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2013
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Volume 21, Number 6
On the 70th Birthday of Nikolaí N. Nekhoroshev

Nikolaí N. Nekhoroshev
Abstract
Citation: Nikolaí N. Nekhoroshev, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 593-598
DOI:10.1134/S1560354716060010
Sevryuk M. B.
Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory
Abstract
We prove a general theorem on the persistence of Whitney $C^\infty$-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where $\dim \text{Fix}\,G < (\text{codim}\,\mathcal{T})/2$, where $\text{Fix}\,G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W. Broer, M.-C. Ciocci, H. Hansmann, and A. Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems.
Keywords: KAM theory, reversible systems, BCHV theorem, reversible context 2, invariant tori, Whitney smooth families
Citation: Sevryuk M. B., Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 599-620
DOI:10.1134/S1560354716060022
Wiggins S.
The Role of Normally Hyperbolic Invariant Manifolds (NHIMs) in the Context of the Phase Space Setting for Chemical Reaction Dynamics
Abstract
In this paper we give an introduction to the notion of a normally hyperbolic invariant manifold (NHIM) and its role in chemical reaction dynamics. We do this by considering simple examples for one-, two-, and three-degree-of-freedom systems where explicit calculations can be carried out for all of the relevant geometrical structures and their properties can be explicitly understood. We specifically emphasize the notion of a NHIM as a “phase space concept”. In particular, we make the observation that the (phase space) NHIM plays the role of “carrying” the (configuration space) properties of a saddle point of the potential energy surface into phase space.
We also consider an explicit example of a 2-degree-of-freedom system where a “global” dividing surface can be constructed using two index one saddles and one index two saddle. Such a dividing surface has arisen in several recent applications and, therefore, such a construction may be of wider interest.
Keywords: normally hyperbolic invariant manifolds, chemical reaction dynamics, dividing surface, phase space transport, index $k$ saddle points
Citation: Wiggins S., The Role of Normally Hyperbolic Invariant Manifolds (NHIMs) in the Context of the Phase Space Setting for Chemical Reaction Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 621-638
DOI:10.1134/S1560354716060034
Bates L.,  Cushman R.
A Generalization of Nekhoroshev’s Theorem
Abstract
Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincar´e theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.
Keywords: periodic orbits, Hamiltonian systems
Citation: Bates L.,  Cushman R., A Generalization of Nekhoroshev’s Theorem, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 639-642
DOI:10.1134/S1560354716060046
Kiesenhofer A.,  Miranda E.
Noncommutative Integrable Systems on $b$-symplectic Manifolds
Abstract
In this paper we study noncommutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a $b$-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure.
Keywords: Poisson manifolds, $b$-symplectic manifolds, noncommutative integrable systems, action-angle coordinates
Citation: Kiesenhofer A.,  Miranda E., Noncommutative Integrable Systems on $b$-symplectic Manifolds, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 643-659
DOI:10.1134/S1560354716060058
Carati A.,  Galgani L.,  Maiocchi A.,  Gangemi F.,  Gangemi R.
Persistence of Regular Motions for Nearly Integrable Hamiltonian Systems in the Thermodynamic Limit
Abstract
A review is given of the studies aimed at extending to the thermodynamic limit stability results of Nekhoroshev type for nearly integrable Hamiltonian systems. The physical relevance of such an extension, i. e., of proving the persistence of regular (or ordered) motions in that limit, is also discussed. This is made in connection both with the old Fermi–Pasta–Ulam problem, which gave origin to such discussions, and with the optical spectral lines, the existence of which was recently proven to be possible in classical models, just in virtue of such a persistence.
Keywords: perturbation theory, thermodynamic limit, optical properties of matter
Citation: Carati A.,  Galgani L.,  Maiocchi A.,  Gangemi F.,  Gangemi R., Persistence of Regular Motions for Nearly Integrable Hamiltonian Systems in the Thermodynamic Limit, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 660-664
DOI:10.1134/S156035471606006X
Bounemoura A.
Generic Perturbations of Linear Integrable Hamiltonian Systems
Abstract
In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of $\varepsilon^{-1}$) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).
Keywords: Hamiltonian perturbation theory, KAM theory, Nekhoroshev theory, Arnold diffusion
Citation: Bounemoura A., Generic Perturbations of Linear Integrable Hamiltonian Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 665-681
DOI:10.1134/S1560354716060071
Tran D. T.,  van der Kamp P. H.,  Quispel G. R. W.
Poisson Brackets of Mappings Obtained as $(q,−p)$ Reductions of Lattice Equations
Abstract
In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The $(q,−p)$ reductions are $(p + q)$-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the $(3,−2)$ reductions of the integrable partial difference equations are Liouville integrable in their own right.
Keywords: lattice equation, periodic reduction, Lagrangian, Poisson bracket
Citation: Tran D. T.,  van der Kamp P. H.,  Quispel G. R. W., Poisson Brackets of Mappings Obtained as $(q,−p)$ Reductions of Lattice Equations, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 682-696
DOI:10.1134/S1560354716060083
Martynchuk N.,  Waalkens H.
Knauf’s Degree and Monodromy in Planar Potential Scattering
Abstract
We consider Hamiltonian systems on $(T^{*}\mathbb R^2, dq \wedge dp)$ defined by a Hamiltonian function $H$ of the “classical” form $H = p^2/2 + V (q)$. A reasonable decay assumption $V(q) \to 0, \, \|q\| \to \infty$, allows one to compare a given distribution of initial conditions at $t = −\infty$ with their final distribution at $t = +\infty$. To describe this Knauf introduced a topological invariant $\text{deg}(E)$, which, for a nontrapping energy $E > 0$, is given by the degree of the scattering map. For rotationally symmetric potentials $V = W(\|q\|)$, scattering monodromy has been introduced independently as another topological invariant. In the present paper we demonstrate that, in the rotationally symmetric case, Knauf’s degree $\text{deg}(E)$ and scattering monodromy are related to one another. Specifically, we show that scattering monodromy is given by the jump of the degree $\text{deg}(E)$, which appears when the nontrapping energy $E$ goes from low to high values.
Keywords: Hamiltonian system, Liouville integrability, nontrapping degree of scattering, scattering monodromy
Citation: Martynchuk N.,  Waalkens H., Knauf’s Degree and Monodromy in Planar Potential Scattering, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 697-706
DOI:10.1134/S1560354716060095
Guzzo M.,  Lega E.
The Nekhoroshev Theorem and the Observation of Long-term Diffusion in Hamiltonian Systems
Abstract
The long-term diffusion properties of the action variables in real analytic quasiintegrable Hamiltonian systems is a largely open problem. The Nekhoroshev theorem provides bounds to such a diffusion as well as a set of techniques, constituting its proof, which have been used to inspect also the instability of the action variables on times longer than the Nekhoroshev stability time. In particular, the separation of the motions in a superposition of a fast drift oscillation and an extremely slow diffusion along the resonances has been observed in several numerical experiments. Global diffusion, which occurs when the range of the slow diffusion largely exceeds the range of fast drift oscillations, needs times larger than the Nekhoroshev stability times to be observed, and despite the power of modern computers, it has been detected only in a small interval of the perturbation parameter, just below the critical threshold of application of the theorem. In this paper we show through an example how sharp this phenomenon is.
Keywords: Hamiltonian systems, Nekhoroshev theorem, long-term stability, diffusion
Citation: Guzzo M.,  Lega E., The Nekhoroshev Theorem and the Observation of Long-term Diffusion in Hamiltonian Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 707-719
DOI:10.1134/S1560354716060101
Sadovskií D. A.
Nekhoroshev’s Approach to Hamiltonian Monodromy
Abstract
Using the hyperbolic circular billiard, introduced in [31] by Delos et al. as possibly the simplest system with Hamiltonian monodromy, we illustrate the method developed by N. N. Nekhoroshev and coauthors [48] to uncover this phenomenon. Nekhoroshev’s very original geometric approach reflects his profound insight into Hamiltonian monodromy as a general topological property of fibrations. We take advantage of the possibility of having closed form elementary function expressions for all quantities in our system in order to provide the most explicit and detailed explanation of Hamiltonian monodromy and its relation to similar phenomena in other domains.
Keywords: integrable fibration, Hamiltonian monodromy, first homology, $A_1$ singularity
Citation: Sadovskií D. A., Nekhoroshev’s Approach to Hamiltonian Monodromy, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 720-758
DOI:10.1134/S1560354716060113
Bizyaev I. A.,  Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups
Abstract
This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector $(3, 6, 14)$, the other is defined by two generatrices and growth vector $(2, 3, 5, 8)$. Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals.
Keywords: sub-Riemannian geometry, Carnot group, Poincaré map, first integrals
Citation: Bizyaev I. A.,  Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Integrability and Nonintegrability of Sub-Riemannian Geodesic Flows on Carnot Groups, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 759-774
DOI:10.1134/S1560354716060125