Volume 19, Number 4

Volume 19, Number 4, 2014
Leonid Pavlovich Shilnikov. Memorial Volume

This special issue of the journal presents a selection of the proceeding papers of the conference "Dynamics, Bifurcations and Strange Attractors" dedicated to the memory of Leonid Pavlovich Shilnikov (1934–2011) to commemorate his contribution to the theory of dynamical systems and bifurcations. The conference was held at Nizhny Novgorod State University, Russia, on July 1–5, 2013.
Afraimovich V. S.,  Gonchenko S. V.,  Lerman L. M.,  Shilnikov A. L.,  Turaev D. V.
This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.
Keywords: Homoclinic chaos, global bifurcations, spiral chaos, strange attractor, saddle-focus, homoclinic loop, saddle-node, saddle-saddle, Lorenz attractor, hyperbolic set
Citation: Afraimovich V. S.,  Gonchenko S. V.,  Lerman L. M.,  Shilnikov A. L.,  Turaev D. V., Scientific Heritage of L.P. Shilnikov, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 435-460
Gonchenko S. V.,  Gordeeva O. V.,  Lukyanov V. I.,  Ovsyannikov I. I.
We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
Keywords: saddle-node, homoclinic tangency, Arnold tongues
Citation: Gonchenko S. V.,  Gordeeva O. V.,  Lukyanov V. I.,  Ovsyannikov I. I., On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 461-473
Morozov A. D.
Bifurcations in degenerate resonance zones for Hamitonian systems with 3/2 degrees of freedom close to nonlinear integrable ones and for symplectic maps of a cylinder are discussed.
Keywords: resonances, degenerate resonances, bifurcations, Hamiltonian systems, averaged systems, separatrix, vortex pairs, symplectic maps
Citation: Morozov A. D., On Bifurcations in Degenerate Resonance Zones, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 474-482
Kruglov V. P.,  Kuznetsov S. P.,  Pikovsky A.
We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Smale–Williams solenoid. Finite-dimensional models are derived as ordinary differential equations for amplitudes of spatial Fourier modes (the 5D and 7D models). Correspondence of the reduced models to the original system is demonstrated numerically. Computational verification of the hyperbolicity criterion is performed for the reduced models: the distribution of angles of intersection for stable and unstable manifolds on the attractor is separated from zero, i.e., the touches are excluded. The example considered gives a partial justification for the old hopes that the chaotic behavior of autonomous distributed systems may be associated with uniformly hyperbolic attractors.
Keywords: Smale–Williams solenoid, hyperbolic attractor, chaos, Swift–Hohenberg equation, Lyapunov exponent
Citation: Kruglov V. P.,  Kuznetsov S. P.,  Pikovsky A., Attractor of Smale–Williams Type in an Autonomous Distributed System, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 483-494
Gonchenko S. V.,  Ovsyannikov I. I.,  Tatjer J. C.
It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.
Keywords: Homoclinic tangency, rescaling, 3D Hénon map, bifurcation, Lorenz-like attractor
Citation: Gonchenko S. V.,  Ovsyannikov I. I.,  Tatjer J. C., Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 495-505
Grines V. Z.,  Levchenko Y. A.,  Medvedev V. S.,  Pochinka O. V.
We prove that each structurally stable diffeomorphism $f$ on a closed 3-manifold $M^3$ with a two-dimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.
Keywords: structural stability, surface basic set, partial hyperbolicity, dynamical coherence
Citation: Grines V. Z.,  Levchenko Y. A.,  Medvedev V. S.,  Pochinka O. V., On the Dynamical Coherence of Structurally Stable 3-diffeomorphisms, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 506-512
Turaev D. V.,  Warner C.,  Zelik S.
A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the three-dimensional space is considered. It is shown that the chaotic behavior of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound.
Keywords: delayed equation, invariant manifold, normal hyperbolicity, billiard
Citation: Turaev D. V.,  Warner C.,  Zelik S., Energy Growth for a Nonlinear Oscillator Coupled to a Monochromatic Wave, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 513-522

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