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On singular orbits and global exponential attractive set of a Lorenz-type system. (English) Zbl 1423.34024

Summary: This paper deals with some unsolved problems of the global dynamics of a three-dimensional (3D) Lorenz-type system: \(\dot{x}=a(y-x)\), \(\dot{y}=cx-xz\), \(\dot{z}=-bz+xy+ex^2\) by constructing a series of Lyapunov functions. The main contribution of the present work is that one not only proves the existence of singularly degenerate heteroclinic cycles, existence and nonexistence of homoclinic orbits for a certain range of the parameters according to some known results and LaSalle theorem but also gives a family of mathematical expressions of global exponential attractive sets for that system with respect to its parameters, which is available only in very few papers as far as one knows. 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.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
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