Exchange of Stability and Bifurcation for Periodic Differential Systems

Abstract

This chapter analyzes exchange of stability and bifurcation for periodic differential systems. The role of an exchange of stability properties in determining bifurcation phenomena was analyzed for periodic systems under particular hypotheses on the Floquet exponents of D u f ( t ,0, μ ). The chapter is concerned with general periodic systems. In contrast with the autonomous case, the natural setting in which to analyze the existence and the stability properties of the bifurcating sets is R × R n , rather than R n . The chapter discusses that the main technique used to determine the existence and the stability properties of the bifurcating sets, Mμ , is to consider a family of autonomous discrete dynamical systems. The chapter also describes the positive prolongational limit set. Some results on a one-parameter family of differential equations in R n are also given in the chapter.