Potential function in dynamical systems and the relation with Lyapunov function

One of the present authors has proposed a novel treatment of stochastic processes leading to the construction of potential functions for dynamics described by stochastic differential equations (SDEs). The approach transforms the deterministic part of the original system into three components: a potential function, a frictional force and a Lorentz force. The potential function drives the dynamics and determines the final steady state distribution that has both local and global meaning. We note that such a potential is closely related to the classical Lyapunov function. In this paper, we first provide a brief review on the decomposition framework and then give a constructive proof on the equivalence of two fundamental concepts: the global Lyapunov function in engineering and the potential function in physics, establishing a bridge between the two distinct fields. This result reveals the physical meaning of Lyapunov functions, thus suggests new approaches on the largely unsolved problem: constructing Lyapunov functions for general nonlinear systems, through the analogy with existing methods on potential functions. In addition, we show another connection that the Lyapunov equation is a reduced form of the generalized Einstein relation for linear systems. By inheriting from a physical treatment of stochastic processes, this work demonstrates a stochastic view of deterministic systems together with the deterministic rules that govern stochastic behaviors.