A FURTHER UNDERSTANDING OF PHASE SPACE PARTITION DIAGRAMS

This paper deals with dynamic systems described by sets of ordinary differential equations. As described in a previous paper [Thomas & Kaufman, 2005], phase space can be partitioned according to the signs of the eigenvalues of the Jacobian matrix and the slopes of its eigenvectors, using "frontiers" described by simple analytic expressions. In the present paper, we develop this approach with special emphasis on: — a preliminary, qualitative, yet quite efficient, description based simply on the sign patterns of elements of the Jacobian matrix — the identification and characterization of two additional frontiers based on the relative slopes of the eigenvectors (in two dimensions only) — the applicability to systems of higher dimensionalities — programs (in Mathematica) that provide immediately the phase space frontier diagrams from the ODE's for two- (Appendix 2) and n-dimensional systems (Appendix 3) For systems of sufficiently low dimensionality, the resulting diagrams give a global view of the structure of phase space, partitioned into domains each consistent as regards to the nature of any steady state that might be present in that domain. Fortunately, the dimensionality of phase space frontier diagrams depends not on the overall dimensionality of the system but on the number of variables that carry nonlinearity; an n-variable system in which only two variables carry nonlinearity can be described by a two-dimensional diagram.