On modified time delay hyperchaotic complex Lü system

Time delays are often considered as sources of complex behaviors in dynamical systems. Much progress has been made in the research of time delay systems with real variables. In this article, we will focus our study on time delay complex systems. This paper investigates a modified time delay hyperchaotic complex Lü system. This system is constructed by including the constant delay to one of its states. The behaviors of our time delay system are greatly different from those of the original system without delay. By setting the parameters, we discuss the effect of delay variation on system stability. Numerically, we calculate the range of system parameters at which chaotic and hyperchaotic attractors of different order exist. We found that our system has hyperchaotic attractors of order 2,3,…,6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 2,3,\ldots ,6$$\end{document}. However, the modified complex Lü system without delay has only hyperchaotic attractors of order 2. Different forms of modified time delay hyperchaotic complex Lü system are constructed by including the delay into different states of this system. Chaos synchronization in modified time delay hyperchaotic complex Lü system is investigated. The active control method based on Lyapunov–Krasovskii function is used to synchronize the hyperchaotic attractors. In particular, studying the time evolution of errors, we show that this technique is very effective for controlling the behavior of our system.