Christophe LETELLIER
14/10/2023

René Lozi

**Abstract**

Since two decades, Fractional Differential Equations (FDE) are more and more used to model a large variety of phenomena in the nature. Their ability to model better than Ordinary Differential Equations (ODE) is due in particular to the ‘memory’ of the initial conditions. The counterpart of this ‘memory’ is that FDE cannot exhibit exact periodic solutions and hence Hopf bifurcation. However, in some situations numerical simulations show similarities with such bifurcation. Therefore we introduce the concept of Hopf-like bifurcation to study the emergence of mixed-mode oscillations and canard explosion, in a planar fractional order FitzHugh-Nagumo model (FFHN). In this aim an algorithm, called *Global-Local Canard Explosion Search Algorithm* (GLCESA) is developed and used to investigate the existence of canard oscillations in the neighborhoods of Hopf-like bifurcation points. The appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a 2-dimensional fractional-order autonomous dynamical system, with a fractional order close to one.