Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case

1990 ◽  
Vol 65 (1) ◽  
pp. 4-35 ◽  
Author(s):  
L. H. Eliasson
Keyword(s):  
Hamiltonian Systems ◽  
Normal Forms ◽  
Elliptic Case

scholarly journals Generic super-exponential stability of invariant tori in Hamiltonian systems

2010 ◽  
Vol 31 (5) ◽  
pp. 1287-1303 ◽  
Author(s):  
ABED BOUNEMOURA
Keyword(s):  
Hamiltonian System ◽  
Exponential Stability ◽  
Fixed Points ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Invariant Tori ◽  
The Other ◽  
New Method ◽  
Stable Invariant

AbstractIn this article, we consider solutions that start close to some linearly stable invariant tori in an analytic Hamiltonian system, and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms with a new method for obtaining generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will focus mainly on the neighbourhood of elliptic fixed points, since with our approach the other cases can be treated in a very similar way.

scholarly journals Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

2018 ◽  
Vol 28 (13) ◽  
pp. 1850168
Author(s):  
Ting Chen ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Linear Change ◽  
Rational Potential ◽  
Change Of Variables ◽  
Poincaré Disk

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

Interaction of Lower and Higher Order Hamiltonian Resonances

2018 ◽  
Vol 28 (08) ◽  
pp. 1850097 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Nonlinear Stability ◽  
Normal Forms ◽  
Double Resonance ◽  
Higher Order ◽  
Order Resonance ◽  
Resonance Zone ◽  
Stability Results ◽  
Low Order

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof [Formula: see text] resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

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