Four-Dimensional Integrable Hamiltonian
                    Systems with Simple Singular Points (Topological
                    Aspects)

Lerman, L. M.; Umanskiy, Yan L.

doi:10.1090/mmono/176

Cited by 22 publications

(63 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Hence, as shown in [33,29] and more recently in [25], for small n−s, singularity theory may be used to classify all generic bifurcations (or all generic bifurcations under given symmetries [25]) in the s DOF subsystem. Here we consider, in addition to the above bifurcations, extrema in the action variables of the Hamiltonian function evaluated along the singularity surfaces.…”

mentioning

confidence: 98%

“…If these iso-energy level sets have different numbers of components, then there exists a singular level set on this energy surface which is not smoothly conjugate to a collection of n-tori. Under quite general conditions, Lerman and Umanskii [33] show that by using the reduction procedure [1] and Nehorošev results [42], such a connected singular level set may be expressed locally as an n − s dimensional torus with frequencies depending on n − s actions crossed with a fixed point and its asymptotic manifolds in the remaining s DOF subsystem (s ≤ n). This s DOF subsystem is called the normal system and its structure generally depends on the n − s actions.…”

mentioning

confidence: 99%

“…(See exact statements in the formulation section below. For a complete treatment see [33].) The regular n-tori correspond to the case s = 0.…”

mentioning

confidence: 99%

“…The larger the n − s, the more possibilities to transfer between the different cases, namely, to encounter bifurcations. In this context, the Lerman and Umanskii work [33] is mainly concerned with n = s = 2, and the Lerman work [29] is mainly concerned with n = s = 3 (yet it includes the generic bifurcation diagrams for the s = 1, 2 cases), whereas the works of Oshemkov [43], Fomenko [16], and Bolsinov and Fomenko [4] are mainly concerned with n = 2, s = 1. Here, we consider s = 1 and n ≥ 2, studying the implications of phase space bifurcations and resonances as a source for instabilities in the near-integrable systems.…”

mentioning

confidence: 99%

“…Traditional spaces for such representation are the frequency domain [27], the space of constants of motion (e.g., [2,33,40,18,17]), the energy-momentum space, and the momentum space (e.g., [2,1,6,7,47,10,33,40]). Such presentations are all equivalent near regular level sets, where action-angle coordinates may be introduced.…”

mentioning

confidence: 99%

See 4 more Smart Citations

On Energy Surfaces and the Resonance Web

Litvak-Hinenzon¹,

Rom‐Kedar²

2004

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

Abstract.A framework for understanding the global structure of near-integrable n DOF Hamiltonian systems is proposed. To this aim two tools are developed-the energy-momentum bifurcation diagrams and the branched surfaces. Their use is demonstrated on a few near-integrable 3 DOF systems. For these systems possible sources of instabilities are identified in the diagrams, and the corresponding energy surfaces are presented in the frequency space and by the branched surfaces. The main results of this formulation are theorems which describe the connection between changes in the topology of the energy surfaces and the existence of resonant lower dimensional tori. Key words. near-integrable Hamiltonians, parabolic resonances AMS subject classifications. 70H08, 37J20DOI. 10.1137/0306001061. Introduction. The study of the structure of energy surfaces of integrable systems and the study of resonances and instabilities in near-integrable systems developed into vast disparate research fields. The relation between the two received very little attention. Indeed, near regular level sets of the integrable Hamiltonian, the standard Arnold-resonance web structure appears, and the relation between the two fields reduces to the study of Arnold conjecture regarding instabilities in phase space. Here, we demonstrate that near singular level sets of the integrable Hamiltonian much information regarding possible instabilities of the near-integrable case may be deduced from the structure of the energy surface and its relations with resonance surfaces. We suggest that by adding some information to the traditional energy-momentum diagrams [7], which we name energy-momentum bifurcation diagrams (EMBD), one achieves a global qualitative understanding of the near-integrable dynamics. We relate the geometric properties of the surfaces corresponding to lower dimensional tori in this diagram to both bifurcations in the energy surface topology and the appearance of lower dimensional resonant tori.Recall that energy surfaces of generic integrable Hamiltonian systems are foliated almost everywhere by n-tori, 1 which may be expressed locally as a product of n circles on which the dynamics reduces to simple rotations (the action-angle coordinates). A given compact regular level set (the set of phase space points with given values of the constants of motion) may be composed of several such tori. The energy surface is composed of all level sets with the same

show abstract

mentioning

confidence: 98%

mentioning

confidence: 99%

“…(See exact statements in the formulation section below. For a complete treatment see [33].) The regular n-tori correspond to the case s = 0.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

On Energy Surfaces and the Resonance Web

Litvak-Hinenzon¹,

Rom‐Kedar²

2004

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type

Ngọc

2000

Comm. Pure Appl. Math.

View full text Add to dashboard Cite

We present a detailed study, in the semi-classical regime h → 0, of microlocal properties of systems of two commuting h-pseudo-differential operators P 1 (h), P 2 (h) such that the joint principal symbol p = (p 1 , p 2 ) has a special kind of singularity called a focus-focus singularity. Typical examples include the quantum spherical pendulum or the quantum Champagne bottle.In the spirit of Colin de Verdière and Parisse [11,12,13], we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type.These conditions are used to give a precise description of the joint spectrum of such systems, including the phenomenon of quantum monodromy and different formulations of the counting function for the joint eigenvalues close to the singularity, in which a logarithm of the semi-classical constant h appears. Thanks to numerical computations done by M.S. Child for the case of the Champagne bottle, we are able to accurately illustrate our statements.

show abstract

Energy Surfaces and Hierarchies of Bifurcations

Shlizerman

Rom‐Kedar

Chaotic Dynamics and Transport in Classical and Quantum Systems

View full text Add to dashboard Cite

Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects)

Cited by 22 publications

References 0 publications

On Energy Surfaces and the Resonance Web

On Energy Surfaces and the Resonance Web

Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type

Energy Surfaces and Hierarchies of Bifurcations

Contact Info

Product

Resources

About