Vol. 2, No. 3

Vol. 2, No. 3, 2006

Anishchenko V. S.,  Nikolaev S. M.
We propose a new autonomous dynamical system of dimension $N = 4$ that demonstrates the regime of stable two-frequency motions. It is shown that system of two generators of quasiperiodic motions with symmetric coupling can realize motions on four-dimensional torus with resonant structures on it in the form of three- and two-dimensional torus. We show that with increase of noise intensity the higher the dimension of torus the faster it is destroyed.
Keywords: quasiperiodic motions, synchronization, chaos
Citation: Anishchenko V. S.,  Nikolaev S. M., Stability, synchronization and destruction of quasiperiodic motions, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 267-278
Borisov A. V.,  Mamaev I. S.
We present a reduction-of-order procedure in the problem of motion of two bodies on the Lobatchevsky plane $H^2$. The bodies interact with a potential that depends only on the distance between the bodies (this holds for an analog of the Newtonian potential). In earlier works, this reduction procedure was used to analyze the motion of two bodies on the sphere
Keywords: Lobatchevsky plane, first integral, reduction-of-order procedure, potential of interaction
Citation: Borisov A. V.,  Mamaev I. S., Reduction in the two-body problem on the Lobatchevsky plane, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 279-285
Dumachev V. N.
On the bais of Liouville theorem the generalization of the Nambu mechanics is considered. Is shown, that Poisson manifolds of $n$-dimensional multi-symplectic phase space have inducting by $(n-1)$ Hamilton $k$-vectors fields, each of which requires of $(k)$-hamiltonians.
Keywords: Liouville theorem, Hamilton vectors fields
Citation: Dumachev V. N., Phase flows in $J^n(π)$, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 287-292
Rozenblat G. M.
We study the motion (rolling motion) of a flat plate whose boundary is an arbitrary convex curve along a straight line. During the motion the plate is always in contact with the supporting line and subject to a dry friction. Plus, the plate is acted on by an arbitrary plane system of forces and at the point of contact only the unilateral constraint is assumed. All possible transitions from a rolling motion with slipping to a pure rolling without slipping and vice versa are classified. Necessary conditions for the plate to remain in contact with the line are obtained. The results obtained are used to study 1) the motion of a non-uniform circular disk, subject to gravity, on a rough horizontal straight line in the vertical plane and 2) the motion of a slender rod, subject to gravity, on a rough straight line.
Keywords: rolling motion, dry friction, unilateral constraint
Citation: Rozenblat G. M., On the motion of a two-dimensional rigid body on a rough straight line, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 293-306
Kuptsov P. V.,  Kuznetsov S. P.
Amplitude equations are obtained for a system of two coupled van der Pol oscillators that has been recently suggested as a simple system with hyperbolic chaotic attractor allowing physical realization. We demonstrate that an approximate model based on the amplitude equations preserves basic features of a hyperbolic dynamics of the initial system. For two coupled amplitude equations models having the hyperbolic attractors a transition to synchronous chaos is studied. Phenomena typically accompanying this transition, as riddling and bubbling, are shown to manifest themselves in a specific way and can be observed only in a small vicinity of a critical point. Also, a structure of many-dimensional attractor of the system is described in a region below the synchronization point.
Keywords: hyperbolic chaos, strange Smale-Williams attractor, chaotic synchronization, amplitude equations
Citation: Kuptsov P. V.,  Kuznetsov S. P., Transition to a synchronous chaos regime in a system of coupled non-autonomous oscillators presented in terms of amplitude equations, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 307-331
Borisov A. V.,  Kilin A. A.,  Mamaev I. S.
We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball in the upmost, downmost and saddle point.
Keywords: nonholonomic constraint, stationary rotations, stability
Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Stability of steady rotations in the non-holonomic Routh problem, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 333-345
Tsiganov A. V.
Elliptic coordinates on the dual space to the Lie algebra $e(3)$ are introduced. On the zero level of the Casimir function, these coordinates coincide with the standard elliptic coordinates on the cotangent bundle to the two-dimensional sphere. The possibility of use of these coordinates in the theory of integrable systems is discussed.
Keywords: elliptic coordinates, integrable systems, separation of variables
Citation: Tsiganov A. V., On elliptic coordinates on the Lie algebra $e(3)$, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 347-352
Bord E. G.
The evolution of the symmetrical disturbances of the vortex poligon is studied. The return of the unstable solution to their initial position is found. The case of eight vortices is considered separately. For this case second order differential equation system is found. The vortex pair realization is explained.
Keywords: vortex polygon, local instability, nonlinear oscillator
Citation: Bord E. G., On the nonlinear disturbances of vortex poligon, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 353-360
Albouy A.
Citation: Albouy A., Mutual Distances in Celestial Mechanics, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 3, pp. 361-386

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