Nonlinear Fokker-Planck Equations with and without Bifurcations and Generalized Thermostatistics

Nonlinear Fokker-Planck equations exhibiting bifurcation phenomena are studied within the framework of generalized thermostatistics. Liapunov functions are constructed that take the form of free energy involving the generalized entropies of Tsallis, and H-theorems are shown to hold. The H-theorems ensure, instead of uniqueness of the equilibrium distribution, global stability of the systems. A local stability analysis is conducted, and the second-order variations of the Liapunov functions are computed to find their relevant part whose sign governs the stability of the equilibrium distributions of the systems. PACS numbers: 05.70, 05.20.