scholarly journals Nonexistence of invariant tori transverse to foliations: An application of converse KAM theory

2021 ◽  
Vol 31 (1) ◽  
pp. 013124
Author(s):  
Nathan Duignan ◽  
James D. Meiss
Keyword(s):  
Invariant Tori ◽  
Kam Theory

scholarly journals Partial Preservation of Frequencies and Floquet Exponents of Invariant Tori in the Reversible KAM Context 2

2017 ◽  
Vol 63 (3) ◽  
pp. 516-541
Author(s):  
M B Sevryuk
Keyword(s):  
Fixed Point ◽  
Invariant Torus ◽  
Variational Equation ◽  
Invariant Tori ◽  
Kam Theory ◽  
Coefficient Matrix ◽  
Nondegeneracy Conditions

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman’s method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

scholarly journals THE QUASI-PERIODIC REVERSIBLE HOPF BIFURCATION

2007 ◽  
Vol 17 (08) ◽  
pp. 2605-2623 ◽  
Author(s):  
HENK W. BROER ◽  
M. CRISTINA CIOCCI ◽  
HEINZ HANßMANN
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Point ◽  
Invariant Tori ◽  
Kam Theory ◽  
Cantor Sets ◽  
Reversible Systems ◽  
Periodic Dynamics

We consider the perturbed quasi-periodic dynamics of a family of reversible systems with normally 1:1 resonant invariant tori. We focus on the generic quasi-periodic reversible Hopf bifurcation and address the persistence problem for integrable quasi-periodic tori near the bifurcation point. Using KAM theory, we describe how the resulting invariant tori of maximal and lower dimensions are parameterized by Cantor sets.

scholarly journals Analytic Lagrangian tori for the planetary many-body problem

2009 ◽  
Vol 29 (3) ◽  
pp. 849-873 ◽  
Author(s):  
LUIGI CHIERCHIA ◽  
FABIO PUSATERI
Keyword(s):  
Body Problem ◽  
Invariant Tori ◽  
Kam Theory ◽  
Many Body ◽  
Complete Proof ◽  
Small Denominators ◽  
C Dynamics ◽  
Lagrangian Tori ◽  
Real Analytic ◽  
Nearly Integrable Hamiltonian Systems

AbstractIn 2004, Féjoz [Démonstration du ‘théoréme d’Arnold’ sur la stabilité du système planétaire (d’après M. Herman). Ergod. Th. & Dynam. Sys.24(5) (2004), 1521–1582], completing investigations of Herman’s [Démonstration d’un théoréme de V.I. Arnold. Séminaire de Systémes Dynamiques et manuscripts, 1998], gave a complete proof of ‘Arnold’s Theorem’ [V. I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspekhi Mat. Nauk. 18(6(114)) (1963), 91–192] on the planetary many-body problem, establishing, in particular, the existence of a positive measure set of smooth (C∞) Lagrangian invariant tori for the planetary many-body problem. Here, using Rüßmann’s 2001 KAM theory [H. Rüßmann. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. R. & C. Dynamics2(6) (2001), 119–203], we prove the above result in the real-analytic class.

scholarly journals Complex Dynamics and Statistics of 1-D Hamiltonian Lattices: Long Range Interactions and Supratransmission

2020 ◽  
Vol 23 (2) ◽  
pp. 133-148
Author(s):  
Anastasios Bountis
Keyword(s):  
Degrees Of Freedom ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Kam Theory ◽  
Small Perturbations ◽  
Physical Systems ◽  
Long Range Interactions ◽  
The 1960S ◽  
Ordered Motion

In this paper, I review a number of results that my co-workers and I have obtained in the field of 1-Dimensional (1D) Hamiltonian lattices. This field has grown in recent years, due to its importance in revealing many phenomena that concern the occurrence of chaotic behavior in conservative physical systems with a high number of degrees of freedom. After the establishment of the Kolomogorov-Arnol'd-Moser (KAM) theory in the 1960s, a wealth of results were obtained about such systems as small perturbations of completely integrable Ndegree- of-freedom Hamiltonians, where ordered motion is dominant in the form of invariant tori. Since the 1980s, however, and particularly in the last two decades, there has been great progress in understanding the properties of Hamiltonian 1D lattices far from the KAM regime, where "weak" and "strong" forms of chaos begin to play an increasingly significant role. It is the purpose of this review to address and highlight some of these advances, in which the author has made several contributions concerning the dynamics and statistics of these lattices.

Introduction

2020 ◽  
pp. 3-16
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Energy Surface ◽  
Initial Conditions ◽  
Invariant Tori ◽  
Kam Theory ◽  
Ergodic Hypothesis ◽  
Open Set ◽  
Nearly Integrable Systems

This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

scholarly journals TWISTLESS KAM TORI, QUASI FLAT HOMOCLINIC INTERSECTIONS, AND OTHER CANCELLATIONS IN THE PERTURBATION SERIES OF CERTAIN COMPLETELY INTEGRABLE HAMILTONIAN SYSTEMS: A REVIEW

1994 ◽  
Vol 06 (03) ◽  
pp. 343-411 ◽  
Author(s):  
GIOVANNI GALLAVOTTI
Keyword(s):  
Perturbation Theory ◽  
Invariant Tori ◽  
Kam Theory ◽  
Perturbation Expansion ◽  
Rotation Velocity ◽  
Perturbation Series ◽  
Kam Tori ◽  
Stable And Unstable Manifolds ◽  
Rotation Numbers ◽  
The Stability

Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the invariant tori of the rotators system at fixed rotation numbers: we exhibit cancellations, to all orders of perturbation theory, that allow proving the stability and analyticity of the diophantine tori. We find in this way a proof of the KAM theorem by direct bounds of the k-th order coefficient of the perturbation expansion of the parametric equations of the tori in terms of their average anomalies: this extends Siegel's approach, from the linearization of analytic maps to the KAM theory; the convergence radius does not depend, in this case, on the twist strength, which could even vanish ("twistless KAM tori"). The same ideas apply to the case in which the potential couples the pendulum and the rotators: in this case the invariant tori with diophantine rotation numbers are unstable and have stable and unstable manifolds ("whiskers"): instead of studying the perturbation theory of the invariant tori we look for the cancellations that must be present because the homoclinic intersections of the whiskers are "quasi flat", if the rotation velocity of the quasi periodic motion on the tori is large. We rederive in this way the result that, under suitable conditions, the homoclinic splitting is smaller than any power in the period of the forcing and find the exact asymptotics in the two dimensional cases (e.g. in the case of a periodically forced pendulum). The technique can be applied to study other quantities: we mention, as another example, the homoclinic scattering phase shifts.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

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