Towards Accurate PWL Approximations of Parameter-Dependent Nonlinear Dynamical Systems With Equilibria and Limit Cycles

Abstract

This paper deals with piecewise-linear (PWL) approximations of nonlinear dynamical systems dependent on parameters and allowing the presence of few equilibria and/or limit cycles only. A method to derive the parameters of the PWL model is proposed that is based on the minimization of functionals defined to take into account a priori some dynamical features of the systems to be approximated. The method is validated by applying it to two simple dynamical systems, i.e., the cusp bifurcation normal form and the supercritical Hopf bifurcation normal form. The robustness of the approximations is checked, with a view to circuit implementations.

A method based on a genetic algorithm to find PWL approximations of multivariate nonlinear functions

“…Nevertheless, in the proposed example the nonlinear functions define the vector field of a dynamical system and the results are qualitatively good. For the approximation of nonlinear dynamical systems with equilibria and limit cycles (like the HH model), better results -in terms of either accuracy of the dynamical behaviours for a fixed PWL model complexity or, vice versa, simplicity of the PWL approximations for a fixed accuracy -could be achieved by defining an error functional and a quality factor tailored to the specific system to be approximated [8].…”

“…When one considers PWL approximations of vector fields governing nonlinear dynamical systems [7], further constraints, usually leading to a more complex expression for the cost function E, must be taken into account [8].…”

Synchronization properties in networks of Hindmarsh-Rose neurons and their PWL approximations with linear symmetric coupling

“…(10) In principle, there are many possible choices for the basis functions, more convenient either from a numerical or a circuit implementation standpoint [4]. Whatever basis is used to compute the coefficients, one can easily derive the coefficients of any other basis through a simple matrix product.…”

“…The chosen approximation is the simplest one presented in [6], which is suitable for analog implementation and where L = 16. Owing to the space limitations, we refer the reader interested in the PWL approximation to papers [4], [6] and references therein.…”

“…In the last few years, a piecewise-linear (PWL) approximation/synthesis technique has been applied towards the implementation of nonlinear dynamical systems [3], [4] and in particular the Hindmarsh-Rose (HR) neuron model [5], [6]. The dynamics of the single PWL model has been verified to be qualitatively and quantitatively very similar to those of the HR model.…”