Regular and Chaotic Dynamics

, Volume 19, Issue 4, pp 483–494 | Cite as

Attractor of Smale - Williams type in an autonomous distributed system

  • Vyacheslav P. KruglovEmail author
  • Sergey P. Kuznetsov
  • Arkady Pikovsky


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 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.


Smale -Williams solenoid hyperbolic attractor chaos Swift -Hohenberg equation Lyapunov exponent 

MSC2010 numbers

37D45 37D20 35B36 


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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Vyacheslav P. Kruglov
    • 1
    • 2
    Email author
  • Sergey P. Kuznetsov
    • 1
    • 3
  • Arkady Pikovsky
    • 2
  1. 1.Saratov State UniversitySaratovRussia
  2. 2.Department of Physics and AstronomyUniversity of PotsdamPotsdam-GolmGermany
  3. 3.Saratov BranchKotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratovRussia

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