Characterization of the Natural Measure by Unstable Periodic Orbits in Chaotic Attractors

1997 ◽  
Vol 79 (4) ◽  
pp. 649-652 ◽  
Author(s):  
Ying-Cheng Lai ◽  
Yoshihiko Nagai ◽  
Celso Grebogi
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Natural Measure

Characterization of blowout bifurcation by unstable periodic orbits

1997 ◽  
Vol 55 (2) ◽  
pp. R1251-R1254 ◽  
Author(s):  
Yoshihiko Nagai ◽  
Ying-Cheng Lai
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits

Unstable periodic orbits and the dimension of chaotic attractors

1987 ◽  
Vol 36 (7) ◽  
pp. 3522-3524 ◽  
Author(s):  
Celso Grebogi ◽  
Edward Ott ◽  
James A. Yorke
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits

Characterization of the Lorentz attractor by unstable periodic orbits

Nonlinearity ◽  
1993 ◽  
Vol 6 (2) ◽  
pp. 251-258 ◽  
Author(s):  
V Franceschini ◽  
C Giberti ◽  
Zhiming Zheng
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits

Unstable periodic orbits and the dimensions of multifractal chaotic attractors

1988 ◽  
Vol 37 (5) ◽  
pp. 1711-1724 ◽  
Author(s):  
Celso Grebogi ◽  
Edward Ott ◽  
James A. Yorke
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits

CHARACTERISATION OF CHAOS IN CHUA'S OSCILLATOR IN TERMS OF UNSTABLE PERIODIC ORBITS

1993 ◽  
Vol 03 (02) ◽  
pp. 411-429 ◽  
Author(s):  
MACIEJ J. OGORZAŁEK ◽  
ZBIGNIEW GALIAS
Keyword(s):  
Periodic Orbits ◽  
Picture Book ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Chua’S Oscillator ◽  
Canonical Chua’S Circuit

We present a picture book of unstable periodic orbits embedded in typical chaotic attractors found in the canonical Chua's circuit. These include spiral Chua's, double-scroll Chua's and double hook attractors. The "skeleton" of unstable periodic orbits is specific for the considered attractor and provides an invariant characterisation of its geometry.

Topological characterization of deterministic chaos: enforcing orientation preservation

2007 ◽  
Vol 366 (1865) ◽  
pp. 559-567 ◽  
Author(s):  
Marc Lefranc ◽  
Pierre-Emmanuel Morant ◽  
Michel Nizette
Keyword(s):  
Periodic Orbits ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Three Dimensions ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Triangulated Surfaces ◽  
Theoretic Characterization

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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