Chaotic Dynamics

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When we say “chaos”, we usually imagine a very complex scene with many different elements that move in different directions, collide with each other, and appear and disappear randomly. Thus, according to everyday intuition, the system’s complexity (e.g., many degrees of freedom) is an important attribute of chaos. It seems reasonable to think that in the opposite case, for example, a system with only a few degrees of freedom, the dynamical behavior must be simple and predictable. In fact, this point of view is Laplacian determinism.

The discovery of dynamical chaos has destroyed this traditional view. Dynamical chaos is a phenomenon that can be described by mathematical models for many natural systems, for example, physical, chemical, biological, and social, which evolve in time according to a deterministic rule and demonstrate capricious and seemingly unpredictable behavior.

To illustrate such behavior, consider a few examples.

Hyperion: Using Newton’s laws, one can compute relatively easily all future solar eclipses not only for the next few hundred years but also for thousands and millions of years into the future. This is indicative of a real predictability of the system’s dynamical behavior. But even in the solar system, there exists an object with unpredictable behavior: a small irregularly shaped moon of Saturn, Hyperion. Its orbit is regular and elliptic, but its altitude in the orbit is not. Hyperion is tumbling in a complex and irregular pattern while obeying the laws of gravitational dynamics. Hyperion may not be the only example of chaotic motion in the solar system. Recent studies indicate that chaotic behavior possibly exists in Jovian planets (Murray & Holman, 1999), resulting from the overlap of components of the mean motion resonance among Jupiter, Saturn, and Uranus. Chaos in Hamiltonian systems,.........

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