Topological Characterization of Deterministic Chaos: Enforcing Orientation Preservation
The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-t...
| Veröffentlicht in: | Philosophical Transactions: Mathematical, Physical and Engineering Sciences. - The Royal Society. - 366(2008), 1865, Seite 559-567 |
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| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
2008
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| Zugriff auf das übergeordnete Werk: | Philosophical Transactions: Mathematical, Physical and Engineering Sciences |
| Schlagworte: | Chaos Unstable periodic orbits Knot theory Topology Entropy Applied sciences Philosophy Physical sciences Mathematics Behavioral sciences |
| Zusammenfassung: | The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map. |
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| ISSN: | 1364503X |