Eur. Phys. J. Plus (2017) 132: 461
https://doi.org/10.1140/epjp/i2017-11739-6

Regular Article

Bifurcations and chaos of time delay Lorenz system with dimension 2n+1

¹ Department of Mathematics, Faculty of Science, Assiut University, 71516, Assiut, Egypt
² Department of Mathematics, Faculty of Science, Sohag University, 82524, Sohag, Egypt
³ Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia

^* e-mail: gmahmoud@aun.edu.eg

Received: 12 August 2017
Accepted: 3 October 2017
Published online: 7 November 2017

Abstract

The aim of this paper is to introduce a generalized form of the Lorenz system with time delay. Instead of considering each state variable of the Lorenz system belonging to , the paper considers two of them belonging to . Hence the Lorenz system has (2n+1) dimension. This system appears in several applied sciences such as engineering, physics and networks. The stability of the trivial and nontrivial fixed points and the existence of Hopf bifurcations are studied analytically. Using the normal form theory and center manifold argument, the direction and the stability of the bifurcating periodic solutions are determined. Finally, numerical simulations are calculated to confirm our theoretical results. The paper concludes that the dynamics of this system are rich. Additionally, the values of the delay parameter at which chaotic and hyperchaotic solutions exist for different values of n using Lyapunov exponents and Kolmogorov-Sinai entropy are calculated numerically.