Bifurcations of symmetric periodic orbits near equilibrium in reversible systems

Abstract

Consider a family of reversible systems (x) over dot = f(x, mu) with the origin being an equilibrium for each mu. Suppose D(x)f(0, 0) has only purely imaginary eigenvalues +/-iw(1),..., +/-iw(k). We investigate the typical bifurcations of symmetric periodic solutions near the origin. A suitable complex basis is chosen so that D(x)f(0, 0) and the involution are in respective simple form. Incorporated with putting f into normal form, a modified version of Lyapunov-Schmidt reduction can be applied to obtain the reduced bifurcation equations. We then focus on the cases in resonance, that is, w(j) = n(j)w(0), where w(0) is a nonzero real number and n(j) is an integer for each j. Some codimension-two bifurcations are illustrated for the system in non-semisimple resonance with n(j) = 1, 2. A few codimension-one cases are also given for comparison with earlier works by other researchers.