PHYSICAL REVIEW E 87, 062917 (2013) Autonomous and forced dynamics of oscillator ensembles with global nonlinear coupling: An experimental study Amirkhan A. Temirbayev, 1 Yerkebulan D. Nalibayev, 1 Zeinulla Zh. Zhanabaev, 1 Vladimir I. Ponomarenko, 2 and Michael Rosenblum 3 1 Physical-Technical Department, al-Farabi Kazakh National University, al-Farabi Avenue 71, 050040 Almaty, Kazakhstan 2 Institute of Radio Engineering and Electronics, Saratov Branch, Russian Academy of Sciences, 38 Zelyonaya Street, Saratov, 410019, Russia 3 Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Strasse 24/25, D-14476 Potsdam-Golm, Germany (Received 18 February 2013; revised manuscript received 14 May 2013; published 25 June 2013) We perform experiments with 72 electronic limit-cycle oscillators, globally coupled via a linear or nonlinear feedback loop. While in the linear case we observe a standard Kuramoto-like synchronization transition, in the nonlinear case, with increase of the coupling strength, we first observe a transition to full synchrony and then a desynchronization transition to a quasiperiodic state. However, in this state the ensemble remains coherent so that the amplitude of the mean field is nonzero, but the frequency of the mean field is larger than frequencies of all oscillators. Next, we analyze effects of common periodic forcing of the linearly or nonlinearly coupled ensemble and demonstrate regimes when the mean field is entrained by the force whereas the oscillators are not. DOI: 10.1103/PhysRevE.87.062917 PACS number(s): 05.45.Xt, 05.65.+b I. INTRODUCTION An ensemble of many interacting oscillatory units is a popular model, widely used for description of collective dynamics of such various objects as lasers and Josephson junctions, spontaneously beating atrial cells and firing and/or bursting neurons, pedestrians on the footbridges and hand- clapping individuals in a large audience, electrochemical oscillators, metronomes, and many others. Quite often the networks of such elements can be approximately considered as fully connected, with the same strength of interaction within each pair of elements. In this case one speaks of the global or mean-field coupling. Analysis of collective behavior of globally coupled systems is not only important for applications but also poses a number of problems which are highly nontrivial from the standpoint of nonlinear dynamics. Due to these reasons, this topic has remained a focus of interest in the past three decades. Basic theory and further references can be found in Refs. [1–3]. The main effect of global coupling is emergence of a collective mode, or mean field, due to synchronization of some or all elements of the population. The degree of the collective synchrony is reflected in the amplitude of the collective mode; this amplitude is often called the order parameter. Typically, the order parameter increases with the interaction strength, if the latter is larger than a certain threshold value. This effect is well understood within the framework of the Kuramoto-Sakaguchi model [4,5] of sine-coupled phase oscillators, which is analyt- ically solvable in the limit case of infinitely large ensemble. The character of the Kuramoto transition from the incoherent state, where the order parameter is zero, to the partially or fully synchronous state with nonzero mean field depends on the distribution of the natural frequencies within the population; this transition can be either smooth [4,6] or abrupt [7]. The described scenario is not universal, however: Consideration of more complicated oscillators and/or general coupling results in such effects as clustering [8], chaotization of the mean field [9,10], and appearance of robust heteroclinic network attractors [8,11]. Another subject of recent interest is partial synchrony in networks of identical integrate-and-fire units, coupled via the so-called α function, imitating the synaptic delay [10,12]. This model exhibits a collective mode that is not synchronized with individual units, while the synchronous state is unstable. A similar regime was numerically observed for a model of active mechanical oscillators, coupled via an inertial load [13]. Coherent but not synchronous dynamics in ensembles of nonlinearly coupled Stuart-Landau oscillators was demonstrated numerically and analyzed theoretically in the framework of phase approximation in Refs. [14–16]. The latter system demonstrates self-organized quasiperiodic dynamics (SOQ); in this state the frequency of the mean field differs from the frequencies of all oscillators and the dependence of the order parameter on the coupling strength is nonmonotonic. Experimental investigation of such regimes is the primary goal of this paper. In spite of the high interest in the field, there are relatively few experimental studies of the dynamics of globally cou- pled systems. Before reviewing these studies, we mention a number of observations of synchronous collective dynamics in systems, where the coupling is assumed to be of the all-to-all type, although it is most likely not homogeneous. This includes observations of the synchronous emission of optical or acoustical pulses by groups of insects [17], rhythmical hand clapping in opera houses [18], glycolytic oscillation in populations of yeast cells [19], etc. A well-known example is pedestrian synchrony on the London Millennium Bridge; the experiments with the pedestrian groups of different sizes demonstrated that collective synchrony is a threshold phe- nomenon [20], in correspondence with the theoretical results for globally coupled oscillators [4,21]. Next, we mention a brilliant demonstration of collective synchrony in a very simple experiment with metronomes, performed within a framework of student research [22]. Well-controlled experiments on arrays of 64 globally coupled electrochemical oscillators verified 062917-1 1539-3755/2013/87(6)/062917(11) ©2013 American Physical Society