Research

A mutually synchronized network has no reference clock and therefore no single point of failure. Real infrastructure still has to be steered onto an external time standard, so we asked what happens when a reference is injected into one node only. Three 24 GHz oscillators were coupled in a ring and in a chain, forced by an external oscillator, and the range of reference frequencies the network still follows was measured. The measured entrainment ranges agree with the nonlinear model, which means a flat-hierarchy network can be locked to an outside standard without giving up the robustness that made it attractive.

Making integrated circuits spreads their components. In a clock network that means detuned intrinsic frequencies and unequal coupling strengths, delays and filter cutoffs, and the assumption is usually that this can only hurt. Using a phase model, tested against experiments and circuit-level simulations of delay-coupled phase-locked loops, we find the opposite for networks with a flat hierarchy. Heterogeneity can be used: it buys better perturbation decay rates, it stabilizes synchronized states, and it tunes the phase differences between clocks past anything a homogeneous choice of parameters reaches. That turns a manufacturing tolerance into a design parameter, and the argument is not restricted to electronics.

How far apart can two clocks sit and still synchronize themselves, with no reference between them? From the closed-loop transfer function of two mutually delay-coupled phase-locked loops and the Nyquist criterion we compute the critical delay at which the in-phase and anti-phase states lose stability, and the range of feed-forward loop gains that keeps them stable at a given delay. Measurements at 24 GHz confirm it, for cross-coupling delays from the nanosecond to the microsecond domain, which is far longer than the period of the oscillation itself. The longest of them corresponds to coupling over 500 metres.

Established network synchronization models in electrical engineering study mutual coupling through a linearized coupling function. That is convenient, and it hides the phenomena that appear once the delay is large. Here we review that work and present a general model which keeps the nonlinearity and the finite cross-coupling delay, so that multistability and the delay-induced stabilization of synchronized states can be predicted rather than discovered. We derive for which system parameters a synchronized state is linearly stable. The key finding is that mutual synchronization yields stable in-phase and anti-phase states even at large delays, with an explicit condition under which stability is guaranteed. This is the reference paper of the line of work below, and the one to read first.

Oscillators with an inert response are the normal case, from biology to engineering, and they admit synchronized states whose frequency is not constant. Such states show up as side bands in the spectrum or as chaotic dynamics. Their stability had only been accessible numerically, because the delay makes the problem multistable. We derive criteria and conditions that decide linear stability analytically, directly from the system parameters, for arbitrary network topologies with identical oscillators and identical delays. It is the criterion the later work in this list leans on, and the reason a claim of stability there can be checked rather than simulated.

The Kuramoto model with inertia describes a network of phase-locked loops as well as it describes mechanical rotors, and putting a delay into the interaction changes the transition it undergoes. In the limit of infinitely many oscillators we compute how the synchronized phase bifurcates from the incoherent one, through an unstable-manifold expansion of the kinetic equation for the single-oscillator distribution. The bifurcation can be supercritical or subcritical, and which one occurs is decided jointly by the delay and the inertia. Direct numerical integration at large but finite oscillator number confirms the result and shows the subtleties a finite network adds.

Self-organized synchronization is everywhere in nature and almost nowhere in engineering. Here we take it seriously as a clocking principle for distributed electronics, where a common time reference is what makes an antenna array or a multi-core processor work at all. We develop a nonlinear phase description of mutually coupled digital phase-locked loops that keeps both the filter impulse response and the transmission delay, and it yields analytic expressions for the collective frequency, the stability, and the time scale of synchronization. Filtering turns out to introduce stability transitions that do not exist without it. Experiments on networks of off-the-shelf DPLL chips confirm the predicted existence, frequency and stability quantitatively, which is the part that made the theory usable.

The first paper of this line, and the one that fixed the model. It sets up a phase description of coupled phase-locked loops that includes the filter kernels and the delayed transmission of the signal, and asks whether such a network can clock itself. The central result runs against intuition: the transmission delay is what makes stable synchronized states possible, and PLLs coupled instantaneously do not tend to synchronize at all. Filtering and delay together set the collective frequency and the time scale on which the network locks. Delay stops being the thing you design around here, and becomes the thing you design with.

At sufficiently large cross-coupling delay a pair of phase-locked loops has several synchronized states available to it, with different frequencies and phase relations. Which one it reaches depends on where it starts, and that is multistability. Here the basin of stability of the in-phase and anti-phase states is measured on two 24 GHz nodes, modified so that the transient during synchronization can be recorded, and compared against time-domain simulation. The transients are reproduced qualitatively and quantitatively. Multistability becomes something a designer can predict instead of something a network occasionally does.

The step from a benchtop demonstration to microwave frequency. Two phase-locked loops with 24 GHz voltage-controlled oscillators were coupled bidirectionally, using a PLL architecture built for mutual rather than hierarchical coupling. The existing phase-domain model was extended to include the oscillator's nonlinear response to the tuning signal, after which the frequencies and phase relations of the self-organized states are predicted precisely rather than approximately. Measurements at several delays and division factors agree, and for delays up to 14 ns self-organized synchronization is shown to be feasible at microwave frequencies.

Classical phase-locked loop theory assumes a single input and no delay. This paper generalizes it to include time delays and mutual coupling, for loops of arbitrary order and arbitrary number of inputs, and gives two methods for finding locked states and their transient dynamics. For entrainment the generalized and the classical theory overlap, which is the consistency check. Measurements on a classical PLL entrained by a clock confirm the predicted phase relations, the dependence of the loop gain on component characteristics and delay, and both the decay rate and the decay frequency of a perturbation.

The companion to the ICC paper of the same year, presented at a workshop on biological and multi-scale communication. It makes the analogy explicit that the rest of this work usually leaves implicit: the delay-coupled oscillator mechanism that keeps cells in a developing embryo in step is the same mechanism that can clock an antenna array, and the architecture it suggests is flat and reference-free.

Massive multiple-input multiple-output (MIMO) promises large gains over current wireless standards, and it needs the carrier signals of every antenna unit in a large array to share a phase reference. We show that mutually coupled digital phase-locked loops can deliver in- phase synchronous clocking across such an array in the presence of transmission delay. From the phase model with filtering and delay we work out how the collective frequency and the time scales of synchronization follow from the system specification, and we confirm it experimentally as a proof of principle.

A real coupling delay is not one number. Here we study identical coupled oscillators with a distribution of delay times and separate what the distribution controls from what it does not. For arbitrary network topology, the frequency and the stability of the fully synchronized state depend only on the mean of the delay distribution. The shape does not change the state, but it does change how the system reaches it: in the presence of coupling delays the synchronization rate can be maximal at a particular coupling strength. The question came from vertebrate segmentation rather than from electronics, which is why the result is stated for any topology.

The thesis behind the preprint above, written at the Max Planck Institute for the Physics of Complex Systems and TU Dresden. It develops the phase-oscillator description of coupled systems in which the coupling delay is distributed rather than fixed, and works through what that distribution does to synchronized states. Everything else on this page grew out of it.

Where the delay came in. The thesis treats differential equations that carry a delay and a noise term at once, and asks what can be said about the stability of their solutions and about the statistics those solutions have. Both ingredients stayed: every clock in the work above is delayed, and every real one is noisy.