A piecewise linear approximation method for the evaluation of Lyapunov Exponents of polynomial nonlinear systems

Lyapunov exponents of a dynamical system give information about its long-term evolution. Exponents estimation is not an easy task; it is computationally costly and, in presence of chaotic dynamics, it exhibits numerical difficulties. In a previous paper, the authors proposed an algorithm for Lyapunov exponents estimation in piecewise linear systems that strongly reduces the computational time. In this paper, the algorithm is applied also to chaotic systems with polynomial nonlinearity. Firstly, a suitable piecewise linear approximation for the polynomial nonlinear function is evaluated by means of a Multi-Layer Perceptron neural network with linear and saturating linear transfer functions. Then, the linearity of the new state equation and of the variational equation, obtained resorting to the piecewise linear approximation of the nonlinear function, are exploited to accurately evaluate Lyapunov exponents of the approximated system with a reduced execution time.