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.
In this paper, we consider the complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response, which is the discrete version of the continuous system in (Nonlinear Anal., Real World Appl. 16:235-249, 2014). First, by giving several examples to display the limitations and errors of the local stability of the equilibrium point p obtained in (Nonlinear Anal., Real World Appl. 16:235-249, 2014), we provide an easily verified and complete discrimination criterion for the local stability of this equilibrium. Then we study some properties of its discrete version, especially for the stability and bifurcation for the equilibrium point E 1 , which has not been considered in any literature to the best of our knowledge. By using the center manifold theorem and bifurcation theory, we consider the flip bifurcation of this system at E 1 and obtain the stability of the closed orbits bifurcated from E 1 . The numerical simulations not only show the correctness of our theoretical analysis, but also we find some new and interesting dynamics of this system.
We devote to studying the problem for the existence of homoclinic and heteroclinic orbits of Unified Lorenz-Type System (ULTS). Other than the known results that the ULTS has two homoclinic orbits to E 0 = (0, 0, 0) for b = −2a 1 , d = −a 1 , a 2 1 + a 2 c > 0, e < 0 and two heteroclinic orbits to E 1,2 = (± − , a1d−a2c a2e ) while no homoclinic orbit when a 1 < 0, e < 0, a 1 + d < 0, a 2 = 0, a 2 c − a 1 d > 0, b + 2a 1 ≥ 0.
This note revisits an extended Lorenz system, which was presented in the paper entitled “Hopf bifurcations in an extended Lorenz system” by Zhou et al. . On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.
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