Taming chaos by weak resonant perturbations is a hot topic in recent years. Vast numerical studies have been done, and it has been clarified that taming chaos is generated in many equations. In this chapter, we explain the mechanism for taming chaos by the use of degeneration technique. Our model studied in this chapter is a forced Rayleigh oscillator with a diode. By weak resonant perturbations, chaos is tamed. Two routes to taming chaos are observed. One route is as follows. A wide band chaotic attractor is suddenly changed to a periodic attractor. The other one is as follows. A wide band chaotic attractor is tamed to a narrow band chaotic attractor, and tamed to a periodic attractor. To analyze these phenomena in detail, we will consider the case where the diode in the circuit operate as a switch. In this case, the Poincar map is derived rigorously as a one-dimensional mapping. By the analysis of the map, it is clarified theoretically that mechanism of the first taming chaos is a saddle node bifurcation, and that the second one is a inverse version of crisis and a inverse period doubling bifurcation.

21.1 Introduction

Taming chaos by weak resonant perturbations is a hot topic in recent years ^[1]- ^[11]. Let us consider a non-autonomous differential equation which generates chaos. The amplitude of the first forcing term is chosen such that this equation generates...