Homoclinic tangle in tokamak divertors

2014 ◽  
Vol 378 (32-33) ◽  
pp. 2410-2416 ◽  
Author(s):  
Alkesh Punjabi ◽  
Allen Boozer
Keyword(s):  
Homoclinic Tangle

Scaling invariance of the homoclinic tangle

2002 ◽  
Vol 66 (4) ◽  
Author(s):  
L. Kuznetsov ◽  
G. M. Zaslavsky
Keyword(s):  
Scaling Invariance ◽  
Homoclinic Tangle

scholarly journals Homoclinic Tangle on the Edge of Shear Turbulence

2011 ◽  
Vol 107 (11) ◽  
Author(s):  
Lennaert van Veen ◽  
Genta Kawahara
Keyword(s):  
Homoclinic Tangle ◽  
Shear Turbulence

Chaotic phenomena triggering the escape from a potential well

1989 ◽  
Vol 421 (1861) ◽  
pp. 195-225 ◽  
Keyword(s):  
Harmonic Balance ◽  
Scaling Laws ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Potential Well ◽  
Homoclinic Tangle ◽  
Balance Analysis ◽  
Melnikov Analysis ◽  
Chaotic Transients ◽  
The Basin Of Attraction

This paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold. The aim is to show how the well-known atoms of dissipative dynamics (saddle-node folds, period-doubling flips, cascades to chaos, boundary crises, etc.) assemble to form molecules of overall response (hierarchies of cusps, incomplete Feigenbaum trees, etc.). Particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle, the latter being accurately predicted by a Melnikov analysis. After escape, chaotic transients are shown to conform to recent scaling laws. Analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible.

Recurrence time in the homoclinic tangle

1995 ◽  
Vol 63 (2) ◽  
pp. 189-197 ◽  
Author(s):  
G. Contopoulos ◽  
C. Polymilis
Keyword(s):  
Recurrence Time ◽  
Homoclinic Tangle

SYMBOLIC DYNAMICS FROM HOMOCLINIC TANGLES

2002 ◽  
Vol 12 (03) ◽  
pp. 605-617 ◽  
Author(s):  
PIETER COLLINS
Keyword(s):  
Lower Bound ◽  
Symbolic Dynamics ◽  
Numerical Techniques ◽  
Symbol Space ◽  
Stable And Unstable Manifolds ◽  
Subshift Of Finite Type ◽  
Homoclinic Tangle ◽  
Shift Map ◽  
Homoclinic Tangles ◽  
Unstable Manifolds

We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numerical techniques. The symbol space is naturally given by components of the complement of the stable and unstable manifolds. The shift map defining the dynamics is a factor of a subshift of finite type, and is obtained from a graph related to the tangle. The entropy of this shift map is a lower bound for the topological entropy of the planar diffeomorphism. We give examples arising from the Hénon family.

scholarly journals PROBING THE LOCAL DYNAMICS OF PERIODIC ORBITS BY THE GENERALIZED ALIGNMENT INDEX (GALI) METHOD

2012 ◽  
Vol 22 (09) ◽  
pp. 1250218 ◽  
Author(s):  
T. MANOS ◽  
CH. SKOKOS ◽  
CH. ANTONOPOULOS
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Power Laws ◽  
Oscillatory Behavior ◽  
Unstable Periodic Orbits ◽  
Symplectic Maps ◽  
Local Dynamics ◽  
Hamiltonian Flows ◽  
Homoclinic Tangle ◽  
Stable Periodic

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper, we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits, the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around nonzero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using the components of deviation vectors orthogonal to the direction of motion for the computation of GALIs, the indices of stable periodic orbits behave for flows as they do for maps.

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