Collective modes in parametrically excited oscillator arrays

– We consider a population of parametrically excited globally coupled oscillators in a weakly nonlinear state. The instabilities of collective modes lead to a traveling-wave regime, where intensities of oscillations of each oscillator vary periodically in time. For large excitation amplitudes a frozen state with nearly uniform oscillation intensities is observed. Ensembles of coupled oscillators demonstrate extremely rich behaviors [1–3]. They appear in descriptions of Josephson junctions [4], multimode lasers [5], and charge density waves [6]. In the living world one uses similar models to describe chirps of grasshoppers [7], neurons [8–10] and yeast cells [11]. One of the most interesting effect in these ensembles is the appearance of collective modes, when at least a part of the oscillators is synchronized and the mean field exhibits nontrivial dynamics. Probably, the most impressive demonstration of this is a rhythmic flashing of fireflies [12]. In the theory, the onset of collective behavior is known as the Kuramoto transition [1], and it is a prominent example of nonequilibrium phase transitions. Recently, populations of parametrically excited oscillators attracted particular interest [13– 16]. One possible realization of such a system is an array of Josephson junctions [17]; below we also describe a simple mechanical example of such an ensemble. In [13–15] a linear analysis of instabilities in the ensemble has been performed in a model where the parametric modulation is a piecewise function of time. Parametric excitation by a sinusoidal field in a linear chain of overdamped oscillators was described in [16]. In this paper we study instabilities and collective modes in globally coupled weakly nonlinear oscillators parametrically excited by a sinusoidal signal. Using the method of averaging, we obtain equations for slowly varying amplitudes. The analysis of these equations allows us to find linear instabilities as well as to analyse nonlinear modes developing from these instabilities. The analytical results are confirmed by numerical experiments. As a prototypic model we consider a population of N coupled weakly nonlinear oscillators subject to a parametric excitation. The governing equations are ẍi + 2γẋi + ω 0(1 + ξi(t))xi + T (xi, ẋi) = − κ N N ∑ j=1 (xi − xj)− 2σ N N ∑