Dynamical invariant calculations involving evolution equations with discontinuities

Abstract

Many models of physical systems involving electronic circuit elements [6], population dynamics [5] involve evolution equations with discontinuities. The key to understand such systems is to hope that the discontinuity does not adversely affect the integration process. There are also three variable chaotic dynamical system examples, such as the Sprott systems for deriving jerky dynamics that have also become of interest [10]. In order to calculate dynamical invariants in chaotic systems such as characteristic exponents and fractal dimensions we often need to find the Jacobian; this often requires attempting to differentiate discontinuous functions. Therefore finding a suitable continuous approximation to the discontinuities becomes important. In previous communications, two example systems had been used with two parametrizations for approximating discontinuous functions with continuous ones, one of which is the same as that used in the literature. In this work, we will use further examples to optimize the parameters of the continuous approximation to discontinuities using different examples in order to test the degree of applicability of this approach. Where possible, the invariants calculated by this method will be compared to the corresponding invariants calculated from its time series.