Abstract
The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.
Highlights
Over the past decades, chaos has received extensive attention from scientific communities such as physics, biology, chemistry, social sciences, and engineering
Since a planar autonomous polynomial system cannot generate chaos, three dimensions are needed for an autonomous system to generate chaos, and the simplest possible form of chaotic systems is 3D autonomous quadratic dynamical systems
We extend the classical Si’lnikov homoclinic theorem to another critical case where a 3D autonomous polynomial system with one equilibrium at which the eigenvalues of the Jacobian are given by λ and ±iω
Summary
Chaos has received extensive attention from scientific communities such as physics, biology, chemistry, social sciences, and engineering. For high-dimensional continuous dynamical systems, there is the famous Si’lnikov theorem [3,4,5,6] Despite these endeavors, there have not yet been systematic results on how to reasonably classify chaos in autonomous systems, and the related studies are few. Even for a simple class of 3D autonomous quadratic dynamical systems, finding out all simplest possible forms of chaotic systems is especially important with significant impacts on both basic research and engineering applications. We know that the linear parts of a nonlinear system can only influence the local dynamical behavior and chaos usually is determined by the nonlinear parts These two algebraic criteria can classify a large set of chaotic systems, they could not reveal the geometric structure and the formation mechanism of chaos well [12]. (4) chaos of other types except for the previous three types
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