A ring of N identical phase oscillators with interactions between L -nearest neighbors is considered, where L ranges from 1 (local coupling) to N /2 (global coupling). The coupling function is a simple sinusoid, as in the Kuramoto model, but with a minus sign which has a profound influence on its behavior. Without the limitation of the generality, the frequency of the free-running oscillators can be set to zero. The resulting system is of gradient type, and therefore, all its solutions converge to an equilibrium point. All so-called q -twisted states, where the phase difference between neighboring oscillators on the ring is 2 πq / N , are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley (Chaos 16 , 015103 (2006)) In addition, we prove that when a twisted state is asymptotically stable for the infinite system, it is also asymptotically stable for sufficiently large N . Note that for smaller N , the same q -twisted states may become unstable and other q -twisted states may become stable. Finally, the existence of additional equilibrium states, called here multi-twisted states, is shown by numerical simulation. The phase difference between neighboring oscillators is approximately 2 πq / N in one sector of the ring, -2 πq / N in another sector, and it has intermediate values between the two sectors. Our numerical investigation suggests that the number of different stable multi-twisted states grows exponentially as N → ∞ . It is possible to interpret the equilibrium points of the coupled phase oscillator network as trajectories of a discrete-time translational dynamical system where the space-variable (position on the ring) plays the role of time. The q -twisted states are then fixed points, and the multi-twisted states are periodic solutions of period N that are close to a heteroclinic cycle. Due to the apparently exponentially fast growing number of such stable periodic solutions, the system shows spatial chaos as N → ∞ .
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Multistability of twisted states in non-locally coupled Kuramoto-type models
Girnyk, Taras; Hasler, Martin; Maistrenko, Yuriy
Follow Chaos: An Interdisciplinary Journal of Nonlinear Science
, Volume 22 (1)
American Institute of Physics – Mar 1, 2012
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