Abstract:
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.
This work was supported by RFBR grant No 11-02-91334 and DFG grant No PI 220/14-1.
V. P.K. acknowledges support from DAAD in the framework of the program Forschungsstipendien
f¨ur Doktoranden und Nachwuchswissenschaftler.
Citation:
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, Arkady Pikovsky, “Attractor of Smale–Williams Type in an Autonomous Distributed System”, Regul. Chaotic Dyn., 19:4 (2014), 483–494
\Bibitem{KruKuzPik14}
\by Vyacheslav~P.~Kruglov, Sergey~P.~Kuznetsov, Arkady~Pikovsky
\paper Attractor of Smale–Williams Type in an Autonomous Distributed System
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 4
\pages 483--494
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\crossref{https://doi.org/10.1134/S1560354714040042}
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Linking options:
https://www.mathnet.ru/eng/rcd175
https://www.mathnet.ru/eng/rcd/v19/i4/p483
This publication is cited in the following 10 articles:
Vyacheslav Kruglov, Igor Sataev, “On hyperbolic attractors in a modified complex Shimizu–Morioka system”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 33:6 (2023)
Kruglov V.P. Kuptsov V P., “Theoretical Models of Physical Systems With Rough Chaos”, Izv. Vyss. Uchebn. Zaved.-Prikl. Nelineynaya Din., 29:1 (2021), 35–77
Kuznetsov S.P., Kruglov V.P., Sataev I.R., “Smale-Williams Solenoids in Autonomous System With Saddle Equilibrium”, Chaos, 31:1 (2021), 013140
S. P. Kuznetsov, “Some Lattice Models with Hyperbolic Chaotic Attractors”, Rus. J. Nonlin. Dyn., 16:1 (2020), 13–21
S. P. Kuznetsov, “Generation of Robust Hyperbolic Chaos in CNN”, Rus. J. Nonlin. Dyn., 15:2 (2019), 109–124
Stavros Anastassiou, Anastasios Bountis, Arnd Bäcker, “Recent Results on the Dynamics of Higher-dimensional Hénon Maps”, Regul. Chaotic Dyn., 23:2 (2018), 161–177
S. P. Kuznetsov, V. P. Kruglov, “On some simple examples of mechanical systems with hyperbolic chaos”, Proc. Steklov Inst. Math., 297 (2017), 208–234
D. Angeli, M. A. Corticelli, A. Fichera, A. Pagano, “Features of a blue-sky transition in an autonomous convective flow”, Int. Commun. Heat Mass Transf., 88 (2017), 139–147
A. Yu. Zhalnin, “Ot kvazigarmonicheskikh ostsillyatsii k neironnym spaikam i berstam: raznoobrazie rezhimov giperbolicheskogo khaosa na osnove attraktora Smeila – Vilyamsa”, Nelineinaya dinam., 12:1 (2016), 53–73
V. P. Kruglov, A. S. Kuznetsov, S. P. Kuznetsov, “Giperbolicheskii khaos v sistemakh s parametricheskim vozbuzhdeniem patternov stoyachikh voln”, Nelineinaya dinam., 10:3 (2014), 265–277
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