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Nonlinear Dynamics

, Volume 67, Issue 1, pp 413–424 | Cite as

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

  • Diogo C. Soriano
  • Filipe I. Fazanaro
  • Ricardo Suyama
  • José Raimundo de Oliveira
  • Romis Attux
  • Marconi K. Madrid
Original Paper

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.

Keywords

Chaos Lyapunov spectrum estimation FitzHugh–Nagumo neuronal model Discontinuous excitation Nonsmooth models 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Diogo C. Soriano
    • 1
  • Filipe I. Fazanaro
    • 2
  • Ricardo Suyama
    • 3
  • José Raimundo de Oliveira
    • 2
  • Romis Attux
    • 1
  • Marconi K. Madrid
    • 4
  1. 1.Laboratory of Signal Processing for Communications (DSPCom), Department of Computer Engineering and Industrial Automation (DCA), School of Electrical and Computer Engineering (FEEC)UNICAMPCampinasBrazil
  2. 2.Modular Robotic Systems Laboratory (LSMR), DCA, FEECUNICAMPCampinasBrazil
  3. 3.Engineering, Modeling and Applied Social Sciences CenterUFABCSanto AndréBrazil
  4. 4.LSMR, Department of Energy Control and Systems (DSCE), FEECUNICAMPCampinasBrazil

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