Automatic phase space analysis of dynamical systems

Abstract

Research on automating the analysis of nonlinear ordinary differential equations by combining artificial intelligence techniques with existing numerical software is discussed. A program called POINCARE that analyzes one-parameter autonomous planar systems at the level of experts through a combination of theoretical dynamics, numerical simulation, and geometrical reasoning is described. The input is the system, a bounding box for the system state, a bounding interval for the parameter, and error tolerances. POINCARE partitions the parameter interval into open subintervals of equivalent behavior bounded by bifurcation points, classifies the bifurcation points, and constructs representative phase diagrams for the subintervals. It detects local and global generic one-parameter bifurcations. It constructs the phase diagrams by identifying fixed points, saddle manifolds, and limit cycles and partitioning the remaining trajectories into open regions of uniform asymptotic behavior. It is explained how to extend POINCARE to two-parameter families of periodically driven planar equations. The basic algorithms carry over after some modifications and extensions. A survey of open issues in the automatic analysis of other dynamical systems is given.