Automatic analysis of one-parameter planar ordinary differential equations by intelligent numeric simulation

Abstract

This paper describes research on automating the analysis of physical systems modeled by ordinary differential equations. It describes 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 geometric reasoning. 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 obtains the region boundaries by numeric simulation, guided by theoretical knowledge. It determines the behavior near fixed points from the Jacobian of the system, scans outward to find basins of attractors and limit cycles, and stops at the bounding box.