with some constant γ not much greater (sometimes even strictly smaller) than is basically exp((j+1)γT),j=0,1,…,p+k. The proof is based on a Pascal triangle property (see Inequality (33)) of higher order chain rule folmulas. Applications to the numerics of global stable/unstable manifolds of hyperbolic equilibria are given.

A computer-assisted proof of Sigma_3-chaos in the forced damped pendulum equation

The present paper is devoted to studying Hubbards pendulum equation

Linearizing the expanding part of noninvertible mappings

ddot{x} + 10^{-1} dot{x} + sin(x) = cos(t)

Partial linearization for noninvertible mappings

. Using rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is raised from the level of experimentally observed facts to the level of a theorem completely proved. A special family of solutions is shown to be chaotic in the sense that, on consecutive time intervals

Long transient oscillations in a class of cooperative cellular neural networks

(2kpi, 2(k+1)pi)

An experimental study on long transient oscillations in cooperative CNN rings

(

Interpolation in dynamic equations on time scales

k in mathbb{Z}