Bifurcations in the periodic orbit sum: multiple windings

Abstract

Periodic orbits that participate in a bifurcation contribute collectively to the periodic orbit sum for the quantum density of states. The contributions of multiple windings of isolated orbits are easily obtained from powers of the stability matrix, but it is generally hard to compose the actions that determine the contributions of higher windings of a bifurcation. Here we derive an approximate relation between the amplitude of the contributions of different windings for the saddle-centre bifurcation and the period-doubling bifurcation.