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Vol. 7, No. 1

Vol. 7, No. 1, 2011

Gonchenko S. V.,  Sten'kin O. V.
Abstract
It has been established by Gavrilov and Shilnikov in [1] that, at the bifurcation boundary separating Morse-Smale systems from systems with complicated dynamics, there are systems with homoclinic tangencies. Moreover, when crossing this boundary, infinitely many periodic orbits appear immediately, just by «explosion». Newhouse and Palis have shown in [2] that in this case there are infinitely many intervals of values of the splitting parameter corresponding to hyperbolic systems. In the present paper, we show that such hyperbolicity intervals have natural bifurcation boundaries, so that the phenomenon of homoclinic Ω-explosion gains, in a sense, complete description in the case of two-dimensional diffeomorphisms.
Keywords: homoclinic tangency, heteroclinic cycle, Ω-explosion, hyperbolic set
Citation: Gonchenko S. V.,  Sten'kin O. V., Homoclinic Ω-explosion: hyperbolicity intervals and their boundaries, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 3-24
DOI:10.20537/nd1101001
Kharlamov M. P.
Abstract
This work continues the author’s article in Rus. J. Nonlinear Dynamics (2010, v. 6, N. 4) and contains applications of the Boolean functions method to investigation of the admissible regions and the phase topology of three algebraically solvable systems in the problem of motion of the Kowalevski top in the double force field.
Keywords: algebraic separation of variables, integral manifolds, Boolean functions, topological analysis
Citation: Kharlamov M. P., Topological analysis and Boolean functions: II. Application to new algebraic solutions, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 25-51
DOI:10.20537/nd1101002
Khudobakhshov V. A.,  Tsiganov A. V.
Abstract
New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail.We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.
Keywords: integrable systems, separation of variables, Abel equations
Citation: Khudobakhshov V. A.,  Tsiganov A. V., On quadratures of integrable systems on a sphere with higher degree integrals of motion, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 53-74
DOI:10.20537/nd1101003
Gledzer A. E.
Abstract
Passive particles advection is considered in the vicinity of hyperbolic stationary point of the separatrix destroyed by insteady perturbations. For different frequencies of the disturbancies the trajectories of advected particles are investigated analytically and numerically. The approximate criteria of capture and release of particles are obtained. The results are linked with known law for the stochastic layer width near separatrix. The obtained criteria are connected with analytical Melnikov’s integral.
Keywords: chaotic dynamics, vortex structures, stochastic layer
Citation: Gledzer A. E., On the Lagrangian transport near oscillating vortex in running flow, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 75-100
DOI:10.20537/nd1101004
Kozlov V. V.
Abstract
The Kac circular model is a discrete dynamical system which has the property of recurrence and reversibility. Within the framework of this model M.Kac formulated necessary conditions for irreversibility over «short» time intervals to take place and demonstrated Boltzmann’s most important exploration methods and ideas, outlining their advantages and limitations. We study the circular model within the realm of the theory of Gibbs ensembles and offer a new approach to a rigorous proof of the «zeroth» law of thermodynamics basing on the analysis of weak convergence of probability distributions.
Keywords: reversibility, stochastic equilibrium, weak convergence
Citation: Kozlov V. V., Statistical irreversibility of the Kac reversible circular model, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 101-117
DOI:10.20537/nd1101005
Borisov A. V.,  Mamaev I. S.,  Vaskina A. V.
Abstract
This paper presents a topological approach to the search and stability analysis of relative equilibria of three point vortices of equal intensities. It is shown that the equations of motion can be reduced by one degree of freedom. We have found two new stationary configurations (isosceles and non-symmetrical collinear) and studied their bifurcations and stability.
Keywords: point vortex, reduction, bifurcational diagram, relative equilibriums, stability, periodic solutions
Citation: Borisov A. V.,  Mamaev I. S.,  Vaskina A. V., Stability of new relative equilibria of the system of three point vortices in a circular domain, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 119-138
DOI:10.20537/nd1101006
Farkas Z.,  Bartels G.,  Unger T.,  Wolf D.
Abstract
The tangential motion at the contact of two solid objects is studied. It consists of a sliding and a spinning degree of freedom (no rolling). We show that the friction force and torque are inherently coupled. As a simple test system, a sliding and spinning disk on a horizontal flat surface is considered. We calculate, and also measure, how the disk slows down and find that it always stops its sliding and spinning motion at the same moment. We discuss the impact of this coupling between friction force and torque on the physics of granular materials.
Citation: Farkas Z.,  Bartels G.,  Unger T.,  Wolf D., Frictional coupling between sliding and spinning motion, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 139-146
DOI:10.20537/nd1101007
Zhuravlev V. F.
Abstract
Citation: Zhuravlev V. F., Comments on "Lagrangian mechanics and dry friction" by V. V. Kozlov, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 147-149
DOI:10.20537/nd1101008
Abstract
Citation: Science and Music (editorial note), Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 151-152
DOI:10.20537/nd1101009
Nierhaus G.
Abstract
Citation: Nierhaus G., Chaos and self-similarity, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 153-175
DOI:10.20537/nd1101010
Abstract
Citation: The Clebsch System. Separation of variables and Explicit Integration? (Collection of papers), Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 177-187
Abstract
Citation: New books of the Scientific and Publishing Center «Regular and Chaotic Dynamics» and Institute of Computer Science (Moscow-Izhevsk). New issues of «Regular and Chaotic Dynamics», Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 1, pp. 189-193

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