ASYMPTOTIC PROBABILITY-DISTRIBUTION FOR A SUPERCRITICAL BIFURCATION SWEPT PERIODICALLY IN TIME

By using a path-integral approach we have studied the asymptotic probability distribution of a periodically swept supercritical bifurcation. The steepest-descent approximation has been used with the corresponding time-dependent Onsager-Machlup Lagrangian of the Fokker-Planck equation. We prove by using the Lyapunov function the uniqueness of the asymptotic time-periodic probability distribution for peridically forced Markov processes; then the mixing property for these types of stochastic processes is proved. An iterative matrix procedure is introduced to calculate the long-time behavior of the probability distribution. Monte Carlo simulations were performed in order to show the agreement between the path-integral approach and the numerical solution of the corresponding periodically forced stochastic differential equation. A discussion on the problem of calculating the weak-noise Graham-Tel invariant measure is presented.