Frequency precision of two-dimensional lattices of coupled oscillators with spiral patterns

Abstract

Two-dimensional lattices of N synchronized oscillators with reactive coupling are considered as high-precision frequency sources in the case where a spiral pattern is formed. The improvement of the frequency precision is shown to be independent of N for large N , unlike the case of purely dissipative coupling where the improvement is proportional to N , but instead depends on just those oscillators in the core of the spiral that acts as the source region of the waves. Our conclusions are based on numerical simulations of up to N = 29 929 oscillators and analytic results for a continuum approximation to the lattice in an infinite system. We derive an expression for the dependence of the frequency precision on the reactive component of the coupling constant, depending on a single parameter given by fitting the frequency of the spiral waves to the numerical simulations.

Building better oscillators using nonlinear dynamics and pattern formation

“…As Winfree anticipated, the result is not this simple in general [26]. In particular, as reviewed subsequently, for oscillator systems such as those based on high-Q resonators where propagation effects are important, the noise reduction and frequency improvement are significantly reduced, and do not in fact scale with the total number of oscillators [3,5]. This is because the synchronized state in these systems consists of phase waves propagating from target or spiral sources, reminiscent of other oscillatory pattern forming systems such as chemical systems.…”

“…Analysing the synchronized states such as shown in figure 4 using eqs (27) and (29) for the generalized coupling function, we find the phase sensitivity vector e † 0 to be 'localized' to a core region of N S oscillators forming the sources [3,5]. The noise reduction ratio (eq.…”

“…We briefly describe the calculations here, focussing on a two-dimensional lattice and using a continuum approximation. A more complete analysis based on an approximation of the interaction function on the discrete lattice rather than the continuum approximation, and results for other systems, can be found in the original papers [3,5].…”

“…The spiral source is not completely captured by the approximation of small phase differences used for the target sources, because at the core, neighbouring oscillators necessarily have large phase differences. The spirals are not predicted by the mapping onto the linear Anderson problem, and we must proceed by a different route to understand the properties [5]. Unlike the target source, spiral sources exist even in the completely uniform lattice, and it is simplest to analyse their behaviour in this limit.…”

“…Our focus is on oscillators built from nanomechanical devices, but the ideas apply generally. This paper is a summary of work published in a number of papers [1][2][3][4][5][6]; more details can be found in those publications. There is also experimental work supporting some of the results [7,8].…”