Ši’lnikov chaos in the generalized Lorenz canonical form of dynamical systems. (English) Zbl 1142.70012
Summary: This paper studies the generalized Lorenz canonical form of dynamical systems introduced by S. Čelikovský and G. Chen [Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, No. 8, 1789–1812 (2002; Zbl 1043.37023)]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The Ši’lnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have Ši’lnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of Ši’lnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.
MSC:
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
| 70K44 | Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics |
| 37N05 | Dynamical systems in classical and celestial mechanics |
Citations:
Zbl 1043.37023References:
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