Limit Cycle, Potential Landscape, and Complex Dynamics

The existence of potential landscape in complex dynamics, though very appealing, has been controversial. Recently one of the present authors proposed a general construction. Because of technical di-culty, such a construction has not been explicitly demonstrated for complicated dynamics, such as chaotic ones. As a step towards such goal, here we demonstrate that the potential function can coexist with limit cycles in nonlinear and dissipative dynamical systems. The potential function indeed drives dynamics and determines the flnal steady state distribution similar to the usual situation in physics, that is, it has both local and global meanings. Our demonstration consists of three steps: We flrst show the existence of limit from a typical physics setting by an explicit construction in two dimensions. When approaching to the limit cycle, the strength of the potential gradient goes to zero; the magnetic fleld goes to zero in the same order of the potential gradient and changes sign at the limit cycle; and the friction goes to zero at higher order than that of potential gradient. The dynamics at the limit cycle is conserved in this limit. The difiusion matrix is nevertheless flnite at the limit cycle. Second, based on such physics knowledge we can construct the potential in the dynamics with limit cycle in a typical dynamical systems setting. Third, we argue that such a construction can be in principle carried out in a general situation combined with on a novel method of dealing with stochastic difierential equation. This novel method is difierent from both Ito and Stratonovich calculus shown explicitly in the present article.