Regularization of the Trajectories of Dynamical Systems by Adjusting Parameters

Abstract

Abstract — A gradient learning method to regulate the trajectories of some nonlinear chaotic systems is proposed. The method is motivated by the gradient descent learning algorithms for neural networks. It is based on two systems: dynamic optimization system and system for finding sensitivities. Numerical results of several examples are presented, which convincingly illustrate the efficiency of the method. Keywords — Chaos, Dynamical Systems, Learning, Neural Networks I. I NTRODUCTION H E study of controlling chaos in non-linear deterministic systems has been in focus during the last two decades. The ability to bring chaotic dynamical systems to regular motions is an important subject. The existing chaos control algorithms can be classified mainly into two categories: feed-back methods [1 - 4] and non-feedback methods [5, 6]. In [7, 8] the authors showed that adding noise to the excitation frequency of the Duffing system the initially chaotic motion becomes regular. In the present paper we propose a gradient learning method for adjusting the system parameters so that the trajectory of the system has certain specified properties. In the theory of neural networks the gradient descent learning requires that the performance of a dynamical system is assessable through certain error function which measures the discrepancy between the trajectories of the dynamical system and the desired behavior [9]. During gradient learning the interconnection weights between neurons are iteratively adjusted to reduce the error. In the present work we consider the system coefficients as adjustable parameters to obtain the desired behavior of the dynamical system. II. PROBLEMSTATEMENTConsider the dynamical system F u w I u t