Description
This work develops a globally convergent, data driven algorithm for estimating the Lyapunov exponents of a dynamical system. This work rigorously tests our algorithm with two dynamical systems: The Lorenz equation and the Rossler equation. Additionally the algorithm is used to approximate the Lyapunov exponents of two additional dynamical systems, the Chua equation, and the Li equation Li et al. [14]. The time series used as data points for the dynamical system was obtained by numerical integration with the fourth-order Runge-Kutta scheme. Our algorithm then interpolates the time series using radial basis functions. Three basic functions Gaussian, multiquadrics, and inverse were tested using our algorithm. The shape parameter of the basic function was optimized using an algorithm developed by Rippa [21]. Our interpolation algorithm is used to produce a numerical approximation of the given dynamical system’s Jacobian. The Lyapunov exponents of the system are calculated by performing a QR-decomposition on the Jacobian approximation. We conducted several trials with Lorenz and Rossler by varying the simulation parameters t, dt, and nearest neighbors. Our trials provided good approximations of the Lorenz Lyapunov exponents except for two trials where t = 90 and one where dt = 0.1. Our algorithm also produced good approximations of the Lyapunov exponents for the Chua and Li equations. However, the algorithm struggled providing accurate approximations of the minimal Lyapunov exponent for Rossler but provided accurate approximations for Rossler’s maximal Lyapunov exponenent.
