Volume 10 (2021)

1.  Differential Calculus of Multidual Functions  Full text (PDF)

Farid Messelmi

Pages: 1-15

Abstract:   The purpose of this paper is to contribute to the development of a main theory of differential calculus in the algebra of multidual numbers. We start by recalling, according the work [7], some results regarding multidual analysis. Inspired from complex analysis, we define the concepts of anti-hyperholomorphicity and k-anti-hyperholomorphicity for multidual functions making use the introduced notion of conjugate variables. Moreover, we introduce the concepts of co-hyperholomorphicity and k-co-hyperholomorphicity by the means of the generalized Dirac operators.

2.  Using Normal Form Theory to Identify the Cyclic Patterns of Population Dynamics  Full text (PDF)

Tahmineh Azizi

Pages: 16-29

Abstract:  In this paper, we study a three-parameter family of discrete time population models which describes a host-parasitoid interaction, and generalizes a model that originally was proposed by Nicholson and Bailey. In contrast to the last case, where there was a density dependent factor exp(r(1-H(n)=k-P(n))) in the equation of host population, the general case of this family consists of exp((r(1-P(n))^(2 a-1)-H(n)/k)) displays a somewhat almost similar range of behaviors for a in Z+. In particular, this general model also undergoes a Neimark Sacker bifurcation that produces an attracting invariant closed curve in some areas of the parameter space. We perform the local stability analysis to discover the local dynamics of equilibrium points. Using the normal form theorem, we will present the topological normal form for Neimark-Sacker bifurcation. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic
oscillations between host and parasitoid populations.

3.  Constructing some discrete 4-D hyperchaotic systems Full text (PDF)

N. K. K. Dukuza

Pages: 30-40

Abstract:  Modeling real life phenomena often leads to complex nonlinear dynamics such as bifurcation and chaos. The study of such problems has attracted interest of many scientists over the past decades. In this paper, we present a method for constructing some discrete four dimensional (4-D) hyperchaotic systems. A nonclassical procedure for discretising autonomous 4-D continuous hyperchaotic systems is applied; a parameter is introduced in this process. By adjusting this parameter, until we obtain exactly two equal-positive Lyapunov exponents, a new discrete 4-D hyperchaotic system is realised. We prove that these discrete systems are bounded-input bounded-output (BIBO) stable. Our illustrative results show that the constructed discrete systems and their continuous counterparts have similar phase portraits.

4.  Synchronized population models in biology Full text (PDF)

Tahmineh Azizi

Pages: 41-53

Abstract:  Chaotic behaviors and synchronized cycles of population models has been investigated by many researchers in di erent areas. In this paper, we use a recently developed method to synchronize discrete-time dynamical system. Using this coupling approach, we are able to nd a threshold to completely synchronize a dynamical system and we will apply this result on a quadratic population model. This model reveals di erent types of dynamics depending on parameter values from stable equilibrium to periodic behavior and chaos. This rigorous method helpes us to synchronize successfully the chaotic attractors of the original system and its coupled one. Finally, we will use different numerical tools such as the mean phase difference and the mean amplitude difference, time series and bifurcation diagram for different threshold and parameter values to test the analytic results.

5.  On some universal dynamics of a 2-D Hénon-like mapping with an unknown bounded function Full text (PDF)

Zeraoulia Elhadj

Pages: 54-63

Abstract:  This paper investigates the dynamics of a 2-D H enon-like mapping with an unknown bounded function. The values of parameters and the range of initial conditions for which the dynamics of this equation is bounded or unbounded are rigorously derived. The results given here are universal and do not depend on the expression of the nonlinearity in the considered map.