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Register a new userHamiltonian control of Kuramoto oscillators
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Many coordination phenomena are based on a synchronisation process, whose global behaviour emerges from the interactions among the individual parts. Often in Nature, such self-organising mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are however cases where synchronisation acts against the stability of the system; for instance in the case of engineered structures, resonances among sub parts can destabilise the whole system. In this Letter we propose an innovative control method to tackle the synchronisation process based on the use of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronisation. We present our results on the paradigmatic Kuramoto model but the applicability domain is far more large.
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We show that an introduction of a phase parameter ($alpha$), with $0 le alpha le pi/2$, in the interlayer coupling terms of multiplex networks of Kuramoto oscillators can induce explosive synchronization (ES) in the multiplexed layers. Along with the {alpha} values, the hysteresis width is determined by the interlayer coupling strength and the frequency mismatch between the mirror (inter-connected) nodes. A mean-field analysis is performed to support the numerical results. Similar to the earlier works, we find that the suppression of synchronization is accountable for the origin of ES. The robustness of ES against changes in the network topology and frequency distribution is tested. Finally, taking a suggestion from the synchronized state of the multiplex networks, we extend the results to the classical concept of the single-layer networks in which some specific links are assigned a phase-shifted coupling. Different methods have been introduced in the past years to incite ES in coupled oscillators; our results indicate that a phase-shifted coupling can also be one such method to achieve ES.
We study the effects of Janus oscillators in a system of phase oscillators in which the coupling constants take both positive and negative values. Janus oscillators may also form a cluster when the other ones are ordered and we calculate numerically the traveling speed of three clusters emerging in the system and average separations between them as well as the order parameters for three groups of oscillators, as the coupling constants and the fractions of positive and Janus oscillators are varied. An expression explaining the dependence of the traveling speed on these parameters is obtained and observed to fit well the numerical data. With the help of this, we describe how Janus oscillators affect the traveling of the clusters in the system.
The Kuramoto-Sakaguchi model for coupled phase oscillators with phase-frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the collective dynamics of finite size Kuramoto-Sakaguchi models. We find that the inclusion of the effects of rogue oscillators is essential to obtain an accurate description, in contrast to the original Kuramoto model where we show that their effects can be ignored. We further introduce a more accurate ansatz function to describe the shape of synchronized oscillators. Our results from this extended collective coordinate approach reduce in the thermodynamic limit to the well-known mean-field consistency relations. For finite networks we show that our model reduction describes the collective behavior accurately, reproducing the order parameter, the mean frequency of the synchronized cluster, and the size of the cluster at given coupling strength, as well as the critical coupling strength for partial and for global synchronization.
We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit $N to infty$, is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large $N$, and highlight the subtleties arising from having a finite number of oscillators.
We study the global bifurcations of frequency weighted Kuramoto model in low-dimension for network of fully connected oscillators. To study the effect of non-zero-centered frequency distribution, we consider two symmetric Lorentzians as an example. We derive the stability diagram of the system and show that the infinite-dimensional problem reduces to a flow in four dimensions. Using the system symmetries, it can be further reduced to two dimensions. Using this analytic framework, we obtain bifurcation boundaries of the system, which is compatible with our numeric simulations. We show that the system has three types of transitions to synchronized state for different parameters of the frequency distribution: (1) a two-step transition, representative of standing waves, (2) a continuous transition, as in the classical Kuramoto model, and (3) a first-order transition with hysteresis. Numerical simulations are also conducted to confirm analytic results.
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