On the local stability of limit cycles

Orbital stability of limit cycles is the result of the competing local tend encies of perturbations from the cycle to decay (during phases of local sta bility) and to grow (during phases of local instability), averaged over a c ycle. We examine this coexistence of attractive and repulsive phases on lim it cycles, including the local rates of expansion and contraction of phase space volumes. This is done in a frame of reference that moves along the or bit, to partially decouple motions tangential and perpendicular to the cycl e. Dynamical systems used for illustration are the generalized Bonhoeffer-v an-der-Pol and Rossler models, both far from and near to different types of bifurcations. Finally, it is shown that the nonuniformity of local stabili ty in phase space affects the response of limit cycle oscillators to pertur bations and gives rise to their phase-dependent response. (C) 1999 American Institute of Physics. [S1054-1500(99)02002-9].