The Role Of Geometry In Computational Dynamics

. This paper is an informal discussion of how geometry and numerical analysis are intertwined in the computational study of dynamical systems and their bifurcations. We use the example of determining the phase portrait of planar vector fields to illustrate the more general and philosophical attitudes that constitute our main thesis. Few mathematical details are included. 1. Introduction There are two fundamental aspects of dynamical systems theory that lead us to reliance upon computers. First, "most" nonlinear vector fields cannot be integrated explicitly, so numerical methods are absolutely essential to obtain quantitative information about the solutions to particular systems. This is usually done through the iterative computation of approximate solutions by numerical integration algorithms. Second, computers work well for simulating trajectories of vector fields of modest complexity. Modern workstations have an architecture that emphasizes very rapid floating point calculations on d..