CONSISTENCY OF LOCAL DYNAMICS AND BIFURCATION OF CONTINUOUS-TIME DYNAMICAL-SYSTEMS AND THEIR NUMERICAL DISCRETIZATIONS

Numerical integration methods for solving differential equations natur ally give rise to difference equations which in many cases can be subs equently converted into iterative maps. In this paper we study some co nsistency problems of local dynamics and bifurcation between the conti nuous-time (CT) dynamical systems defined by the differential equation s and the discrete-time (DT) dynamical systems resulting from numerica l methods of solving the differential equations. We first formulate th e concepts of dynamical and bifurcational consistencies, and then pres ent qualitative and quantitative results on the discretization step si ze and bifurcation parameter for general one-step methods of order p a nd specific methods like the Euler, backward Euler, explicit and impli cit Runge-Kutta methods, so that the local dynamics and low-dimensiona l bifurcations (e.g., the saddle-node and Hopf bifurcations) of the CT systems are inherited exactly by the DT systems.