Authors: Ilham Djellit; Brahim Kilani; Julien Clinton Sprott
Addresses: Laboratory of Mathematics, Dynamics and Modelization, Department of Mathematics, Faculty of Sciences, University Badji Mokhtar, Annaba 23000, Algeria ' Laboratory of Mathematics, Dynamics and Modelization, Department of Mathematics, Faculty of Sciences, University Badji Mokhtar, Annaba 23000, Algeria ' Department of Physics, University of Wisconsin, 1150 University Avenue, Madison WI 53706, USA
Abstract: The Lorenz model is of interest because of its abundant bifurcations and dynamical phenomena, due largely to the presence of critical sets or non-definition sets. The model is investigated as a three-parameter quadratic family. This paper further develops and refines a study of its basins of attraction and it is explained by using two types of non-classical singularity sets. This has an important impact on the number of preimages and shows the essential role played by the vanishing denominator in the inverses. A deeper analysis of the global dynamic properties of the model in the parameter ranges, where three steady states exist, reveals the role of symmetry with an interesting and complex dynamic structure.
Keywords: bifurcation; critical sets; vanishing denominator; broken symmetry; modified Lorenz models; three-parameter quadratic families; dynamic properties; dynamical systems.
International Journal of Dynamical Systems and Differential Equations, 2015 Vol.5 No.2, pp.136 - 148
Received: 13 Nov 2013
Accepted: 14 Nov 2014
Published online: 15 Jun 2015 *