TY  - JOUR
A1  - Kruglov, Vyacheslav P.
A1  - Kuznetsov, Sergey P.
A1  - Pikovskij, Arkadij
T1  - Attractor of Smale - Williams type in an autonomous distributed system
JF  - Regular and chaotic dynamics : international scientific journal
N2  - 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 with uniformly hyperbolic attractors.
KW  - Smale - Williams solenoid
KW  - hyperbolic attractor
KW  - chaos
KW  - Swift - Hohenberg equation
KW  - Lyapunov exponent
Y1  - 2014
U6  - https://doi.org/10.1134/S1560354714040042
SN  - 1560-3547
SN  - 1468-4845
VL  - 19
IS  - 4
SP  - 483
EP  - 494
PB  - Pleiades Publ.
CY  - New York
ER  -