Obtaining initial parameter estimates for chaotic dynamical systems using linear associative memories

摘要

Parameter estimation problems for nonlinear dynamical systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive approximation schemes generally require good initial parameter estimates in order to converge to the truth. The linear associative memory method has demonstrated its effectiveness in obtaining useful initial parameter estimates for simple nonlinear dynamical systems. No work, however, has yet been done to apply this method to a chaotic system. This paper initiates such a study using the logistic map, which is capable of generating mathematical chaos. Supervised training was conducted between system parameters and system outputs to construct optimal memory matrices. Untrained system outputs were then used together with the memory matrices to estimate system parameters. Very accurate parameter estimates were obtained for noise-free system outputs. Good parameter estimates were obtained for system outputs corrupted by noise. A “rule of thumb” is suggested that can be used to aid in a successful search for true parameter values if the initial training range is not located “near” them.