TY  - JOUR
A1  - Pikovskij, Arkadij
T1  - Chimeras on a social-type network
JF  - Mathematical modelling of natural phenomena : MMNP
N2  - We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.
KW  - Network
KW  - Chimera
KW  - correlations
KW  - Lyapunov exponent
Y1  - 2021
U6  - https://doi.org/10.1051/mmnp/2021012
SN  - 0973-5348
SN  - 1760-6101
VL  - 16
PB  - EDP Sciences
CY  - Les Ulis
ER  - 
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  - 
TY  - JOUR
A1  - Mulansky, Mario
A1  - Ahnert, Karsten
A1  - Pikovskij, Arkadij
A1  - Shepelyansky, Dima L.
T1  - Strong and weak chaos in weakly nonintegrable many-body hamiltonian systems
JF  - Journal of statistical physics
N2  - We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.
KW  - Lyapunov exponent
KW  - Arnold diffusion
KW  - Chaos spreading
Y1  - 2011
U6  - https://doi.org/10.1007/s10955-011-0335-3
SN  - 0022-4715
VL  - 145
IS  - 5
SP  - 1256
EP  - 1274
PB  - Springer
CY  - New York
ER  -