A new topological technique for characterizing homoclinic tangles

Kevin A. Mitchell; John B. Delos

doi:10.1016/j.physd.2006.07.027

A new topological technique for characterizing homoclinic tangles

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2006.07.027 ◽

2006 ◽

Vol 221 (2) ◽

pp. 170-187 ◽

Cited By ~ 30

Author(s):

Kevin A. Mitchell ◽

John B. Delos

Keyword(s):

Homoclinic Tangles

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Geometrical and dynamical properties of homoclinic tangles in a simple Hamiltonian system

Physical Review E ◽

10.1103/physreve.47.1546 ◽

1993 ◽

Vol 47 (3) ◽

pp. 1546-1557 ◽

Cited By ~ 13

Author(s):

G. Contopoulos ◽

C. Polymilis

Keyword(s):

Hamiltonian System ◽

Homoclinic Tangles ◽

Dynamical Properties

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Homoclinic tangles and transport in two-dimensional, time-quasiperiodic vector fields

Global Dynamics, Phase Space Transport, Orbits Homoclinic to Resonances, and Applications - Fields Institute Monographs ◽

10.1090/fim/001/04 ◽

1993 ◽

pp. 29-46

Keyword(s):

Vector Fields ◽

Two Dimensional ◽

Homoclinic Tangles

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Markov shifts and topological entropy of families of homoclinic tangles

Journal of Differential Equations ◽

10.1016/j.jde.2019.01.002 ◽

2019 ◽

Vol 266 (12) ◽

pp. 8492-8518

Author(s):

Bráulio Garcia ◽

Valentín Mendoza

Keyword(s):

Topological Entropy ◽

Markov Shifts ◽

Homoclinic Tangles

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CHAOS AND COMPLEXITY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401001992 ◽

2001 ◽

Vol 11 (01) ◽

pp. 19-26 ◽

Cited By ~ 11

Author(s):

RAY BROWN ◽

ROBERT BEREZDIVIN ◽

LEON O. CHUA

Keyword(s):

Dynamical Systems ◽

Lyapunov Exponents ◽

Chaotic Dynamics ◽

High Dimensional ◽

Topological Conjugacy ◽

Integer Lattices ◽

Homoclinic Tangles ◽

Dimensional Complexity ◽

Finite Integer ◽

The Relationship

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

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Forced oscillations of a self-oscillating bimolecular surface reaction model

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0066 ◽

1988 ◽

Vol 417 (1853) ◽

pp. 363-388 ◽

Cited By ~ 11

Keyword(s):

Surface Reaction ◽

Period Doubling ◽

Reaction Model ◽

Forced Oscillations ◽

Hopf Bifurcations ◽

Codimension Two ◽

Homoclinic Tangles ◽

Two Parameter ◽

Parameter Bifurcation ◽

Surface Reaction Model

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

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