Dynamics of the general Lorenz family

Abstract

The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lü system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.

A note on “Introduction and synchronization of a five-term chaotic system with an absolute-value term” in [Nonlinear Dyn. 73 (2013) 311–323] by Pyung Hun Chang and Dongwon Kim

“…As a matter of fact, if the conclusion in Lemma 4.3 holds, then S ± are global asymptotically stable according to the procedures of proof and similar statements in the references [8][9][10][11][12][13][14][15][16]. However, S ± are unstable when a > 0 on the ground of Lemma 3.2.…”

“…In 2006, using the definitions of ω-limit set and α-limit set, together with Lyapunov functional, Li et al [8] rigorously proved the Chen system to have two and only two heteroclinic orbits under certain conditions. Inspired by these ideas, many other Lorenz-type systems [9][10][11][12][13][14][15][16] have been proved to possess the same property when some conditions hold.…”

Analysis of global dynamics in an unusual 3D chaotic system

“…China e-mail: liuyongjianmaths@126.com simple nonlinearities [3][4][5][6][7][8][9][10][11][12][13]. It is very important to note that some 3D autonomous chaotic systems have three particular fixed points: one saddle and two unstable saddle-foci (for example, the Lorenz system [1], the Chen system [3], the Lü system [4], and the conjugate Lorenz-type system [14]).…”

“…Equating the coefficients of the powers of z 3 in Eqs. (7) shows that 2 3 , h 2 ≡ 0 and the (unique) local center manifold at (0, 0, 0) is the graph of the function (z 1 ,…”

Complex dynamics in a 5-D hyper-chaotic attractor with four-wing, one equilibrium and multiple chaotic attractors

“…These systems were interesting and explained some considerations for constructing new chaotic attractors. Hence, other researchers tried to introduce new chaotic systems, such as Qi, Yang, Liu, Wang and so on [23][24][25][26].…”