Homoclinic Orbits in the Complex Domain

1997 ◽  
Vol 07 (02) ◽  
pp. 253-274 ◽  
Author(s):  
V. F. Lazutkin ◽  
C. Simó
Keyword(s):  
Homoclinic Orbits ◽  
Homoclinic Tangency ◽  
Complex Domain ◽  
Standard Map ◽  
Homoclinic Points ◽  
Promising Tool ◽  
Theoretical Support ◽  
Area Preserving Maps ◽  
Area Preserving

We consider the standard map, as a paradigm of area preserving map, when the variables are taken as complex. We study how to detect the complex homoclinic points, which cannot dissappear under a homoclinic tangency. This seems a promising tool to understand the stochastic zones of area preserving maps. The paper is mainly phenomenological and includes theoretical support to the observed phenomena. Several conjectures are stated.

Hamiltonian Chaos

2019 ◽  
pp. 154-176
Author(s):  
David D. Nolte
Keyword(s):  
Quantum Chaos ◽  
Kam Theory ◽  
Rigid Rotator ◽  
Standard Map ◽  
Web Map ◽  
Classical Chaos ◽  
Hamiltonian Chaos ◽  
Area Preserving Maps ◽  
Area Preserving ◽  
The Web

Nondissipative or Hamiltonian systems are also capable of chaos as phase space volume is twisted and folded in area-preserving maps like the Standard Map. When nonintegrable terms are added to a potential function, Hamiltonian chaos emerges. The Standard Map (also known as the Chirikov map) for a periodically kicked rigid rotator provides a simple model with which to explore the emergence of Hamiltonian chaos as well as the KAM theory of islands of stability. A periodically kicked harmonic oscillator displays extended chaos in the web map. Hamiltonian classical chaos makes a direct connection to quantum chaos, which is illustrated using the chaotic stadium, for which quantum scars are associated with periodic classical orbits in the stadium.

scholarly journals Extensive Numerical Results for Integrable Case of Standard Map

2020 ◽  
Vol 23 (2) ◽  
pp. 149-152
Author(s):  
Ugur Tirnakli ◽  
Constantino Tsallis
Keyword(s):  
Mechanical Characterization ◽  
Initial Conditions ◽  
Random Variable ◽  
Integrable Case ◽  
Standard Map ◽  
Region Of Stability ◽  
Area Preserving Maps ◽  
Statistical Mechanical ◽  
Area Preserving ◽  
Special Case

In recent years, conservative dynamical systems have become a vivid area of research from the statistical mechanical characterization viewpoint. With this respect, several areapreserving maps have been studied. It has been numerically shown that the probability distribution of the sum of the suitable random variable of these systems can be well approximated by a Gaussian (q-Gaussian) when the initial conditions are randomly selected from the chaotic sea (region of stability islands) in the available phase space. In this study, we will summarize these results and discuss a special case for the standard map, a paradigmatic example of area-preserving maps, for which the map is totally integrable.

Hamiltonian suspension of perturbed Poincaré sections and an application

2014 ◽  
Vol 157 (1) ◽  
pp. 101-112 ◽  
Author(s):  
MÁRIO BESSA ◽  
JOÁO LOPES DIAS
Keyword(s):  
Poincaré Map ◽  
Energy Surface ◽  
Homoclinic Tangency ◽  
Poincare Map ◽  
Poincaré Sections ◽  
Hamiltonian Flows ◽  
Area Preserving Maps ◽  
Poincare Sections ◽  
Area Preserving

AbstractWe construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltonian defined on a symplectic 4-manifold M and any point p ∈ M, there exists a C2-close Hamiltonian whose regular energy surface through p is either Anosov or contains a homoclinic tangency.

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