Using Hamiltonian control to desynchronize Kuramoto oscillators

Abstract

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.PACS numbers: 05.45. Xt, 02.30.Yy, 45.20.Jj Synchronisation is one of the most important example of collective behaviour in Nature, being at the basis of many processes in living beings [1][2][3]. Indeed, their activity is regulated by (almost) periodic processes of different duration that must run at unison to determine a collective behaviour [4,5] enable to sustain life. For this reason synchronisation is widespread and it has been studied in many research domains such as biology (flashing of fireflies [6] and the cricket chirping [7] during the mating season), chemistry (glycolytic oscillations in populations of yeast cells [8]), physics (arrays of coupled lasers [9] and the superconducting Josephson junctions [10]), just to mention few of them. One of the most representative case being the heart [11], an organ of vital importance for all species in the animal kingdom; the heart is composed by a collection of individual cells, myocytes, whose complex interactions among them are responsible for the ability to pump blood in the circulatory system. Initially, at embryonic state, such cells do not interact each other and their beats are independent, only after a couple of days, the myocytes form interconnected sheets of cells that help them beat in unison. The absence of such synchronisation phenomenon in human induces cardiac arrhythmia and artificial pacemakers are necessary to recover the normal behaviour.Despite the very different nature of the systems exhibiting synchronisation phenomena, most of the main features are quite universal and can thus be described using the paradigmatic Kuramoto model (KM) [12][13][14][15] of coupled non-linear oscillators. Once the N oscillators are set on top of a complex network [16], the KM can be described byφwhere K is the interaction strength, ω i are the natural frequencies of the oscillators drawn from some distribu- Let us observe that there are cases where such collective rhythm has a negative impact on the organism life [17]; for instance, it has been observed, that certain psychomotor symptoms, e.g. tremors, are results of an abnormal synchronisation phenomenon in the activity of the responsible neuronal zo...

Decentralized Heading Control With Rate Constraints Using Pulse-Coupled Oscillators

“…, both of which may be time-varying, cause the phase update rule in (15) to be strictly increasing over the entire interval [0, 2π) for all time. Remark 3.…”

“…As shown previously, for a robotic network following the control given in (2), the effect of the heading rate constraint to the standard PRF in (9) can be modeled as a PRF in the form of (14) such that the forward coupling function and the backward coupling function cause the phase update rule in (15) to be strictly increasing over the entire interval [0, 2π) at each firing instance. Thus, we can present the following main theoretical result.…”

“…Remark 7. To achieve heading desynchronization, each robot must change its heading at the same rate, ω max , so that the phase update rule in (15) is strictly increasing. If non-identical robots with different maximum rates of rotation are used in the same network, then the network must use some rotation rate that is no larger than the smallest maximum rate.…”

“…In this paper, we will consider an approach to achieve heading control in a decentralized network with a constraint on the rate of heading change using hybrid dynamics inspired by oscillator networks. Recent research on complex oscillator networks, such as [15], [16], has studied their collective behavior and the ability to control populations of oscillators. However, those results require continuous coupling between oscillators, and do not lend themselves well to the discrete nature of communication in robotic networks.…”

Dynamical phase transitions in generalized Kuramoto model with distributed Sakaguchi phase

“…In this work, We shall use the technique to study the case for annealed disorder only, the case for quenched disorder can be studied later. The main problem in the latter case is that the coupled random variable α ij or α(x, x ′ ) in the system is not a separable function of its arguments in our work, which is the case for most of the studies done so far in analyzing Ott-Antonsen models for disordered Kuramoto systems [75,78,79,80,96,127,128]. So, we proceed with the case of annealed disorder.…”