Bifurcations of Periodic Equations

In this chapter, we investigate the bifurcation behavior of periodic solutions of 1-periodic differential equations. We first point out that, in terms of Poincaré maps, the study of local bifurcations of periodic solutions is equivalent to the study of bifurcations of fixed points of monotone maps given in Section 3.3. Next, we develop selected ideas from the “method of averaging” and show how to compute higher- order derivatives of the Poincaré map about a fixed point. For practical purposes, we accomplish this solely in terms of the vector field and the 1-periodic solution. We then employ these results to determine local bifurcations of nonhyperbolic 1-periodic solutions of 1-periodic differential equations depending on a scalar parameter. We should warn you that this chapter, by necessity, is more technical than the previous ones.