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 observe some of the global convergence characteristics of these techniques. 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 fixed point techniques applied to a single nonlinear oscillator with a single set of parameters. Clearly, there are many issues here, including proximity to bifurcations, order of the dynamical system, temporal convergence characteristics, i.e. CPU time, and so on, but it is instructive to observe a snapshot of the basins of attraction, the boundaries of which path-following routines seek to avoid when a parameter is changed.