Limit cycle and conserved dynamics

We demonstrate here that the potential can coexist with limit cycle in nonlinear dissipative dynamics, where the potential plays the driving role for dynamics and determines the final steady state distribution in a manner similar to other situations in physics. First, we show the existence of limit cycle from a typical physics setting by an explicit construction: the potential is of the Mexican-hat shape, the strength of the magnetic field scales with that of the potential gradient near the limit cycle, and the friction goes to zero faster than that of potential gradient when approaching to the limit cycle. The dynamics at the limit cycle is conserved in this limit. The diffusion matrix is nevertheless finite at the limit cycle. Second, based on the physics knowledge, we 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 a novel method. Our present result may be useful in many applications, such as in the discussion of metastability of limit cycle and in the construction of Hopfield potential in the neural network computation.