Vol. 2, No. 1

Vol. 2, No. 1, 2006

Gonchenko S. V.,  Sten'kin O. V.,  Shilnikov L. P.
Let a $C^r$-smooth $r \geqslant 5$ two-dimensional diffeomorphism $f$ have a non-transversal heteroclinic cycle containing several saddle periodic and heteroclinic orbits and, besides, some of the heteroclinic orbits are non-transversal, i.e. at the points of these orbits the invariant manifolds of the corresponding saddles intersect non-transversally. Suppose that a cycle contains at least two saddle periodic orbits such that the saddle value (the absolute value of product of multipliers) of one orbit is less than 1 and it is greater than 1 for the other orbit. We prove that in any neighbourhood (in $C^r$-topology) of $f$ in the space of $C^r$-diffeomorphisms, there are open regions (so-called Newhouse regions with heteroclinic tangencies) where diffeomorphisms with infinitely many stable and unstable invariant circles are dense. For three-dimensional flows, this result implies the existence of Newhouse regions where flows having infinitely many stable and unstable invariant two-dimensional tori are dense.
Keywords: nontransversal heteroclinic cycle, Newhouse region, invariant circle
Citation: Gonchenko S. V.,  Sten'kin O. V.,  Shilnikov L. P., On the existence of infinitely many stable and unstable invariant tori for systems from Newhouse regions with heteroclinic tangencies, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 3-25
Sokolovskiy M. A.,  Verron J.
The paper explores the properties of motion of $A+1$ point vortices with $A$ planes of symmetry immersed into a two-layer fluid. The central vortex is supposed to be in the upper layer while the other $A$ vortices have equal intensity and form a regular $A$-gon configuration in the lower layer. For $A\geqslant2$, we study possible stationary motions. For $A=2$, using methods of qualitative analysis, we classify the motions of this vortical structure and obtain preliminary numerical results concerned with stability of symmetrical configurations.
Keywords: two-layer fluid, point vortex, vortex structures, choreography, phase portrait
Citation: Sokolovskiy M. A.,  Verron J., Some properties of motion of $A+1$ vortices in a two-layer rotating fluid, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 27-54
Krasnopolskaya T. S.,  Shvets A. Y.
We propose a new model and study the properties of a piezoceramic transducer interacting with an excitation device of limited power-supply. A special attention is given to the examination of the origin and stages of development of deterministic chaos in this system. It is shown that a great variety of effects typical for problems of chaotic dynamics is inherent in the system. The presence of several types of chaotic attractors is established, and moreover the existence of hyper-chaos is revealed. Various scenarios of transition from regular regimes to chaotic ones are explored. The phase portraits, Poincare surfaces of section and maps of some chaotic attractors are investigated. For some of the chaotic attractors, the spectral densities and distributions of invariant measures are obtained and explored.
Keywords: limited excitation, piezo-effect, chaotic attractor
Citation: Krasnopolskaya T. S.,  Shvets A. Y., Deterministic Chaos in Generator-Piezoceramic Transduser System, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 55-74
Butko Y. A.
We consider an infinite-dimensional Schrodinger equation with a scalar and vector potential in a Hilbert space. The vector potential plays the same role as a magnetic field in the finite-dimensional case. We have proved the existence of the solution to the Cauchy problem. The solution is local in time and space variables and is expressed by a probabilistic formula that mimics the Feynman-Kac-Ito formula.
Keywords: infinite dimensional Schrodinger equation, stochastic integrals, vector potential, Feynman-Kac-Ito formula, functional integrals
Citation: Butko Y. A., The Feynman-Kac-Ito formula for an infinite-dimensional Schrodinger equation with a scalar and vector potential, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 75-87
Kholostova O. V.
Motions of a non-autonomous time-periodic Hamiltonian system with one degree of freedom are considered. The Hamiltonian of the system contains a small parameter. The origin of the phase space is a linearly stable equilibrium of the unperturbed or complete system. It is supposed that the degeneration takes place in the unperturbed system with regard for terms of order less than five (the frequency of small nonlinear oscillations does not depend on the amplitude), and a resonance (up to the sixth order inclusively) occurs. For each resonance case a model Hamiltonian is constructed, and a qualitative investigation of motion of the model system is carried out. Using Poincare’s theory of periodic motions and KAM-theory we solve rigorously the problem of existence, bifurcations and stability of periodic motions of the initial system. The motions we study are analytical with respect to fractional (for resonances up to the forth order inclusively) or integer (resonances of fifth and sixth orders) degrees of the small parameter. As an illustration, we analyze resonance periodic motions of a spherical pendulum and a Lagrange top with a vibrating point of suspension in the presence of the degeneration considered.
Keywords: hamiltonian system, stability, resonance, Poincare's theory of periodical motions, KAM-theory
Citation: Kholostova O. V., On bifurcations and stability of resonance periodic motions of hamiltonian systems with one degree of freedom caused by degeneration of the hamiltonian, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 89-110
Aref H.
The concept was developed some twenty years ago as an outgrowth of work on advection by interacting point vortices. The term ’chaotic advection’ was first introduced in the title of an abstract for the 35th annual meeting of the APS Division of Fluid Dynamics (DFD) in 1982. The main reference, a Journal of Fluid Mechanics paper published in 1984, may be the true ’birthdate’ of the term. Earlier work from the 1960s by Arnol’d and Henon on advection by steady 3D flows already contained closely related ideas and results but was not widely appreciated. The present paper, based on the 2000 Otto Laporte Memorial Lecture delivered at the 53rd APS/DFD annual meeting, traces these and other precursors and the development of chaotic advection over the past two decades. Some exciting recent developments, such as application to fluid mixing in MEMS, and to materials processing, and the introduction of topological methods of analysis, are highlighted. On balance, chaotic advection is now established as a subtopic of fluid mechanics with wide ramifications and continued promise for theory, experiment and applications.
Keywords: Otto Laporte, chaotic advection, stirring, mixing, agitator
Citation: Aref H., The development of chaotic advection, Rus. J. Nonlin. Dyn., 2006, Vol. 2, No. 1, pp. 111-133

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