Hierarchical Abstraction of Phase Response Curves of Synchronized Systems of Coupled Oscillators

We prove that a group of injection-locked oscillators, each modelled using a nonlinear phase macromodel, responds as a single oscillator to small external perturbations. More precisely, we show that any group of injection-locked oscillators has a single effective PRC [1] or PPV [2], [3] that characterises its phase/timing response to small external perturbations. This result constitutes a foundation for understanding and predicting synchronization/timing hierarchically in large, complex systems that arise in nature and engineering. I. PRC/PPV PHASE MACROMODELS Given an ODE or DAE description d dt ~q(~x(t)) +~f (~x)+~b(t) =~0 (1) of an oscillator with an orbitally stable T -periodic autonomous solution ~xs(t), it can be shown [2], [4] that the timing jitter or phase characteristics of the oscillator, under the influence of small perturbations~b(t), can be captured by the nonlinear scalar differential equation d dt α(t) =~v1 (t +α(t)) ·~b(t), (2) where the quantity ~v1(·), a T -periodic function of time, is known as the Phase Response Curve (PRC) [1] or Perturbation Projection Vector (PPV) [2], [3]. For convenience, we scale the time axis to normalize all periods to 1. Define a 1-periodic version of the steady state solution to be ~xp(t) =~xs(tT ), (3) and a 1-periodic version of the PPV to be ~p(t) =~v1(tT ). (4) Using these 1-periodic quantities and defining f , 1 T , (2) can be expressed as d dt α(t) = ~pT ( f t + f α(t)) ·~b(t). (5) Defining phase to be φ(t) = f t + f α(t), (6) (5) becomes d dt φ(t) = f + f~pT (φ(t)) ·~b(t). (7) x(t), the solution of (1), can often be approximated usefully by a phase-shifted version of its unperturbed periodic solution, i.e., ~x(t)≃~xs(T φ(t)) =~xp(φ(t)). (8) (2) (or equivalently, (7)) is termed the PPV equation or PPV phase macromodel. In the absence of any perturbation~b(t), note that α(t)≡ 0 (w.l.o.g),~x(t) =~xs(t) =~xp( f t) and φ(t) = f t. We will call the latter the phase of natural oscillation and denote it by φ⋄(t), f t. II. DERIVATION OF HIERARCHICAL PPV MACROMODEL A. Coupled Phase System and its Properties 1) Coupled system of PPV phase macromodels: Consider a group of N ≥ 2 coupled oscillators (Figure 1). We model each oscillator by its PPV equation (7): d dt φi(t) = fi + fi~pi (φi(t)) ·~bi(t), i = 1, · · · ,N, (9) Fig. 1. Oscillator system with internal coupling and external inputs. where i-subscripted quantities refer to the ith oscillator. Inputs to each oscillator are drawn from two sources (as depicted in Figure 1): 1) internal couplings with other oscillators, and 2) external sources.~bi(t) can therefore be written as ~bi(t) =~ai(t)+ N ∑ j 6=i j=1 ~bi j ( φ j(t) ) , (10) where ~ai(t) is the external input (i.e., from outside the group of N oscillators) to the ith oscillator, and~bi j ( φ j(t) ) represents the influence of the jth oscillator on the ith. We make the natural assumption that the ~bi j(·) are 1-periodic — i.e., that each oscillator generates outputs that follow its own phase and timing properties; it is these outputs that couple internally to the inputs of other oscillators. Note that as i varies, the dimensions of ~pi(t), ~ai(t) and ~bi j can differ, since they depend on the size of the ith oscillator’s differential equations. The system of N equations (9) can be written in vector ODE form as d dt ~φ(t) =~gφ ( ~φ(t) ) +~bφ ( ~φ(t), t ) , (11)