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An unusual 3D autonomous quadratic chaotic system with two stable node-foci. (English) Zbl 1193.34091

Summary: This paper reports the finding of an unusual three-dimensional autonomous quadratic Lorenz-like chaotic system which, surprisingly, has two stable node-type of foci as its only equilibria. The new system contains the diffusionless Lorenz system and the Burke-Shaw system, and some others, as special cases. The algebraic form of the new chaotic system is similar to the other Lorenz-type systems, but they are topologically nonequivalent. To further analyze the new system, some dynamical behaviors such as Hopf bifurcation and singularly degenerate heteroclinic and homoclinic orbits, are rigorously proved with simulation verification. Moreover, it is proved that the new system with some specified parameter values has Silnikov-type homoclinic and heteroclinic chaos.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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