Simple polynomial classes of chaotic jerky dynamics. (English) Zbl 0993.37019
Summary: Third-order explicit autonomous differential equations, commonly called jerky dynamics, constitute a powerful approach to understand the properties of functionally very simple but nonlinear three-dimensional dynamical systems that can exhibit chaotic long-time behavior. In this paper, we investigate the dynamics that can be generated by the two simplest polynomial jerky dynamics that, up to these days, are known to show chaotic behavior in some parameter range. After deriving several analytical properties of these systems, we systematically determine the dependence of the long-time dynamical behavior on the system parameters by numerical evaluation of Lyapunov spectra. Some features of the systems that are related to the dependence on initial conditions are also addressed. The observed dynamical complexity of the two systems is discussed in connection with the existence of homoclinic orbits.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
Keywords:
autonomous differential equations; chaotic long-time behavior; polynomial jerky dynamics; numerical evaluation of Lyapunov spectra; dynamical complexity; existence of homoclinic orbitsReferences:
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