Holger R Dullin and Vu Ngoc San 2004 Nonlinearity 17 1777 doi:10.1088/0951-7715/17/5/012
Vanishing twist near focus–focus points
Holger R Dullin1,3 and Vu Ngoc San2
Show affiliationsRecommended by A I Neishtadt
We show that near a simple focus–focus value in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy–momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy–momentum map that is transversal to the line of constant energy. In contrast to this, we also show that the frequency map is non-degenerate for every point in a neighbourhood of a simple focus–focus point.
- PACS
- MSC
-
37E45 Rotation numbers and vectors
37J40 Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
37J35 Completely integrable systems, topological structure of phase space, integration methods
65F15 Eigenvalues, eigenvectors
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (See also 53D20)
- Subjects
- Dates
-
Issue 5 (September 2004)
Received 2 July 2003, in final form 5 March 2004
Published 12 July 2004
-
Vanishing twist near focus–focus points
Holger R Dullin and Vu Ngoc San 2004 Nonlinearity 17 1777
-
Effects of noise on elliptic bursters
Jianzhong Su et al 2004 Nonlinearity 17 133
-
Singular and regular solutions of a nonlinear parabolic system
Petr Plechác and Vladimír Sverák 2003 Nonlinearity 16 2083
-
On global solutions for the Constantin–Lax–Majda equation with a generalized viscosity term
Takashi Sakajo 2003 Nonlinearity 16 1319
-
Orbit bifurcations and wavefunction autocorrelations
A Bäcker et al 2002 Nonlinearity 15 1417
-
Symmetries and reversing symmetries of toral automorphisms
Michael Baake and John A G Roberts 2001 Nonlinearity 14 R1
-
About ergodicity in the family of limaçon billiards
Holger R Dullin and Arnd Bäcker 2001 Nonlinearity 14 1673
-
Generic twistless bifurcations
H R Dullin et al 2000 Nonlinearity 13 203
-
The baby Skyrme models and their multi-skyrmions
Tom Weidig 1999 Nonlinearity 12 1489
-
Computing connectedness: An exercise in computational topology
V Robins et al 1998 Nonlinearity 11 913
Related review articles
What's this?- The nonlinear Schrödinger equation with a random potential: results and puzzles
- Tibetan singing bowls
- Dimension of non-conformal repellers: a survey
, and Chern–Simons–Higgs solitons on 
: dimensional reduction of Chern–Pontryagin densities