Consistency of local dynamics and bifurcation of continuous-time dynamical systems and their numerical discretizations

Abstract

Numerical integration methods for solving differential equations naturally give rise to difference equations which in many cases can be subsequently converted into iterative maps. In this paper we study some consistency problems of local dynamics and bifurcation between the continuous-time (CT) dynamical systems defined by the differential equations and the discrete-time (DT) dynamical systems resulting from numerical methods of solving the differential equations. We first formulate the concepts of dynamical and bifurcational consistencies, and then present qualitative and quantitative results on the discretization step size and bifurcation parameter for general one-step methods of order p and specific methods like the Euler, backward Euler, explicit and implicit Runge-Kutta methods, so that the local dynamics and low-dimensional bifurcations (e.g., the saddle-node and Hopf bifurcations) of the CT systems are inherited exactly by the DT systems.