Two kinds of singularities in planar differential systems and the response to external forces

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Abstract
A Lyapunov function is defined for two-dimensional differential systems. The relationship between the Lyapunov function and the stationary probability distribution of the corresponding Fokker-Planck equations in the weak-noise limit is analyzed. Based on the structure of the Lyapunov function we classify two types of singularities. The classification of different kinds of singularities is shown to be significant for the intrinsical dynamics of the deterministic flow. The bifurcation behavior of periodically forced planar systems that contain distinctive kinds of singularities in the autonomous cases is investigated. It is found that the responses of planar systems to the external force, i.e., the resulting bifurcation sequences, are substantially different if the trajectories circle around different types of singularities.