A Potential Function Approach To Study Stability Of Dissipative Dynamical Systems And Its Applications

Stochastic differential equations can be used to describe the dynamical evolution of a number of systems in Biology, Physics and Chemistry. These systems doesn't satisfy the detailed balance condition. The construction of potential function for a general stochastic differential systems without the detailed balance condition has great application values in the study of non-equilibrium statistic physics and systems bio-medicine. The present work uses the potential function to investigate the dynamical properties of deterministic dissipative systems. There still lacks a general method of using a scalar "energy" function to be taken as the Hamiltonian which is able to de-termine the dynamics under the action of dissipation in Mechanics. In this report we constructed a Lyapunov-like potential function, using a 4-dimensional rotating sys-tem with regular and positional dissipation as a model. We were able to first show an asymptotically equilibrium as a stable fixed point in our potential function, equivalent to a Lyapunov function. We then constructed the potential functions for other fixed points, unstable, saddle or stable in the Lyapunov sense. Therefore, the constructed potential function can be taken as the generalized Hamiltonian of the dissipative gy-roscopic system which is able to completely determine the dynamics. By numerical simulation, we verified the steady distribution of the stochastic system in the asymp-totically stable domain is the Boltzmann-Gibbs distribution, and discovered that there were four potential valleys near a 4-dimensional saddle. At last, we applied the po-tential function to address two remaining physical problems:we explained dissipa-tion induced paradox by giving an explicit version on the singularity of the stability boundary; we speculated the dynamical conditions for the inversion of the Tippe top. At last, we discuss the application value of the potential function for bio-networks by constructed the potential landscape for a gene regulatory network, p53-MDM2 model, having a visible and systematic understanding of its dynamical properties in the bi-stable state.