On the analytic structure of two-degree-of-freedom Hamiltonian systems around an elliptic fixed point

Nonlinearity ◽  
1996 ◽  
Vol 9 (4) ◽  
pp. 877-886 ◽  
Author(s):  
Tassos C Bountis ◽  
Vassilios M Rothos
Keyword(s):  
Fixed Point ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Analytic Structure ◽  
Two Degree Of Freedom

A NONINTEGRABILITY CRITERION FOR ADIABATIC SYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1829-1833
Author(s):  
DESPINA VOYATZI ◽  
EFI MELETLIDOU
Keyword(s):  
Fixed Point ◽  
Time Dependence ◽  
Hamiltonian Systems ◽  
Additional Assumption ◽  
High Order ◽  
Degree Of Freedom ◽  
Unstable Fixed Point

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on ε. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.

The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom

1998 ◽  
Vol 18 (4) ◽  
pp. 1007-1018 ◽  
Author(s):  
RAFAEL ORTEGA
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Time Dependent ◽  
Stable Fixed Point ◽  
Stable Periodic ◽  
Area Preserving ◽  
Fixed Period

Let $F:{\Bbb R}^2 \to {\Bbb R}^2$ be a mapping that is analytic and area preserving. If $F\neq \hbox{\it identity}$, then every stable fixed point is isolated.This result can be applied to prove that the number of stable periodic solutions of a fixed period of certain Hamiltonian systems is finite.

Complexity in spatio-temporal dynamics

1995 ◽  
Vol 125 (5) ◽  
pp. 959-974 ◽  
Author(s):  
D. L. Rod ◽  
B. D. Sleeman
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Temporal Dynamics ◽  
Degree Of Freedom ◽  
Kovacic Algorithm ◽  
Spatio Temporal ◽  
Two Degree Of Freedom

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

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