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.

scholarly journals Phase space structure and fractal trajectories in 1½ degree of freedom Hamiltonian systems whose time dependence is quasiperiodic

1998 ◽  
Vol 5 (2) ◽  
pp. 69-74 ◽  
Author(s):  
M. G. Brown
Keyword(s):  
Phase Space ◽  
Time Dependence ◽  
Hamiltonian Systems ◽  
Fluid Flows ◽  
Space Structure ◽  
Degree Of Freedom ◽  
Periodic Functions ◽  
Phase Space Structure ◽  
Fractal Properties ◽  
Incompressible Fluid Flows

Abstract. We consider particle motion in nonautonomous 1 degree of freedom Hamiltonian systems for which H(p,q,t) depends on N periodic functions of t with incommensurable frequencies. It is shown that in near-integrable systems of this type, phase space is partitioned into nonintersecting regular and chaotic regions. In this respect there is no different between the N = 1 (periodic time dependence) and the N = 2, 3, ... (quasi-periodic time dependence) problems. An important consequence of this phase space structure is that the mechanism that leads to fractal properties of chaotic trajectories in systems with N = 1 also applies to the larger class of problems treated here. Implications of the results presented to studies of ray dynamics in two-dimensional incompressible fluid flows are discussed.

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.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals Jachymski-Matkowski-Świątkowski's fixed point theorem

2019 ◽  
Vol 13 (2) ◽  
pp. 632-642
Author(s):  
Tomonari Suzuki
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Additional Assumption ◽  
Semimetric Spaces

We improve Jachymski-Matkowski-?wi?tkowski's fixed point theorem for contractions in semimetric spaces with some additional assumption. We prove another fixed point theorem for contractions.

scholarly journals Arbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systems

2019 ◽  
Vol 23 (3) ◽  
pp. 703-725 ◽  
Author(s):  
Xiuyan Li ◽  
Chiping Zhang ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Hamiltonian Systems ◽  
High Order

scholarly journals Results in wt-Distance over b-Metric Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 220 ◽  
Author(s):  
Erdal Karapınar ◽  
Cristian Chifu
Keyword(s):  
Fixed Point ◽  
Additional Assumption ◽  
Metric Spaces

In this manuscript, we introduce Meir-Keeler type contractions and Geraghty type contractions in the setting of the w t -distances over b-metric spaces. We examine the existence of a fixed point for such mappings. Under some additional assumption, we proved the uniqueness of the found fixed point.

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