Lyapunov Exponents of a Discontinuous 4D Hyperchaotic System of Integer or Fractional Order

Marius-F Danca

doi:10.3390/e20050337

Abstract

In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer or fractional order cannot be numerically integrated using methods for continuous differential equations. A possible approach for discontinuous systems is presented. To integrate the initial value problem of fractional order or integer order, the discontinuous system is continuously approximated via Filippov’s regularization and Cellina’s Theorem. The Lyapunov exponents of the approximated system of integer or fractional order are represented as a function of two variables: as a function of two parameters, or as a function of the fractional order and one parameter, respectively. The obtained three-dimensional representation leads to comprehensive conclusions regarding the nature, differences and sign of the Lyapunov exponents in both integer order and fractional order cases.

Highlights

Systems with discontinuous right-hand side modeled as Initial Value Problems (IVPs) are mostly ideal, since switch-type functions like sgn are used, where the hysteresis or delay of the real switching operation is not considered, or the regularization represents a good approach for numerical integration of the underlying problems
Determination of Lyapunov Exponents (LEs) of discontinuous systems of integer order (IO) or FO requires the numerical integration of the underlying IVP, which cannot be realized with the classical methods for continuous differential equations of IO or FO, respectively
By a careful analysis, if one considers the vertical section with the plane k = 4.1 (Figure 5c), the dynamics of LEs as function of q reveals that, for q close to 1 (q > 0.95), there exists zero LE

Summary

Introduction

Systems with discontinuous right-hand side modeled as Initial Value Problems (IVPs) are mostly ideal, since switch-type functions like sgn are used, where the hysteresis or delay of the real switching operation is not considered, or the regularization represents a good approach for numerical integration of the underlying problems. In this paper the LEs are numerically determined, after the considered IVP of IO or FO is continuously approximated. Determination of LEs of discontinuous systems of IO or FO requires the numerical integration of the underlying IVP, which cannot be realized with the classical methods for continuous differential equations of IO or FO, respectively. To allow the study of numerical LEs, a possible continuous regularization of the right-hand side to overcome the discontinuity problem is presented in this paper.

The Integer Case q d

The Fractional-Order Case

Conclusions