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Volume 21, Number 1

Volume 21, Number 1, 2016

Akbarzadeh R.
Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$
Abstract
In 2001, A. V. Borisov, I. S. Mamaev, and V. V. Sokolov discovered a new integrable case on the Lie algebra $so(4)$. This is a Hamiltonian system with two degrees of freedom, where both the Hamiltonian and the additional integral are homogenous polynomials of degrees 2 and 4, respectively. In this paper, the topology of isoenergy surfaces for the integrable case under consideration on the Lie algebra $so(4)$ and the critical points of the Hamiltonian under consideration for different values of parameters are described and the bifurcation values of the Hamiltonian are constructed. Also, a description of bifurcation complexes and typical forms of the bifurcation diagram of the system are presented.
Keywords: topology, integrable Hamiltonian systems, isoenergy surfaces, critical set, bifurcation diagram, bifurcation complex, periodic trajectory
Citation: Akbarzadeh R., Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 1-17
DOI:10.1134/S1560354716010019
Jiang K.
Local Normal Forms of Smooth Weakly Hyperbolic Integrable Systems
Abstract
In the smooth $(C^{\infty})$ category, a completely integrable system near a nondegenerate singularity is geometrically linearizable if the action generated by the vector fields is weakly hyperbolic. This proves partially a conjecture of Nguyen Tien Zung [11]. The main tool used in the proof is a theorem of Marc Chaperon [3] and the slight hypothesis of weak hyperbolicity is generic when all the eigenvalues of the differentials of the vector fields at the non-degenerate singularity are real.
Keywords: completely integrable systems, geometric linearization, nondegenerate singularity, weak hyperbolicity
Citation: Jiang K., Local Normal Forms of Smooth Weakly Hyperbolic Integrable Systems, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 18-23
DOI:10.1134/S1560354716010020
Kharlamov M. P.,  Ryabov P. E.,  Savushkin A. Y.
Topological Atlas of the Kowalevski–Sokolov Top
Abstract
We investigate the phase topology of the integrable Hamiltonian system on $e(3)$ found by V. V. Sokolov (2001) and generalizing the Kowalevski case. This generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. The relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the isoenergy manifolds of the reduced systems with two degrees of freedom are classified. The set of critical points of the momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the diagram of the momentum map and give a classification of isoenergy and isomomentum diagrams equipped with the description of regular integral manifolds and their bifurcations. We construct the Smale–Fomenko diagrams which, when considered in the enhanced space of the energy-momentum constants and the essential physical parameters, separate 25 different types of topological invariants called the Fomenko graphs. We find all marked loop molecules of rank 0 nondegenerate critical points and of rank 1 degenerate periodic trajectories. Analyzing the cross-sections of the isointegral equipped diagrams, we get a complete list of the Fomenko graphs. The marks on them producing the exact topological invariants of Fomenko–Zieschang can be found from previous investigations of two partial cases with some additions obtained from the loop molecules or by a straightforward calculation using the separation of variables.
Keywords: integrable Hamiltonian systems, relative equilibria, isoenergy surfaces, critical subsystems, bifurcation diagrams, rough topological invariants
Citation: Kharlamov M. P.,  Ryabov P. E.,  Savushkin A. Y., Topological Atlas of the Kowalevski–Sokolov Top, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 24-65
DOI:10.1134/S1560354716010032
Lemarie C.
Some Poisson Structures and Lax Equations Associated with the Toeplitz Lattice and the Schur Lattice
Abstract
The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this paper, we unveil the origins of this Poisson structure and derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety $\mathcal{H}_n$ of $\mathbf{GL}_n(\mathbb{C})$, which we view as a real or complex Poisson–Lie group whose Poisson structure comes from a quadratic $R$-bracket on $\mathfrak{gl}_n(\mathbb{C})$ for a fixed $R$-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on $\mathcal{H}_n$, combined with the properties of the quadratic $R$-bracket allow us to give explicit formulas for the Lax equation. Then we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over $\mathbb{R}$, we can construct a Poisson subvariety $\mathcal{H}_n^\tau$ of $U_n$ which is itself a Poisson–Dirac subvariety of $\mathbf{GL}_n^\mathbb{R}(\mathbb{C})$. We then construct a Hamiltonian for the Poisson structure induced on $\mathcal{H}_n^\tau$, corresponding to another system which derives from the Toeplitz lattice the modified Schur lattice. Thanks to the properties of Poisson–Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.
Keywords: integrable lattices, Poisson–Lie groups, Lax equations
Citation: Lemarie C., Some Poisson Structures and Lax Equations Associated with the Toeplitz Lattice and the Schur Lattice, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 66-96
DOI:10.1134/S1560354716010044
Selskii A.,  Makarov V. A.
Synchronization of Heteroclinic Circuits through Learning in Coupled Neural Networks
Abstract
The synchronization of oscillatory activity in neural networks is usually implemented by coupling the state variables describing neuronal dynamics. Here we study another, but complementary mechanism based on a learning process with memory. A driver network, acting as a teacher, exhibits winner-less competition (WLC) dynamics, while a driven network, a learner, tunes its internal couplings according to the oscillations observed in the teacher. We show that under appropriate training the learner can “copy” the coupling structure and thus synchronize oscillations with the teacher. The replication of the WLC dynamics occurs for intermediate memory lengths only, consequently, the learner network exhibits a phenomenon of learning resonance.
Keywords: synchronization, learning, heteroclinic circuit, neural networks, winner-less competition
Citation: Selskii A.,  Makarov V. A., Synchronization of Heteroclinic Circuits through Learning in Coupled Neural Networks, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 97-106
DOI:10.1134/S1560354716010056
Xu J.,  Lu X.
General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies
Abstract
In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.
Keywords: Hamiltonian system, KAM iteration, invariant tori, nondegeneracy condition
Citation: Xu J.,  Lu X., General KAM Theorems and their Applications to Invariant Tori with Prescribed Frequencies, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 107-125
DOI:10.1134/S1560354716010068
Valeriano L.
Parametric Stability in Robe’s Problem
Abstract
In this work we have given a Hamiltonian formulation to Robe’s problem, obtaining again the classic results. We have computed the resonances existing in the circular case and obtained some information with regard to the linear stability of the central equilibrium of Robe’s problem in the elliptic case. In some critical cases we have constructed, in the parameter plane, the boundary curves that separate the regions of stability and instability.
Keywords: Robe’s problem, restricted three-body problem, parametric stability
Citation: Valeriano L., Parametric Stability in Robe’s Problem, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 126-135
DOI:10.1134/S156035471601007X
Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S.
Dynamics of the Chaplygin Sleigh on a Cylinder
Abstract
This paper is concerned with the motion of the Chaplygin sleigh on the surface of a circular cylinder. In the case of inertial motion, the problem reduces to the study of the dynamical system on a (two-dimensional) torus and to the classification of singular points. Particular cases in which the system admits an invariant measure are found.

In the case of a balanced and dynamically symmetric Chaplygin sleigh moving in a gravitational field it is shown that on the average the system has no drift along the vertical.

Keywords: Chaplygin sleigh, invariant measure, nonholonomic mechanics
Citation: Bizyaev I. A.,  Borisov A. V.,  Mamaev I. S., Dynamics of the Chaplygin Sleigh on a Cylinder, Regular and Chaotic Dynamics, 2016, vol. 21, no. 1, pp. 136-146
DOI:10.1134/S1560354716010081

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