Open AccessMixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.
Author(s) -
René Lozi,
Mohammed-Salah Abdelouahab,
Guanrong Chen
Publication year - 2020
Publication title -
applied mathematics and nonlinear sciences
Language(s) - English
Resource type - Journals
ISSN - 2444-8656
DOI - 10.2478/amns.2020.2.00047
Subject(s) - hopf bifurcation , mode (computer interface) , order (exchange) , bifurcation , amplitude , dynamical systems theory , dynamics (music) , physics , mathematics , statistical physics , computer science , nonlinear system , finance , quantum mechanics , economics , operating system , acoustics
This article highlights particular mixed-mode oscillations (MMO) based on canard explosion observed in a fractional-order Fitzhugh-Nagumo (FFHN) model. In order to rigorously analyze the dynamics of the FFHN model, a recently introduced mathematical notion, the Hopf-like bifurcation (HLB), which provides a precise definition for the change between a fixed point and an S −asymptotically T −periodic solution, is used. The existence of HLB in this FFHN model is proved and the appearance of MMO based on canard explosion in the neighborhoods of such HLB points are numerically investigated using a new algorithm: the global-local canard explosion search algorithm. This MMO is constituted of various patterns of solutions with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied simultaneously. On the basis of such numerical experiment, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional-order close to one. Therefore, this very simple two-dimensional FFHN model, presents an incredible ability to mimic the complex dynamics of neurons.