Normal Forms of Hamiltonian Systems and Stability of Equilibria

2020 ◽  
pp. 117-170
Author(s):  
Hildeberto Eulalio Cabral
Keyword(s):  
Hamiltonian Systems ◽  
Normal Forms ◽  
Stability Of Equilibria

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.

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