Infinitely Many Heteroclinic Orbits of a Complex Lorenz System

Abstract

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.

Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

“…The proof of Proposition 6 follows from the Routh-Hurwitz criterion and is similar to the one [ [7,12,20,21,23,25,40,41,44,45,49]. In this effort, one in this section tries to give some kind of the forming mechanism of the hyperchaotic attractor [15, p. 867] of the system (1) with (a 1 , a 2 , c, d, e, f, b, p, q) = (−12, 12, 23, −1, −1, 1, 2.1, −6, −0.2) combining numerical techniques and the dynamics of S z .…”

“…Existence of herteroclinic orbit. Utilizing two different Lyapunov functions, concepts of both α-limit set and ω-limit set [5,16,17,19,20,21,24,34,35,39,40,41,44,45], this section devotes to investigating the existence of heteroclinic orbits of the system (1), aiming at complementing and extending the obtained results in [5,Theorem 5,p.579]. The fundamental work of the proof is how to construct suitable Lyapunov functions for the system (1).…”

“…respectively. Proceeding as in [5,16,17,19,20,21,24,34,35,39,40,41,44,45], the following results are derived for the both cases 2a 1 + b > 0 and 2a 1 + b = 0.…”

Dynamical Analysis and Simulation of a New Lorenz-Like Chaotic System

“…Ever since Lorenz discovered the chaotic phenomena in a simple 3D nonlinear ODE in 1963, more and more specialists devoted themselves to chaotic dynamics. Some of them were interested in finding different kinds of new chaotic models and discussed their dynamical properties, for example, Lü system [9], Chen system [10], Lorenz-type system [11][12][13][14], and other new chaotic system [15][16][17][18][19], while some of them focused their attentions on revisiting the existing chaotic system and explored some new phenomena which had not been found before [20][21][22][23][24][25][26][27][28]. All of these research studies are beneficial to reveal the essence of chaos.…”

A true three-scroll chaotic attractor coined

“…Moreover, Figure 1-2 illustrate that the aforementioned three-scroll chaotic attractor coexists with the single saddle-node E 0 . Secondly, since Kokubu and Roussarie [9] introduced the concept of a singularly degenerate heteroclinic cycle that consists of an invariant set formed by a line of equilibria together with a heteroclinic orbit connecting two of the equilibria, many other researchers [6,15,18,22,36,37,38,39,40,41,42,43,44,45,46] have begun to study it, due to the broken of it could create classical and conjugate Lorenzlike attractors, four-wing attractors and so on. When studying a four-dimensional hyperchaotic Lorenz-like system, Wang and Dong [44] recently found that simplex explosions of stable non-isolated equilibria is also one route to chaos/hyperchaos, especially in the case of the original Lorenz attractor and some hyperchaotic attractors.…”