Splitting of Asymptotic Manifolds

Springer Monographs in Mathematics - Introduction to the Perturbation Theory of Hamiltonian Systems ◽

10.1007/978-3-642-03028-4_3 ◽

2009 ◽

pp. 59-74

Author(s):

Dmitry Treschev ◽

Oleg Zubelevich

Keyword(s):

Asymptotic Manifolds

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Self-intersection of complex asymptotic manifolds of hyperbolic equilibrium, position and non-existence of first integrals in the systems of the type of Hamiltonian with two degrees of freedom.

Journal of Radio Electronics ◽

10.30898/1684-1719.2019.5.5 ◽

2019 ◽

Vol 2019 (5) ◽

Author(s):

S.L. Ziglin ◽

Keyword(s):

Equilibrium Position ◽

Degrees Of Freedom ◽

First Integrals ◽

Two Degrees Of Freedom ◽

Hyperbolic Equilibrium ◽

Asymptotic Manifolds

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DIFFUSION OF CHAOTIC ORBITS IN BARRED SPIRAL GALAXIES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411029732 ◽

2011 ◽

Vol 21 (08) ◽

pp. 2221-2233 ◽

Cited By ~ 4

Author(s):

M. HARSOULA ◽

C. KALAPOTHARAKOS ◽

G. CONTOPOULOS

Keyword(s):

Spiral Structure ◽

Initial Conditions ◽

Unstable Periodic Orbits ◽

Chaotic Orbits ◽

The Galaxy ◽

Barred Spiral Galaxy ◽

Age Of The Universe ◽

Barred Spiral Galaxies ◽

The Universe ◽

Asymptotic Manifolds

We study the diffusion of chaotic orbits in an N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions outside corotation support the spiral structure of the galaxy due to the phenomenon of stickiness close and along the unstable asymptotic manifolds of the unstable periodic orbits. These orbits are diffused outwards after about 13 rotations of the bar. During this time, the spiral structure is clearly visible and then it fades out gradually. The diffusion time for the majority of the chaotic orbits with initial conditions inside corotation is much longer than the age of the Universe. These orbits support mainly the outer parts of the bar. However, a part of the chaotic orbits inside corotation are diffused outwards fast and support the spiral structure.

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DEGENERATE BIFURCATIONS OF RESONANT TORI IN HAMILTONIAN SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499000602 ◽

1999 ◽

Vol 09 (05) ◽

pp. 849-863 ◽

Cited By ~ 16

Author(s):

GEORGE VOYATZIS ◽

SIMOS ICHTIAROGLOU

Keyword(s):

Hamiltonian Systems ◽

Integrable Model ◽

Necessary Condition ◽

Symplectic Maps ◽

Integrable Hamiltonian Systems ◽

Small Perturbations ◽

Model Hamiltonian ◽

The Stability ◽

Asymptotic Manifolds ◽

Twist Condition

The twist condition is a necessary condition in integrable Hamiltonian systems and symplectic maps to obtain Poincaré–Birkhoff bifurcations under small perturbations. When this condition does not hold, topological structures other than Poincaré–Birkhoff chains arise in phase space through bifurcations of isolated periodic orbits and reconnections of asymptotic manifolds. In this paper we construct an integrable model Hamiltonian with degeneracies suitable to observe these phenomena close to nontwist resonant tori. The generation of isochronous chains and the stability of their fixed points is determined analytically and a condition for the reconnection is found. Particular examples are given, illustrating the bifurcation and reconnection scenario for several cases of degeneracy.

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