The Structure of Chaos
We describe the structure of chaotic regions in 2 and 3 degrees of freedom. The lobes formed by the asymptotic curves of unstable periodic orbits follow certain rules that allow predictions of their structure. When the energy becomes larger than the escape energy there are some “limiting asymptotic curves” defining the regions of escape. When there are no escapes the asymptotic curves become longer as the energy increases and produce an infinity of new tangencies, followed by the appearance of new stable periodic orbits.
In systems of 3 degrees of freedom the asymptotic curves of complex unstable orbits are spirals, described by a linear theory. In their vicinity there is an infinity of periodic orbits arranged along certain spirals. The asymptotic curves of doubly unstable orbits oscillate in a way different from that of unstable 2-D systems.
KeywordsBlack Hole Periodic Orbit Rotation Number Chaotic Region Unstable Periodic Orbit
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