An efficient algorithm for the evaluation of master stability function in networks of coupled oscillators

Electronics Dept., Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129, Torino, ItalyEmail:{marco.righero, fernando.corinto, mario.biey}@polito.itAbstract—An eﬃcient methodology to study sta-ble in-phase synchronization in networks of identicalnonlinear oscillators is proposed. The problem of in-vestigating synchronization properties is reduced to aneigenvalue problem by means of the joint applicationof the Master Stability Function and the HarmonicBalance technique. The proposed method permits toachieve huge reduction in computational time respectto traditional time-domain approaches. In addition,such method can be extended to study synchronizationpatterns in networks of nonlinear oscillators describedby diﬀerential-algebraic equations.1. IntroductionThe existence of (locally) stable synchronous statesin networks of coupled nonlinear oscillators is mainlyshown by computing the spectrum of Lyapunov Expo-nents (LE). Unfortunately, speciallywhen dealing withcontinuous time high-order systems, such a computa-tion requires long CPU time and may show numericalinstabilities [1–3]. In 1998, Pecora and Caroll [4] proposed a technique,subject to some constraints, to simplify this task. Theproblem of synchrony detection was split in two parts:one related to network topology and the other request-ing the computation of LE of the single (generally low-order) uncoupled oscillator. This second part requiresthe evaluation of the so called Master Stability Func-tion (MSF) [4]. Even if in this case we have to dealwith low ordernonlinearoscillators, the computationaleﬀort remains important because steady-state periodicsolutionsofthe uncoupledoscillatorarerequired. Suchsolutions may be obtained by means of time-domainmethods by discarding transient behavior. On theother hand spectral methods (for instance HarmonicBalance - (HB)) provide an accurate approximation ofsteady-state periodic oscillations in nonlinear oscilla-tors [5]. The main aim of this paper is to present an eﬃcientmethod, based on the joint use of HB and MSF tech-niques, in order to evaluate synchronization propertiesin networks composed of coupled identical nonlinearoscillators with at least a stable limit cycle. In par-ticular, the HB-based method allows one to identifylimit cycles and, beyond the reduction in CPU time,makes it possible to investigate nonlinear oscillatorsdescribed by implicit diﬀerential equations [6]. The manuscript is organized as follows. In SectionII, a brief summary of the MSF approach is presentedand in Section III the new algorithm is described indetails. The impressive reduction in CPU time is pre-sented through examples in Section IV. Some conclu-sions are drawn at the end of the paper.2. The Master Stability FunctionThe Master Stability Function permits to study syn-chronization conditions for networks of coupled non-linear systems [4]. We summarize the main ideas topoint out how spectral methods can be successfullyused toconceiveeﬃcient algorithmsforevaluatingsyn-chronization on limit cycles.We consider networks of N cells described by themodel (n = 1,.,N)x˙