| Scientific computation is one of the most important application fields in computer science, which includes the numerical simulation for the models from all kinds of fields, and the numerical solution of complex theoretical problems. Scientific computation is indispensable for most scientific researchers to attack concrete problems and to analyse the natural phenomena. Recently, in both physics and biology, a series of works attempt to generalize the potential function concept, or the Lyapunov function, for general nonlinear dynamics, such as the bistable systems, the limit cycle systems and the chaotic systems. The generalization is driven by the the theorectical need to understand the nonequilibrium process, because of the fundamental role of the potential function in physics and in Wright’s evolution theory in biology. Besides, another reason for the generalization of the potential function is that the potential function characterizes the dynamical structure, the stability, and the convergence for the system. But, its general existence is controversial because there is no mathematical proof for the existence of the potential function for general nonlinear systems or nongradient systems. Therefore, the constructions for representative models are necessary, which serve as the touch stones for this problem. In this thesis, we propose a novel numerical method to calculate the potential function, together with the mathematical proof for the continuity. We apply this method for the bistable system, the limit cycle system, and the chaotic system. We break through in the computational side against the intrinsic complexity of these attractors and the dimension limitation. In the chaotic dynamics, we exhibit the fractal structure of the potential function. |
