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Condolence note

Professor Alexey V. Borisov, Editor-in-Chief of the journal RCD, suddenly passed away at the age of 55 on January 24, 2021.
This is a tremendous loss for all of us. We will always remember him as a great scientist and an outstanding personality.


Volume 26, Number 4

Markeev A. P.
On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Three-body Problem
Abstract
A study is made of the stability of triangular libration points in the nearly-circular restricted three-body problem in the spatial case. The problem of stability for most (in the sense of Lebesgue measure) initial conditions in the planar case has been investigated earlier. In the spatial case, an identical resonance takes place: for all values of the parameters of the problem the period of Keplerian motion of the two main attracting bodies is equal to the period of small linear oscillations of the third body of negligible mass along the axis perpendicular to the plane of the orbit of the main bodies. In this paper it is assumed that there are no resonances of the planar problem through order six. Using classical perturbation theory, KAM theory and algorithms of computer calculations, stability is proved for most initial conditions and the Nekhoroshev estimate of the time of stability is given for trajectories starting in an addition to the above-mentioned set of most initial conditions.
Keywords: restricted three-body problem, triangular libration points, stability, Arnold diffusion
Citation: Markeev A. P., On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Three-body Problem, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 321-330
DOI:10.1134/S1560354721040018
Kourliouros K.
Sections of Hamiltonian Systems
Abstract
A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e. g., a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular, we give a complete list of exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system.
Keywords: Hamiltonian systems, constraints, singularities, normal forms, functional moduli
Citation: Kourliouros K., Sections of Hamiltonian Systems, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 331-349
DOI:10.1134/S156035472104002X
Pochinka O. V.,  Zinina S. K.
Construction of the Morse – Bott Energy Function for Regular Topological Flows
Abstract
In this paper, we consider regular topological flows on closed $n$-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse – Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.
Keywords: energy function, Morse – Bott energy function, regular topological flow, chain recurrent set, ambient manifold
Citation: Pochinka O. V.,  Zinina S. K., Construction of the Morse – Bott Energy Function for Regular Topological Flows, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 350-369
DOI:10.1134/S1560354721040031
Balabanova N.,  Montaldi J.
Two-body Problem on a Sphere in the Presence of a Uniform Magnetic Field
Abstract
We investigate the motion of one and two charged non-relativistic particles on a sphere in the presence of a magnetic field of uniform strength. For one particle, the motion is always circular, and determined by a simple relation between the velocity and the radius of motion. For two identical particles interacting via a cotangent potential, we show there are two families of relative equilibria, called Type I and Type II. The Type I relative equilibria exist for all strengths of the magnetic field, while those of Type II exist only if the field is sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We also determine the stability of the two families of relative equilibria.
Keywords: Hamiltonian reduction, relative equilibria, stability, bifurcations
Citation: Balabanova N.,  Montaldi J., Two-body Problem on a Sphere in the Presence of a Uniform Magnetic Field, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 370-391
DOI:10.1134/S1560354721040043
Zhuravlev N. B.,  Rossovsky L. E.
Spectral Radius Formula for a Parametric Family of Functional Operators
Abstract
The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing ``chaotic'' dependence on the compression ratio. For example, it turns out that the spectral radius of the operator $$ L_2(\mathbb R^n)\ni u(x)\mapsto u(p^{-1}x+h)-u(p^{-1}x-h)\in L_2(\mathbb R^n),\quad p>1,\quad h\in\mathbb R^n, $$ is equal to $2p^{n/2}$ for transcendental values of $p$, and depends on the coefficients of the minimal polynomial for $p$ in the case where $p$ is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for $p$. The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of~$p$.
Keywords: elliptic functional differential equation, differential-difference equation, rescaling, linear flow on the torus
Citation: Zhuravlev N. B.,  Rossovsky L. E., Spectral Radius Formula for a Parametric Family of Functional Operators, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 392-401
DOI:10.1134/S1560354721040055
Andrade J.,  Vidal C.,  Sierpe C.
Stability of the Relative Equilibria in the Two-body Problem on the Sphere
Abstract
We consider the 2-body problem in the sphere $\mathbb{S}^2$. This problem is modeled by a Hamiltonian system with $4$ degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of $2$ degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and center-saddle bifurcations.
Keywords: two-body-problem on the sphere, Hamiltonian formulation, normal form, resonance, nonlinear stability
Citation: Andrade J.,  Vidal C.,  Sierpe C., Stability of the Relative Equilibria in the Two-body Problem on the Sphere, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 402-438
DOI:10.1134/S1560354721040067
Shibayama M.,  Yamada J.
Nonintegrability of the Reduced Planar Three-body Problem with Generalized Force
Abstract
We consider the planar three-body problem with generalized potentials. Some nonintegrability results for these systems have been obtained by analyzing the variational equations along homothetic solutions. But we cannot apply it to several exceptional cases. For example, in the case of inverse-square potentials, the variational equations along homothetic solutions are solvable. We obtain sufficient conditions for nonintegrability for these exceptional cases by focusing on some particular solutions that are different from homothetic solutions.
Keywords: nonintegrability, three-body problem
Citation: Shibayama M.,  Yamada J., Nonintegrability of the Reduced Planar Three-body Problem with Generalized Force, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 439-455
DOI:10.1134/S1560354721040079
Biswas A.,  Kara A. H.,  Ekici M.,  Zayed E.,  Alzahrani A. K.,  Belic M. R.
Conservation Laws for Solitons in Magneto-optic Waveguides with Anti-cubic and Generalized Anti-cubic Nonlinearities
Abstract
This paper implements a multiplier approach to exhibit conservation laws in magneto-optic waveguides that maintain anti-cubic as well as generalized anti-cubic forms of the nonlinear refractive index. Three conservation laws are retrieved for each form of nonlinearity. They are power, linear momentum and Hamiltonian. The conserved quantities are computed from their respective densities.
Keywords: solitons, conservation law, anti-cubic
Citation: Biswas A.,  Kara A. H.,  Ekici M.,  Zayed E.,  Alzahrani A. K.,  Belic M. R., Conservation Laws for Solitons in Magneto-optic Waveguides with Anti-cubic and Generalized Anti-cubic Nonlinearities, Regular and Chaotic Dynamics, 2021, vol. 26, no. 4, pp. 456-461
DOI:10.1134/S1560354721040080