Density of unstable periodic points in K guarantees an ab... — Carmelics

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Density of unstable periodic points in K guarantees an abundance of aperiodic orbits characteristic of chaos

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Reasons For

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Reason for

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1.Unstable periodic points are points where trajectories from neighboring points exhibit sensitive dependence on initial conditions (WSD)
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2.If the set of unstable periodic points is dense in K, aperiodic orbits of the kind characteristic of chaos will be abundant
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Reasons Against

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Reason against 1 of 2

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1.Density of unstable periodic points is a necessary but not sufficient condition for the abundance of aperiodic orbits characteristic of chaos.
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2.Topological transitivity must also hold for the density condition to guarantee the mixing behavior required for genuine chaotic aperiodicity.
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3.Without topological transitivity, a system can possess dense unstable periodic points yet partition into invariant subsystems with bounded, non-chaotic dynamics.
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