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Complex dynamics of a simple 3D autonomous chaotic system with four-wing. (English) Zbl 1474.34287

Summary: The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Z. Wang et al., Nonlinear Dyn. 60, No. 3, 443–457 (2010; Zbl 1189.34100)] with the entitle “A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems” and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Citations:

Zbl 1189.34100
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References:

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