Algebraic evaluation of linking numbers of unstable periodic orbits in chaotic attractors

1994 ◽  
Vol 49 (5) ◽  
pp. 4693-4695 ◽  
Author(s):  
L. Le Sceller ◽  
C. Letellier ◽  
G. Gouesbet
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers

A GEOMETRIC MODEL FOR THE DUFFING OSCILLATOR

1993 ◽  
Vol 03 (03) ◽  
pp. 685-691 ◽  
Author(s):  
J.W.L. McCALLUM ◽  
R. GILMORE
Keyword(s):  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Attractor ◽  
Geometric Model ◽  
Duffing Oscillator ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Full Period ◽  
Parameter Values

A geometric model for the Duffing oscillator is constructed by analyzing the unstable periodic orbits underlying the chaotic attractors present at particular parameter values. A template is constructed from observations of the motion of the chaotic attractor in a Poincaré section as the section is swept for one full period. The periodic orbits underlying the chaotic attractor are found and their linking numbers are computed. These are compared with the linking numbers from the template and the symbolic dynamics of the orbits are identified. This comparison is used to validate the template identification and label the orbits by their symbolic dynamics.

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

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.

DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

2012 ◽  
Vol 22 (01) ◽  
pp. 1230001
Author(s):  
BENJAMIN COY
Keyword(s):  
Learning Community ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Topological Analysis ◽  
Three Dimensions ◽  
Locally Linear Embedding ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Linear Embedding ◽  
Locally Linear

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

Systematic Computation of the Least Unstable Periodic Orbits in Chaotic Attractors

1998 ◽  
Vol 81 (20) ◽  
pp. 4349-4352 ◽  
Author(s):  
Fotis K. Diakonos ◽  
Peter Schmelcher ◽  
Ofer Biham
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits

Generation of Periodic Orbits from Chaotic Attractors of Weakly Coupled Diode Resonators

1997 ◽  
Vol 07 (04) ◽  
pp. 897-902
Author(s):  
Jong Cheol Shin ◽  
Sook-Il Kwun ◽  
Youngtae Kim
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Attractors ◽  
Unstable Periodic Orbits ◽  
Chaotic Motions ◽  
Small Perturbations ◽  
Controlling Chaos ◽  
Weakly Coupled ◽  
Dynamical Properties

We have designed coupled diode resonators to study the effect of small perturbations due to weak symmetric coupling on chaotic dynamics. Our experiment clearly demonstrated that chaos of the diode resonators was suppressed so that chaotic motions were converted into periodic ones with small modifications to the attractor when an appropriate coupling signal perturbed the diode resonators. Many unstable periodic orbits were stabilized and they were very stable depending on the dynamical properties of the coupling signals. Our results suggest that coupling of signals belonging to the same class is effective in controlling chaos.

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