Infinitely Many Heteroclinic Orbits of a Complex Lorenz System

The existence of heteroclinic orbits of a chaotic system is a difficult yet interesting mathematical problem. Nowadays, a rigorous analytical proof for the existence of a heteroclinic orbit can be carried out only for some special chaotic and hyperchaotic systems, and few results are known for the complex systems. In this paper, by revisiting a complex Lorenz system, it is found that this system possesses an infinite set of heteroclinic orbits to the origin and its circle equilibria. However, it is impossible for the corresponding real Lorenz system to have infinitely many heteroclinic orbits. The theoretical tools for proving the main results are Lyapunov functions and the definitions of [Formula: see text]-limit set and [Formula: see text]-limit set. Numerical simulations show the effectiveness and correctness of the theoretical conclusions. The investigations not only enrich the related results for the complex Lorenz system, but also find the essential difference between the complex Lorenz system and its corresponding real version: the complex Lorenz system has infinitely many heteroclinic orbits whereas its corresponding real one does not.

Hybrid Projective Synchronization of Fractional-Order Chaotic Complex Nonlinear Systems With Time Delays

Time delays are frequently appearing in many real-life phenomena and the presence of time delays in chaotic systems enriches its complexities. The analysis of fractional-order chaotic real nonlinear systems with time delays has a plenty of interesting results but the research on fractional-order chaotic complex nonlinear systems with time delays is in the primary stage. This paper studies the problem of hybrid projective synchronization (HPS) of fractional-order chaotic complex nonlinear systems with time delays. HPS is one of the extensions of projective synchronization, in which different state vectors can be synchronized up to different scaling factors. Based on Laplace transformation and the stability theory of linear fractional-order systems, a suitable nonlinear controller is designed to achieve synchronization between the master and slave fractional-order chaotic complex nonlinear systems with time delays in the sense of HPS with different scaling factors. Finally, the HPS between fractional-order delayed complex Lorenz system and fractional-order delayed complex Chen system and that of fractional-order delayed complex Lorenz system and fractional-order delayed complex Lu system are taken into account to demonstrate the effectiveness and feasibility of the proposed HPS techniques in the numerical example section.