Dynamics of Partial Control of Chaotic Systems

Abstract

In our chaotic lives we usually do not try to specify our plans in great detail, or if we do, we should be prepared to make major modifications. Our plans for what we want to achieve are accompanied with situations we must avoid. Disturbances often disrupt our immediate plans, so we adapt to new situations.We only have partial control over our futures. Partial control aims at providing toy examples of chaotic situations where we try to avoid disasters, constantly revising our trajectories. Moremathematically, partial control of chaotic systems is a newkind of control of chaotic dynamical systems in presence of disturbances. The goal of partial control is to avoid certain undesired behaviors without determining a specific trajectory. The surprising advantage of this control technique is that it sometimes allows the avoidance of the undesired behaviors even if the control applied is smaller than the external disturbances of the dynamical system. A key ingredient of this technique is what we call safe sets. Recently we have found a general algorithm for finding these sets in an arbitrary dynamical system, if they exist. The appearance of these safe sets can be rather complex though they do not appear to have fractal boundaries. In order to understand better the dynamics on these sets, we introduce in this paper a new concept, the asymptotic safe set. Trajectories in the safe set tend asymptotically to the asymptotic safe set. We present two algorithms for finding such sets. We illustrate all these concepts for a time-2π map of the Duffing oscillator. This is joint work with James A. Yorke (USA), Samuel Zambrano (Italy) and Juan Sabuco (Spain).