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2013
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Volume 23, Number 1

de la Llave R.
Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions
Abstract
A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.
Keywords: analytic maps, linearization
Citation: de la Llave R., Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 1-11
DOI:10.1134/S156035471801001X
Karaliolios N.
Local Rigidity of Diophantine Translations in Higher-dimensional Tori
Abstract
We prove a theorem asserting that, given a Diophantine rotation $\alpha $ in a torus $\mathbb{T} ^{d} \equiv \mathbb{R} ^{d} / \mathbb{Z} ^{d}$, any perturbation, small enough in the $C^{\infty}$ topology, that does not destroy all orbits with rotation vector $\alpha$ is actually smoothly conjugate to the rigid rotation. The proof relies on a KAM scheme (named after Kolmogorov–Arnol'd–Moser), where at each step the existence of an invariant measure with rotation vector $\alpha$ assures that we can linearize the equations around the same rotation $\alpha$. The proof of the convergence of the scheme is carried out in the $C^{\infty}$ category.
Keywords: KAM theory, quasi-periodic dynamics, Diophantine translations, local rigidity
Citation: Karaliolios N., Local Rigidity of Diophantine Translations in Higher-dimensional Tori, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 12-25
DOI:10.1134/S1560354718010021
Kozlov V. V.
Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability
Abstract
A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.
Keywords: Hamiltonian system, quadratic integrals, integral cones, degree of instability, quantum systems, Abelian integrals
Citation: Kozlov V. V., Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 26-46
DOI:10.1134/S1560354718010033
Martínez-Torres D.,  Miranda E.
Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds
Abstract
We prove that, for compact regular Poisson manifolds, the zeroth homology group is isomorphic to the top foliated cohomology group, and we give some applications. In particular, we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson manifold is. We use these Poisson homology computations to provide families of perfect Poisson manifolds.
Keywords: Poisson homology, foliated cohomology
Citation: Martínez-Torres D.,  Miranda E., Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 47-53
DOI:10.1134/S1560354718010045
Buhovsky L.,  Kaloshin V.
Nonisometric Domains with the Same Marvizi–Melrose Invariants
Abstract
For any strictly convex planar domain $\Omega \subset \mathbb R^2$ with a $C^\infty$ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose~\cite{MM}. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine $\Omega$ up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains $\Omega$ and $\bar \Omega$ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits $\{S^n\}_{n \geqslant 1}$ (resp. $\{ \bar S^n\}_{n \geqslant 1}$) of period going to infinity such that $ S^n $ and $ \bar S^n $ have the same period and perimeter for each $ n $.
Keywords: convex planar billiards, length spectrum, Laplace spectrum, Marvizi–Melrose spectral invariants
Citation: Buhovsky L.,  Kaloshin V., Nonisometric Domains with the Same Marvizi–Melrose Invariants, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 54-59
DOI:10.1134/S1560354718010057
Carpenter B. K.,  Ezra G. S.,  Farantos S. C.,  Kramer Z. C.,  Wiggins S.
Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points
Abstract
In this paper we analyze a two-degree-of-freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the “roaming mechanism” whose reaction dynamics are of current interest in the chemistry community.
Keywords: Double Morse potential, phase space structure, dynamics, periodic orbit, roaming
Citation: Carpenter B. K.,  Ezra G. S.,  Farantos S. C.,  Kramer Z. C.,  Wiggins S., Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 60-79
DOI:10.1134/S1560354718010069
Andrade J.,  Vidal C.
Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere
Abstract
In this paper we consider a symmetric restricted circular three-body problem on the surface $\mathbb{S}^2$ of constant Gaussian curvature $\kappa=1$. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius $a\in (0,1)$. It is verified that both poles of $\mathbb{S}^2$ are equilibrium points for any value of the parameter $a$. This problem is modeled through a Hamiltonian system of two degrees of freedom depending on the parameter $a$. Using results concerning nonlinear stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances. It is verified that for the north pole there are two values of bifurcation (on the stability) $a=\dfrac{\sqrt{4-\sqrt{2}}}{2}$ and $a=\sqrt{\dfrac{2}{3}}$, while the south pole has one value of bifurcation $a=\dfrac{\sqrt{3}}{2}$.
Keywords: circular restricted three-body problem on surfaces of constant curvature, Hamiltonian formulation, normal form, resonance, nonlinear stability
Citation: Andrade J.,  Vidal C., Stability of the Polar Equilibria in a Restricted Three-body Problem on the Sphere, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 80-101
DOI:10.1134/S1560354718010070
Cruz I.,  Mena-Matos H.,  Sousa-Dias M.
Multiple Reductions, Foliations and the Dynamics of Cluster Maps
Abstract
Reduction of cluster maps via presymplectic and Poisson structures is described in terms of the canonical foliations defined by these structures. In the case where multiple reductions coexist, we establish conditions on the underlying presymplectic and Poisson structures that allow for an interplay between the respective foliations. It is also shown how this interplay may be explored to simplify the analysis and obtain an effective geometric description of the dynamics of the original (not reduced) map. Consequences of the approach we developed to the description of several features of some cluster maps dynamics are illustrated by two examples which include the Somos-5 map and an instance of a Somos-7 map.
Keywords: presymplectic manifolds, Poisson manifolds, foliations, cluster maps
Citation: Cruz I.,  Mena-Matos H.,  Sousa-Dias M., Multiple Reductions, Foliations and the Dynamics of Cluster Maps, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 102-119
DOI:10.1134/S1560354718010082
Stankevich N. V.,  Dvorak A.,  Astakhov V. V.,  Jaros P.,  Kapitaniak M.,  Perlikowski P.,  Kapitaniak T.
Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators
Abstract
The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.
Keywords: chaos, hyperchaos, Toda oscillator
Citation: Stankevich N. V.,  Dvorak A.,  Astakhov V. V.,  Jaros P.,  Kapitaniak M.,  Perlikowski P.,  Kapitaniak T., Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 120-126
DOI:10.1134/S1560354718010094