Canards Existence in 3D & 4D [ Back ]

Date:
23.02.15   
Times:
15:30 to 16:00
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Jean-Marc Ginoux
University:
Université de Toulon

Abstract

The aim of this work is to propose a direct method for determining the condition of existence of “canard solutions” for three and four-dimensional singularly perturbed systems with only one fast variable and four-dimensional singularly perturbed systems with two fast variables in the folded saddle case. Contrary to previous works, this method, which does not require a center manifold reduction nor a blow-up technique i.e. a desingularization procedure for folded singularities, uses the normalized slow dynamics and not the projection of the desingularized vector field. Thus this method enables to state a unique generic condition for the existence of “canard solutions” for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics and not on the projection of the desingularized vector field. Application of this method in the case of one fast variable to the famous three and four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to prove the existence of “canard solutions” in such Memristor Based Chaotic Circuits. Application to this method in the case of two fast variables to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model will enable to prove as many previous works the existence of “canard solutions” in such system.