Analysis of Hyperchaotic Complex Lorenz Systems

Abstract

This paper introduces and analyzes new hyperchaotic complex Lorenz systems. These systems are 6-dimensional systems of real first order autonomous differential equations and their dynamics are very complicated and rich. In this study we extend the idea of adding state feedback control and introduce the complex periodic forces to generate hyperchaotic behaviors. The fractional Lyapunov dimension of the hyperchaotic attractors of these systems is calculated. Bifurcation analysis is used to demonstrate chaotic and hyperchaotic behaviors of our new systems. Dynamical systems where the main variables are complex appear in many important fields of physics and communications.

Dynamical Analysis, Synchronization, Circuit Design, and Secure Communication of a Novel Hyperchaotic System

“…With the parameters chosen as = 0.1, = 4.55, = 1.96, = 0.37, and = 0.5, the corresponding Lyapunov exponents are obtained as follows: 1 = 0.0717, 2 = 0.0209, 3 = −0.4187, and 4 = −1.5439. Thus, the Lyapunov dimension [8,21] of the new hyperchaotic system (5) is also calculated as…”

“…A hyperchaotic system is defined as an attractor with at least two positive Lyapunov exponents and an autonomous system with phase space of dimension at least four [8]. The sum of Lyapunov exponents must be negative to ensure that the system is dissipative [9].…”

Combination Synchronization of Three Identical or Different Nonlinear Complex Hyperchaotic Systems

“…In this section, we take the complex hyperchaotic Lorenz system [6] as an example to investigate the combination synchronization among three identical systems.…”

“…The rich dynamics behaviors of the complex Chen and complex Lü systems were investigated in [4]. By adding state feedback controllers to their complex chaotic systems, complex hyperchaotic Chen, Lorenz and Lü systems were introduced and studied in [5][6][7], respectively. The authors [8] constructed a complex nonlinear hyperchaotic system by adding a cross-product nonlinear term to the complex Lorenz system.…”

Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems

“…Compared to chaotic real systems, chaotic complex systems exhibit more abundant and complicated dynamical behaviors with strong unpredictability, which can be applied to chaos secure communication for the sake of higher signal transmission efficiency and secure performance. [2][3][4] Therefore, much attention and many efforts are devoted to investigate synchronization of chaotic complex systems in recent years, and various synchronization schemes have been proposed and realized successfully, such as complete synchronization, 5,6 antisynchronization, 7,8 lag synchronization, 9,10 phase synchronization, 11 projective synchronization, 12,13 and their extended synchronization schemes. [14][15][16][17][18][19] All of the above-mentioned synchronization methods are designed for one drive system and one response system.…”