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Attractor of Smale - Williams type in an autonomous distributed system
- We consider an autonomous system of partial differential equations for a one-dimensional 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 Poincar, 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 withWe consider an autonomous system of partial differential equations for a one-dimensional 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 Poincar, 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.…
Author details: | Vyacheslav P. Kruglov, Sergey P. Kuznetsov, Arkadij PikovskijORCiDGND |
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DOI: | https://doi.org/10.1134/S1560354714040042 |
ISSN: | 1560-3547 |
ISSN: | 1468-4845 |
Title of parent work (English): | Regular and chaotic dynamics : international scientific journal |
Publisher: | Pleiades Publ. |
Place of publishing: | New York |
Publication type: | Article |
Language: | English |
Year of first publication: | 2014 |
Publication year: | 2014 |
Release date: | 2017/03/27 |
Tag: | Lyapunov exponent; Smale - Williams solenoid; Swift - Hohenberg equation; chaos; hyperbolic attractor |
Volume: | 19 |
Issue: | 4 |
Number of pages: | 12 |
First page: | 483 |
Last Page: | 494 |
Funding institution: | RFBR [11-02-91334]; DFG [PI 220/14-1]; DAAD in the framework of the program Forschungsstipendien fur Doktoranden und Nachwuchswissenschaftler |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
Peer review: | Referiert |