Abstract
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 2πq/N, are equilibrium points, where q is an integer. Their stability in the limit N → ∞ is discussed along the line of Wiley et al. [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 2πq/N in one sector of the ring, −2πq/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 → ∞.
Received Sun Oct 09 00:00:00 UTC 2011
Accepted Tue Dec 27 00:00:00 UTC 2011
Published online Thu Feb 02 00:00:00 UTC 2012
Lead Paragraph:
Oscillators are important dynamical systems. They are ubiquitous in biology, and they are used in many man-made devices. Networks of coupled oscillators are also frequently found in biology, and they start being used in technology.^{4} ^{4} A fundamental question is whether such oscillators synchronize, i.e., whether or not they oscillate at the same frequency. The simplest model for an oscillator is a pair of first order ordinary differential equations, with suitable parameters to ensure oscillatory behavior. A number of them can be aggregated into a network, usually by coupling through one of their dynamical (state-) variables. If the coupling is weak, then the orbit of each oscillator is perturbed only very slightly, whereas its phase is very sensitive to the coupling. Therefore, under weak coupling, the dynamical behavior of the network of oscillators is reproduced very well by a system of differential equations for the phases only, one per oscillator, which is a substantial simplification. The phase of each oscillator, when decoupled from the other oscillators, increases linearly, i.e., its angular frequency is constant. After averaging, the coupling between two oscillators usually obeys the form of a 2π-periodic function of their phase differences. The best known network of phase oscillators is the so-called Kuramoto model where the oscillators are coupled through the sinus of the phase differences in a homogeneous all-to-all fashion. The network studied in this paper differs from the Kuramoto model in three aspects: the coupling is not all-to-all, the coupling function has the opposite sign, and all oscillators have the same free-running frequency. The Noscillators in our network are disposed on a ring and interaction takes place among the L-nearest neighbor oscillators on the ring. The interaction function between two oscillators is proportional to the sinus of the phase difference, with a sign that makes the interaction repulsive for small phase differences. Note that, in the Kuramoto model, the interaction for small phase differences is attractive. Finally, the free-running angular frequencies are all the same, say ω, whereas in the Kuramoto model, they are generally different, but concentrated around a mean frequency. If they were also identical in the Kuramoto model, all oscillators would synchronize and oscillate together stably with angular frequency ω and identical phases. Our system has N dynamical regimes where all oscillators have identical angular frequency and constant phase differences 2πq/N between neighbors on the ring, the so-called q-twisted states. The sign of the interaction in the system does not matter in this respect. It makes a big difference, however, when the stability of the q-twisted states are considered. For attractive coupling, this question is addressed in Ref. 1. There, a rigorous stability analysis of the q-twisted states in the limit as N → ∞ is carried out. We have applied the same type of analysis in our case of repulsive coupling. As a result, we obtain an interval for qr where the infinite N stability conditions are satisfied, and this implies that, for each value of qr in this interval, we can find a sufficiently large N and an L ≈ (rN − 1)/2, such that the q-twisted state in this finite size system is stable. This gives a whole family of stable q-twisted states. Even though, when N is finite, we can only guarantee their stability for large enough N; numerical simulations show that they are also stable even for moderate values of N. In addition to this principal family of stable q-twisted states, there are small islands in (r,q)-parameter plane where the infinite N system and, therefore, also the finite N system for large enough N have stable q-twisted states. Finally, the existence of additional stable dynamical regimes, where the oscillators also have the common angular frequency, but where the phase differences between neighboring oscillators are not uniform around the ring, is shown by numerical simulations. In fact, in one sector of the ring, the phase differences between neighboring oscillators are approximately 2πq/N; in another sector, it is −2πq/N, and in between, it has intermediate values. There is some interesting interpretation of these solutions from the point of view of dynamical systems theory. In particular, our simulations indicate the presence of spatialchaos as N → ∞.
Acknowledgments:
The authors thank M. Wolfrum, B. Fiedler, and A. Pikovskiy for illuminating discussions. T. Girnyk and Yu. Maistrenko thank the Laboratory of Nonlinear Systems, EPFL for hospitality during their stay there.
Article outline:
I. INTRODUCTION
II. MODEL
III. STABILITY ANALYSIS OF THE TWISTED STATES FOR N → ∞
IV. STABILITY ANALYSIS OF THE TWISTED STATES FOR FINITE N
V. COMPARISON OF ANALYTICAL AND NUMERICAL RESULTS
VI. MULTI-TWISTED STATES
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