René Lozi

Université Côte d’Aur, Nice, France

Are chaotic dynamical systems useful for the advancement of science?

Abstract

Since the seminal work of Henri Poincaré on the three-body problem, and more recent research dating back to the second half of the 20th century on chaotic dynamical systems, many applications have emerged in different domains (biology, economics, electronic, cryptography, physics, etc.).

The "butterfly effect" reveled by Edward Lorenz in 1963 and the word "chaos" coined by James A. Yorke in 1975 have brought global awareness of these concepts often not actually understood by the public. However, it is only at the beginning of 90’ that the applications of chaotic properties of dynamical systems were introduced with the pioneering idea of synchronization of two chaotic attractors of Louis M. Pecora and Thomas L. Carroll.

We try to describe the evolution of the last 50 years on the subject and to find out whether applications are useful for the advancement of science, and what might be the future trends for this domain of mathematics.

Saber Elaydi

Trinity University, USA

A Journey into Global Stability: From Monotone to Mixed Monotone and from autonomous to nonautonomous

Abstract

The study of the global stability of fixed and periodic points of monotone maps and triangular maps in one or higher-dimensional spaces has been successful.

For general maps, the use of Liapunov functions has had limited success. Recently, Liapunov functions have been successful in obtaining global stability of the disease-free equilibrium of epidemic models. In this talk, we extend some of these results to mixed monotone maps. A special property of these maps is that they can be embedded in symmetric monotone maps in higher-dimension spaces. The aim here is to investigate the global stability of the interior fixed points of mixed monotone autonomous systems.

The study is then extended to non-autonomous systems that are asymptotically autonomous, and to periodic difference equations. For the periodic systems, we show that a periodic cycle is globally asymptotically stable. These results are then applied to single and multi-species evolutionary competition models such as the Ricker model and the Leslie-Gower model with one trait or multi-traits.

Ali Moussaoui

University of Tlemcen, Tlemcen, Algeria

On some Bioeconomic Models in Fisheries

 Abstract

A short review of Mathematical fishery models is presented with a focus on bio-economical models with a variable market price of the resource. Next, a fishery model describing the variations of the fish stock, the fishing effort, the storage and the price of the resource on the market supposed to depend on supply and demand is presented. The study shows the existence of a catastrophic equilibrium corresponding to the extinction of the resource and one or two sustainable fishery equilibrium points that can coexist under certain conditions. The model shows that storing part of the resource makes it possible to avoid a catastrophic situation with the extinction of the fish stock and to stabilize the fishery in the long term. 

Pedro Lima

Instituto Superior Técnico, University of Lisbon, Portugal

 Numerical solution of the stochastic neural field  equation and applications to working memory

 Abstract

Neural field equations were introduced in the 70 years of last century as a mathematical model for describing neural activity and studying certain processes that occur in the cerebral cortex. One of these processes is working memory, usually defined as the capacity of neurons to transiently hold sensory information to guide forthcoming action.
In this talk, we describe a neural field model which explains how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events Moreover, we investigate how noise-induced perturbations may affect the coding process. From a mathematical point of view, this is obtained by means of a two-dimensional neural field equation, where one dimension represents the nature of the event (for example, the color of a light signal) and the other represents the moment when the signal has occurred. Some numerical experiments are carried out using a computational algorithm for two-dimensional stochastic neural field equations. The numerical results are discussed and their
physical interpretation is explained.

Haci Mehmet Baskonus

Harran University, Turkey

Dumitru Baleanu 

Cankaya University, 06530 Ankara, Turkey