scholarly journals A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

2020 ◽  
Vol 40 (12) ◽  
pp. 6795-6813
Author(s):  
Marian Gidea ◽  
◽  
Rafael de la Llave ◽  
Tere M. Seara ◽  
◽  
...  
Keyword(s):  
Hamiltonian Systems ◽  
Invariant Manifold ◽  
General Mechanism ◽  
Normally Hyperbolic Invariant Manifold

scholarly journals Diffusion along transition chains of invariant tori and Aubry–Mather sets

2012 ◽  
Vol 33 (5) ◽  
pp. 1401-1449 ◽  
Author(s):  
MARIAN GIDEA ◽  
CLARK ROBINSON
Keyword(s):  
Hamiltonian Systems ◽  
Invariant Tori ◽  
Diffusion Problem ◽  
Finite Collection ◽  
Arnold Diffusion ◽  
Twist Map ◽  
One Dimensional ◽  
Transverse Intersection ◽  
Normally Hyperbolic Invariant Manifold ◽  
Area Preserving

AbstractWe describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry–Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry–Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.

The Generalization of the Periodic Orbit Dividing Surface for Hamiltonian Systems with Three or More Degrees of Freedom – II

2021 ◽  
Vol 31 (12) ◽  
pp. 2150188
Author(s):  
Matthaios Katsanikas ◽  
Stephen Wiggins
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbit ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Dimensional Space ◽  
New Method ◽  
Normally Hyperbolic Invariant Manifold ◽  
Dividing Surface ◽  
Dimensional Object ◽  
Three Degrees Of Freedom

We develop a method for the construction of a dividing surface using periodic orbits in Hamiltonian systems with three or more degrees-of-freedom that is an alternative to the method presented in [ Katsanikas & Wiggins, 2021 ]. Similar to that method, for an [Formula: see text] degrees-of-freedom Hamiltonian system, we extend a one-dimensional object (the periodic orbit) to a [Formula: see text] dimensional geometrical object in the energy surface of a [Formula: see text] dimensional space that has the desired properties for a dividing surface. The advantage of this new method is that it avoids the computation of the normally hyperbolic invariant manifold (NHIM) (as the first method did) and it is easier to numerically implement than the first method of constructing periodic orbit dividing surfaces. Moreover, this method has less strict required conditions than the first method for constructing periodic orbit dividing surfaces. We apply the new method to a benchmark example of a Hamiltonian system with three degrees-of-freedom for which we are able to investigate the structure of the dividing surface in detail. We also compare the periodic orbit dividing surfaces constructed in this way with the dividing surfaces that are constructed starting with a NHIM. We show that these periodic orbit dividing surfaces are subsets of the dividing surfaces that are constructed from the NHIM.

scholarly journals A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

2019 ◽  
Vol 73 (1) ◽  
pp. 150-209
Author(s):  
Marian Gidea ◽  
Rafael Llave ◽  
Tere M‐Seara
Keyword(s):  
Hamiltonian Systems ◽  
General Mechanism
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