Résumé: This paper explores the dynamics of 2D spatiotemporal discrete systems, focusing on the stability and bifurcations of periodic solutions, particularly 3-cycles. After introducing the concept of a third-order cycle, we discuss both numerical and analytical techniques used to analyze these cycles, defining four types of 3-periodic points and their associated stability conditions. As a specific case, this study examines a spatiotemporal quadratic map, analyzing the existence of 3-cycles and various bifurcation scenarios, such as fold and flip bifurcations, as well as chaotic behavior. In 2D spatiotemporal systems, quadratic maps intrinsically offer better conditions that favor the emergence of chaos, which is characterized by high sensitivity to initial conditions. The findings emphasize the complexity of these systems and the crucial role of bifurcation curves in understanding stability regions. The paper concludes with key insights and suggestions for future research in this field.

Résumé: This study investigates the dynamic behavior of an SIRS epidemic model in discrete time, focusing primarily on mathematical analysis. We identify two equilibrium points, disease-free and endemic, with our main focus on the stability of the endemic state. Using data from the US Department of Health and optimizing the SIRS model, we estimate model parameters and analyze two types of bifurcations: Flip and Transcritical. Bifurcation diagrams and curves are presented, employing the Carcasses method. for the Flip bifurcation and an implicit function approach for the Transcritical bifurcation. Finally, we apply constrained optimal control to the infection and recruitment rates in the discrete SIRS model. Pontryagin’s maximum principle is employed to determine the optimal controls. Utilizing COVID-19 data from the USA, we showcase the effectiveness of the proposed control strategy in mitigating the pandemic’s spread.

Résumé: This study aims to investigate the well-posedness and stability of a thermoelastic Timoshenko system with non-Fourier heat conduction. Specifically, we analyze the system using the dual-phase-lag (DPL) model, which incorporates two thermal relaxation times, τq and τθ, to model non-instantaneous heat propagation. Applying the semigroup approach, we demonstrate the existence and uniqueness of the solutions. Subsequently, we introduce a novel stability parameter ϰ using the multiplier method. Exponential decay is proven for the case of ϰ=0 with 2⁢τθ>τq. Using Gearhart–Prüss theorem, we show the lack of exponential stability when ϰ≠0 and 2⁢τθ=τq. Numerically, we present a fully discrete approximation using the finite element method and the backward Euler scheme, and we provide some numerical simulations to show the discrete energy decay and the behavior of solutions.

Résumé: In this work, we give theoretical and numerical analyses for local bifurcations of 2D spatiotem- poral discrete systems of the form x_{m+1,n+1} = f(x_{m,n} , x_{m+1,n}), where f is a real nonlinear function, m and n are two independent integer variables, representing respectively a spatial coordinate and the time. On the basis of the spectral theory, we derive the conditions under which the local bifurcations such as flip and fold occur at the fixed points for some parameter values. As a case-study, a quite complex system, a 2D spatiotemporal dynamic given by two coupled logistic maps, named 2D logistic coupled map (2D-LCM) is considered. The proposed map provides a reliable experimental and theoretical basis for identifying some cases of local bifurcations.

Résumé: This note provides a complete description of the spectrum of diagonal perturbation of weighted shift operator acting on a separable Hilbert space.

Résumé: This paper deals with stability and local bifurcations of two-dimensional (2D) spatiotemporal discrete systems. Necessary and sufficient conditions for asymptotic stability of the systems are obtained. They prove to be more accurate than those in the current literature. Some definitions for the bifurcations of 2D spatiotemporal discrete systems are also given, and an illustrative example is provided to explain the new results.

Résumé: In this work, we propose a novel 3D chaotic map obtained by coupling the piecewise and logistic maps. Showing excellent properties, like a high ran- domness, a high complexity and a very long period, this map has enabled us to implement and investigate a new chaotic pseudo-random number generator (CPRNG). The produced pseudo-random numbers exhibit a uni- form distribution and successfully pass the NIST SP 800-22 randomness tests suite. In addition, an applica- tion in the field of color image encryption is proposed where the encryption key is strongly correlated with the plain image and is then used to perform the confusion and diffusion stages. Furthermore, the ability to expand the size of our map has an impact on the complexity of the system and increases the size of the key space, making our cryptosystems more efficient and safer. We also give some statistical tests and computer simulations which confirm that the proposed algorithm has a high level of security.

Résumé: This paper provides a description of the spectrum of diagonal perturbation of weighted n-shift operator acting on a separable Hilbert space.

Résumé: Descent direction methods and trust region methods are usually used to solve the unconstrained optimization problem $\left( p\right)$ \begin{equation*} \left( P\right) \left\{ \underset{x\in \mathbb{R}^{n}}{\min }f\left( x\right) \right. . \end{equation*} In this work, we are interested in convergence results that use trust region methods which employ the conjugate gradient method Day-Yuan version as a subprogram for each iteration. Further, we penalize the quadratic problems with constraints and convert them into series of unconstrained problems.

Résumé: Let be a complex polynomial of fixed degree . In this paper we show that Cauchy’s method may fail to find all zeros of for any initial guess lying in the complex plane and we propose several ways to find all zeros of a given polynomial using scaled Cauchy’s methods.

Résumé: In this work, we consider some dynamical properties and speci c contact bifurcations of a discrete-time predator{prey system having inverses with vanishing denominator. The dynamics is investigated by using concepts of focal points, prefocal curves and bifurcation theory. The system undergoes ip bifurcation and Neimark-Sacker bifurcation. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to con rm further the complexity of the dynamical behaviors as extinction, persistence and permanence.

Résumé: Conjugate gradient method is a root-finding algorithm to non-linear equations. In this paper, we suggest extending this method for a polynomial to the complex plane. Through the experimental and theoretical mathematics method, we drew the following conclusions: (1) the conjugate gradient is a dynamical system with two complex parameters; (2) locally conditions for convergence to any roots of complex functions is given; (3) the conjugate gradient method may fail to converge to all roots for cubic with three simple roots; (4) the boundary of conjugate gradient basins are fractals in some cases, and depends on the parameters; (5) the algorithm is then improved by introducing a method to determine the optimal parameters.

Résumé: The use of the L-BFGS method is very efficient for the resolution of large scale optimization problems. The techniques to update the diagonal matrix seem to play an important role in the performance of overall method. In this work, we introduce some methods for updating the diagonal matrix derived from quasi-Newton formulas (DFP, BFGS). We compare their performances with the Oren-Spedicato update proposed by Liu and Nocedal (1989) and we get considerable amelioration in the total running time. We also study the convergence of L-BFGS method if we use the BFGS and inverse BFGS update of the diagonal matrix on uniformly convex problems.

Résumé: Newton's method is a root-Ønding algorithm and Newton basins is the set of initial guesses that lead to one root of polynomial on the complex plane. A boundary of Newton basins are fractals and called Julia set. In this work we introduce a modiØcation on the algorithm and we show that the fractal aspect of basins is preserved.

Résumé: The theory of critical curves for maps of the plane provides powerful tools for locating the chief characteristic features of a discrete dynamical system in two dimensions: the location of its chaotic attractors, its basin boundaries, and the mechanisms of its bifurcations. Nowadays one begins to recognize the role played by critical curves of maps in the analysis, in the understanding and description of the bifurcations, and transition to chaotic behavior in coupled maps. In this paper we consider some properties of such maps, which possess a chaotic attractor. Some examples are considered in this paper in which we can see the effective role played by such curves in bifurcation theory.

Résumé: The dynamics of complex cubic polynomials have been studied extensively in the recent years. The main interest in this work is to focus on the Julia sets in the dynamical plane, and then is consecrated to the study of several topics in more detail. Newton's method is considered since it is the main tool for finding solutions to equations, which leads to some fantastic images when it is applied to complex functions and gives rise to a chaotic sequence.

Résumé: This work is an extension of the survey on Cayley's problem in case where the conjugate gradient method is used. We show that for certain values of parameters, this method produces beautiful fractal structures.