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A novel 3D system with a single heteroclinic trajectory modelled
Xianyi Li
Keywords:Three-dimensional chaotic system, singularly degenerate heteroclinic cycle, single heteroclinic trajectory, Hopf bifurcation, Lyapunov function
Abstract:
      The study for singular trajectories of 3D ( three-dimensional) chaotic systems is one of recent main interests. To the best of our knowledge, most of Lorenz-type or Lorenz-like systems are always chaotic. And also due to the symmetry of their structure, a pair of symmetric heteroclinic trajectories are always found. The dynamics of their non-isolate equilibria $(0, 0, z)$ is always related to the parameter $z$. Whether or not does there exist a 3D system with a single heteroclinic trajectory such that the dynamics of its non-isolate equilibria $(0, 0, z)$ has nothing to do with the parameter $z$ ? In the present note, based on a known Lorenz-type system, we model a 3D nonlinear system with such natures. To show its unique characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complicate dynamics, mainly for the distribution of its equilibrium point, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, and its homoclinic and heteroclinic trajectories, etc. The main results in this work are to not only rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters, but also show that the properties of its non-isolate equilibria have nothing to do with their coordinate positions. These interesting phenomena have not been previously reported in the Lorenz system family. What's more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, this new found system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the chaotic dynamics.