A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model

• Published in: Nonlinear dynamics Vol. 67; no. 1; pp. 413 - 424
• Main Authors: Soriano, Diogo C., Fazanaro, Filipe I., Suyama, Ricardo, de Oliveira, José Raimundo, Attux, Romis, Madrid, Marconi K.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 2012; Springer Nature B.V
• Subjects: Automotive Engineering; Chaos theory; Classical Mechanics; Control; Discrete time systems; Divergence; Dynamical Systems; Engineering; Equations of motion; Evolution; Liapunov exponents; Linearization; Lyapunov exponents; Mathematical analysis; Mathematical models; Mechanical Engineering; Nonlinear dynamics; Original Paper; Reproduction; Vibration; Chaos; FitzHugh–Nagumo neuronal model; Discontinuous excitation; Lyapunov spectrum estimation; Nonsmooth models
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-011-9989-2

This work presents a new method to calculate the Lyapunov spectrum of dynamical systems based on the time evolution of initially small disturbed copies (“clones”) of the motion equations. In this approach, it is not necessary to construct the tangent space associated with the time evolution of linearized versions of motion equations, being the Lyapunov exponents directly estimated in terms of the rate of convergence or divergence of these disturbed clones with respect to the fiducial trajectory, there being periodic correction via the Gram–Schmidt Reorthonormalization procedure. The proposed method offers the possibility of partial estimation of the Lyapunov spectrum and can also be applied to nonsmooth dynamics, since the linearization procedure is no longer required. The idea is tested for representative continuous- and discrete-time dynamical systems and validated by means of comparison with the classical method to perform this calculation. To illustrate its applicability in the nonsmooth context, the largest Lyapunov exponent of the FitzHugh–Nagumo neuronal model under discontinuous periodic excitation is calculated taking the amplitude of stimulation as control parameter. This analysis reveals some complex behaviours for this simple neuronal model, which motivates relevant discussions about the possible role of chaos in the cognitive process.