ON THE GLOBAL CONVERGENCE CHARACTERISTICS OF NUMERICALLY EVALUATED JACOBIAN MATRICES

Since locating all the fixed points of a nonlinear oscillator involves the numerical solution of simultaneous equations, it is useful to obs erve some of the global convergence characteristics of these technique s. Specifically, the popular Newton or quasi-Newton approaches require numerical evaluation of the Jacobian matrix of the Poincare map. This note focuses attention on the domains of attraction for a number of f ixed point techniques applied to a single nonlinear oscillator with a single set of parameters. Clearly, there are many issues here, includi ng proximity to bifurcations, order of the dynamical system, temporal convergence characteristics, i.e. CPU time, and so on, but it is instr uctive to observe a snapshot of the basins of attraction, the boundari es of which path-following routines seek to avoid when a parameter is changed.