A Simple Approach to Calculation and Control of unstable periodic orbits in Chaotic piecewise-Linear Systems

It is well known that some simple electric circuits behave chaotically, such as the Chua’s circuit [Madan, 1993] hysteresis circuits [Saito, 1990], the Colpitts oscillator [Kennedy, 1994] the buck converter [Yuan et al., 1998] and so on. In these systems, there exist some switching elements, so the overall system equation can be described by piecewise-linear functions. For piecewise-linear systems since exact solutions are obtained as piecewise functions, many analytic issues such as the stability of attractors, bifurcation and existence of chaos can be discussed in a rigorous way. For example, Andronov studied the stability of the limit cycle observed from the piecewise-linear model of a vacuum-tube oscillator. This work is introduced in [Pontryagin, 1962], where the differential equation includes a step-wise (binary level) function which is approximated by some continuous saturation characteristics. Since this model is a second-order autonomous system, the stability of its limit cycle can be studied analytically. However, this particular piecewise-linear system is probably one of the very few examples whose dynamical properties can be obtained in an analytical way. The key point in this analysis is that the switching time, at which the system equation changes nonsmoothly, can be exactly evaluated. Unfortunately, calculating a periodic orbit and analyzing its stability are very difficult because the conditions on the switching times cannot be explicitly derived, especially for higher-order systems [Madan, 1993]. Therefore, one has to resort to appropriate numerical methods, particularly for higher-order systems, for this kind of computation and analysis. And yet, even if an efficient method is available for calculating periodic orbits, a large number of UPOs embedded within a chaos attractor are not able to calculate in general. An attempt was made by Diakonos et al. [1998] to develop a systematic computational scheme for UPOs embedded within a chaotic attractor. Other efficient techniques to obtain UPOs are also