Abdel-Kaddous Taha

Chaos generator for secure transmission using a sine map and an RLC series circuit

This paper deals with the model analysis of a chaos generator for secure transmission. In this proposed system discrete value signals and continuous value signals are used together and are interacting one another. The main advantage of this system is that a discrete chaotic map can be implemented in the digital circuit with many parameters, and initial condition can be set with a great accuracy. Also, the problem of periodicity which is always encountered in any fully discrete systems can be avoided. A chaotic behavior study of the system is then provided, in which bifurcation phenomena are explained and chaotic attractors are shown. Finally, an asymptotic chaotic behavior of the proposed system is derived.

Dynamical Analysis and Big Bang Bifurcations of 1D and 2D Gompertz’s Growth Functions

In this paper, we study the dynamics and bifurcation properties of a three-parameter family of 1D Gompertzs growth functions, which are defined by the population size functions of the Gompertz logistic growth equation. The dynamical behavior is complex leading to a diversified bifurcation structure, leading to the big bang bifurcations of the so-called “box-within-a-box” fractal type. We provide and discuss sufficient conditions for the existence of these bifurcation cascades for 1D Gompertzs growth functions. Moreover, this work concerns the description of some bifurcation properties of a Henons map type embedding: a “continuous” embedding of 1D Gompertzs growth functions into a 2D diffeomorphism. More particularly, properties that characterize the big bang bifurcations are considered in relation with this coupling of two population size functions, varying the embedding parameter. The existence of communication areas of crossroad area type or swallowtails are identified for this 2D diffeomorphism.

From Weak Allee Effect to No Allee Effect in Richards’ Growth Models

Population dynamics have been attracting interest since many years. Among the considered models, the Richards’ equations remain one of the most popular to describe biological growth processes. On the other hand, Allee effect is currently a major focus of ecological research, which occurs when positive density-dependence dominates at low densities. In this chapter, we propose the dynamical study of classes of functions based on Richards’ models describing the existence or not of Allee effect. We investigate bifurcation structures in generalized Richards’ functions and we look for the conditions in the \((\beta, r)\) parameter plane for the existence of a weak Allee effect region. We show that the existence of this region is related with the existence of a dovetail structure. When the Allee limit varies, the weak Allee effect region disappears when the dovetail structure also disappears. Consequently, we deduce the transition from the weak Allee effect to no Allee effect to this family of functions. To support our analysis, we present fold and flip bifurcation curves and numerical simulations of several bifurcation diagrams.

TWO-DIMENSIONAL COUPLED PARAMETRICALLY FORCED MAP

A two-dimensional parametrically forced system constructed from two identical one-dimensional subsystems, whose parameters are forced into periodic varying, with mutually influencing coupling is pr...

Homoclinic and Big Bang Bifurcations of an Embedding of 1D Allee’s Functions into a 2D Diffeomorphism

In this work a thorough study is presented of the bifurcation structure of an embedding of one-dimensional Allee’s functions into a two-dimensional diffeomorphism. A complete classification of the nature and stability of the fixed points, on the contour lines of the two-dimensional diffeomorphism, is provided. A necessary and sufficient condition so that the Allee fixed point is a snapback repeller is established. Sufficient conditions for the occurrence of homoclinic tangencies of a saddle fixed point of the two-dimensional diffeomorphism are also established, associated to the snapback repeller bifurcation of the endomorphism defined by the Allee functions. The main results concern homoclinic and big bang bifurcations of the diffeomorphism as “germinal” bifurcations of the Allee functions. Our results confirm previous predictions of structures of homoclinic and big bang bifurcation curves in dimension one and extend these studies to “local” concepts of Allee effect and big bang bifurcations to this two-dimensional exponential diffeomorphism.

Big Bang Bifurcation Analysis and Allee Effect in Generic Growth Functions

The main purpose of this work is to study the dynamics and bifurcation properties of generic growth functions, which are defined by the population size functions of the generic growth equation. This family of unimodal maps naturally incorporates a principal focus of ecological and biological research: the Allee effect. The analysis of this kind of extinction phenomenon allows to identify a class of Allee’s functions and characterize the corresponding Allee’s effect region and Allee’s bifurcation curve. The bifurcation analysis is founded on the performance of fold and flip bifurcations. The dynamical behavior is rich with abundant complex bifurcation structures, the big bang bifurcations of the so-called “box-within-a-box” fractal type being the most outstanding. Moreover, these bifurcation cascades converge to different big bang bifurcation curves with distinct kinds of boxes, where for the corresponding parameter values several attractors are associated. To the best of our knowledge, these results represent an original contribution to clarify the big bang bifurcation analysis of continuous 1D maps.