Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution of m -PWL Variational Equations

Abstract

An algorithm to compute the Lyapunov exponents of piecewise linear function-based multidirectional multiscroll chaotic oscillators is reported. Based on the $m$ regions in the piecewise linear functions, the suggested algorithm determines the individual expansion rate of Lyapunov exponents from $m$-piecewise linear variational equations and their associated $m$-Jacobian matrices whose entries remain constant during all computation cycles. Additionally, by considering OpAmp-based chaotic oscillators, we study the impact of two analog design procedures on the magnitude of Lyapunov exponents. We focus on analyzing variations of both frequency bandwidth and voltage/current dynamic range of the chaotic signals at electronic system level. As a function of the design parameters, a renormalization factor is proposed to estimate correctly the Lyapunov spectrum. Numerical simulation results in a double-scroll type chaotic oscillator and complex chaotic oscillators generating multidirectional multiscroll chaotic attractors on phase space confirm the usefulness of the reported algorithm.