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Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the conjugate Lorenz-type system. (English) Zbl 1261.34038

Summary: By using the Poincaré compactification in \(\mathbb R^{3}\), a global analysis of the conjugate Lorenz-type system is presented, including the complete description of its dynamic behavior on the sphere at infinity. Combining analytical and numerical techniques, it is shown that, for the parameter value \(b=0\), the system presents an infinite set of singularly degenerate heteroclinic cycles. The chaotic attractors for the system in the case of small \(b>0\) are found numerically, and thus the nearby singularly degenerate heteroclinic cycles. It is hoped that this global study can give a contribution in understanding of the conjugate Lorenz-type system, and will shed some light leading to final revelation of the true geometrical structure and the essence of chaos for the amazing original Lorenz attractor.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
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