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- Information theory, synchronization and topological order in complete dynamical networks of discontinuous mapsPublication . Rocha, J. Leonel; Carvalho, S.This paper is dedicated to the study of information measures, synchronization and a topological order in complete dynamical networks of discontinuous piecewise linear maps with different slopes. It stands out that the networks topologies are characterized by circulant matrices and the conditional Lyapunov exponents are explicitly determined. Some properties of the mutual information rate and the Kolmogorov–Sinai entropy, depending on the synchronization interval, are discussed. A topological order between the complete dynamical networks is presented, which is characterized by the monotony of the network topological entropy. It is proved that if the network topological entropy increases, then the mutual information rate and the Kolmogorov–Sinai entropy increase or decrease, according to the variation of the coupling parameter. Furthermore, various types of computer simulations show the experimental applications of these results and techniques.
- Generalized r-Lambert function in the analysis of fixed points and bifurcations of homographic 2-Ricker mapsPublication . Rocha, J. Leonel; TAHA, Abdel-KaddousThis paper aims to study the nonlinear dynamics and bifurcation structures of a new mathematical model of the γ-Ricker population model with a Holling type II per-capita birth function, where the Allee effect parameter is γ = 2. A generalized r-Lambert function is defined on the 3D parameters space to determine the existence and variation of the number of nonzero fixed points of the homographic 2-Ricker maps considered. The singularity points of the generalized r-Lambert function are identified with the cusp points on a fold bifurcation of the homographic 2-Ricker maps. In this approach, the application of the transcendental generalized r-Lambert function is demonstrated based on the analysis of local and global bifurcation structures of this three-parameter family of homographic maps. Some numerical studies are included to illustrate the theoretical results.
- Dynamics and bifurcations of a map of homographic Ricker typePublication . Rocha, J. Leonel; TAHA, Abdel-Kaddous; Fournier-Prunaret, D.A dynamical system of the type homographic Ricker map is considered; this is a particular case of a new extended gamma-Ricker population model with a Holling type II per-capita birth function. The purpose of this paper is to investigate the nonlinear dynamics and bifurcation structure of the proposed model. The existence, nature and stability of the fixed points of the homographic Ricker map are analyzed, by using a Lambert W function. Fold and flip bifurcation structures of the homographic Ricker map are investigated, in which there are flip codimension-2 bifurcation points and cusp points, while some parameters evolve. Some communication areas and big bang bifurcation curves are also detected. Numerical studies are included.
- Bifurcation analysis of the γ-Ricker population model using the Lambert W functionPublication . Rocha, J. Leonel; TAHA, Abdel-KaddousIn this work, we present the dynamical study and the bifurcation structures of the γ-Ricker population model. Resorting to the Lambert W function, the analytical solutions of the positive fixed point equation for the γ-Ricker population model are explicitly presented and conditions for the existence and stability of these fixed points are established. The main focus of this work is the definition and characterization of the Allee effect bifurcation for the γ-Ricker population model, which is not a pitchfork bifurcation. Consequently, we prove that the phenomenon of Allee effect for the γ-Ricker population model is associated with the asymptotic behavior of the Lambert W function in a neighborhood of zero. The theoretical results describe the global and local bifurcations of the γ-Ricker population model, using the Lambert W function in the presence and absence of the Allee effect. The Allee effect, snapback repeller and big bang bifurcations are investigated in the parameters space considered. Numerical studies are included.