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

Abstract

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.

Characterization of multiscroll attractors using Lyapunov exponents and Lagrangian coherent structures

“…16 aiming to address the information generation and the complexity increase process in a multiscroll attractor model when the number of fundamental scrolls units rises. This task was accomplished by means of the evaluation of the Lyapunov spectrum using a method developed to overcome the drawbacks underlying this model 48 without the need to apply methodologies based on time series analysis or laborious partition of the phase space for obtaining the variational equation in the tangent map approach. 37,38,42,50 It is also important to emphasize that there are different ways of generating multiscroll attractors, 12,14,28,33 being a general approach to analyse all of them virtually impossible.…”

“…This methodology was successfully tested for a number of classical nonlinear dynamical systems and also for discontinuously excited neuronal models for stability analysis purposes. 47,48 Formally, given an N-dimensional dynamical system described by _…”

“…After the set of difference state vectors has been determined, the Gram-Schmidt Reorthonormalization (GSR) procedure 38,48,50 can be applied in order to separate the contribution of the most expansive direction of the dynamics from the remaining ones, which also avoids the collapse of the clones into a single (i.e., the most expansive) direction. This procedure leads to a set of vectors v 1 ; v 2 ; …; v N that correspond to corrected versions of the difference state vectors.…”

“…To deal with problems of this nature, it was previously developed a method called Cloned Dynamics (ClDyn) approach 48 for estimating the Lyapunov spectrum based on the divergence (or the convergence) of small disturbed copies of the original trajectory. This approach does not require the construction of the tangent space associated with the variational equations and, as a consequence, it is attractive for being employed in non-smooth dynamical systems or even for models with a hard mathematical description.…”

Single Gaussian Chaotic Neuron: Numerical Study and Implementation in an Embedded System

“…With this in mind, the signals of the brain generated by neural structures could be chaotic. Soriano et al presents a similar work where the Fitzhugh-Nagumo neural model is analysed using a new method called cloned dynamics approach used for calculating the Lyapunov spectrum of dynamical systems [4].…”