Prof. René Lozi
Université Côte d’Azur, Department of Mathematics, Nice, France
Speech Title: Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model
Abstract:
In this presentation, we investigate the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh-Nagumo (FFHN) model (a model of neuron). To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an S-asymptotically T-periodic solution of the fractional order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB.
The existence of canard oscillations in the neighborhoods of such HLB points is numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, 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 the fractional order close to one. The article demonstrates that the FFHN Model is a very simple 2-dimensional model with an incredible ability to present the complex dynamics of neurons.
Keywords: FitzHugh-Nagumo model; canard explosion; fractional-order system; mixed-mode oscillation; chaos.
Biography:
René LOZI is emeritus Professor at UniversityCôte d’Azur, Dieudonné Center of Mathematics, France. He completed a PhD in 1975 and the French State Thesis (on chaotic dynamical systems) under the supervision of Prof. René Thom (Fields medalist) in 1983.
In 1991, he became Full Professor at University of Nice and IUFM (Institute for teacher trainees). He has served as Director of this institute (2001-2006) and as Vice-Chairman of the French Board of Directors of IUFM (2004- 2006). He is a member of several editorial boards of international journals. In 1977, he discovered a particular mapping of the plane having a strange attractor (now, commonly known as "Lozi map").
Nowadays, his research areas include complexity and emergence theory, dynamical systems, bifurcations, control of chaos, cryptography based on chaos, and recently memristors (physical devices for neurocomputing) and artificial intelligence. He is working in those fields with renowned researchers from many countries. He received the Dr. Zakir Husain Award 2012 from the Indian Society of Industrial and Applied Mathematics during the 12th biannual conference of ISIAM at the University of Punjab, Patiala, (India) in January 2015. He received the C.S. Hsu Award 2024 for distinguished scholars in Nonlinear Dynamics and Control, issued annually by the Nonlinear Science and Complexity Conferences Series, in August 2024 at the Sichan university of Yibin, China.
Since August 2023, he is Vice-President of the International Society of Difference Equations.