On the Dynamics of Coupled Parametrically Forced Oscillators

It is well known that an autonomous dynamical system can have a stable periodic orbit, arising for example through a Hopf bifurcation. When a collection of such oscillators is coupled together, the system can display a number of phase-locked solutions which can be understood in the weak coupling limit by using a phase model. It is also well known that a stable periodic orbit can be found for a parametrically forced dynamical system, with the phase of the periodic orbit being locked to the forcing. Here we discuss the periodic solutions which occur for a collection of such parametrically forced oscillators that are weakly coupled together. INTRODUCTION The scientific study of coupled oscillators started with Christian Huygens’ observations in the seventeenth century of mutual synchronization of pendulum clocks connected by a beam [1,2]. More recently, it has been recognized that mutual synchronization of coupled oscillators the adjustment of rhythms of oscillating objects due to their weak interactions occurs in many biological systems, including neurons during epileptic seizures [3] and pacemaker cells in the human heart [4]. Coupled oscillators have also been studied in detail for technological systems, such as arrays of lasers and superconducting Josephson junctions: see [5], [6], and [7], a recent popular book on the topic, for many biological and technological examples of synchronization for coupled oscillators. We classify as autonomous oscillators those for which the stable oscillations occur for an autonomous dynamical system, that is one for which there are no explicit time-dependent terms in the evolution equation. For example, the oscillations might arise through a Hopf bifurcation, as for the microelectromechanical systems (MEMS) oscillators considered in [8,9]. In the limit of weak coupling, it is possible to reduce the dynamics of coupled autonomous oscillators to a phase model, with a single variable describing the phase of each oscillator with respect to some reference state (see, e.g., [10–13]). This typically leads to models for which the dynamics depend only on the phase differences between different oscillators. It is possible to show that several types of phase-locked solutions, for which the phase of all oscillators increases at the same constant rate, are guaranteed to exist in the weak coupling limit for any generic coupling function when the coupling topology has appropriate symmetry properties [14–17]; for the case of identical all-to-all coupling for N oscillators, these are • in phase solution: all N oscillators have the same phase • two-block solutions: there are two blocks of oscillators, one in which p oscillators share the same phase, and one in which N− p oscillators share the same phase • rotating block solutions: for N = mk, there are m blocks with k oscillators in each block sharing the same phase, with neighboring blocks differing in phase by 2π/m • double rotating block solutions: for N = m(k1+ k2), there are two rotating block solutions, one with m blocks with k1 oscillators in each block sharing the same phase and with neighboring blocks differing in phase by 2π/m, another with m blocks with k2 oscillators in each block sharing the same phase and with neighboring blocks differing in phase by 2π/m, where there is a phase difference 0 < φ < 2π/m between a block with k1 oscillators and the closest phaseadvanced block with k2 oscillators. On the other hand, we classify as non-autonomous oscillators those for which the stable oscillations only occur for a Proceedings of DSCC2008 2008 ASME Dynamic Systems and Control Conference October 20-22, 2008, Ann Arbor, Michigan, USA 1 Copyright © 2008 by ASME DSCC2008-2189 non-autonomous dynamical system, that is one for which there are explicit time-dependent terms such as time-periodic forcing. We will focus on parametrically forced oscillators, which are non-autonomous oscillators for which the forcing enters as a time-varying system parameter. Coupled parametrically forced oscillators arise in MEMS [18–20] and other application areas [21–24], but have not received as much theoretical research attention as coupled autonomous oscillator systems. This paper represents the first steps in developing a comprehensive theory of the dynamics of general weakly coupled non-autonomous oscillators, in the spirit of the theory of general weakly coupled autonomous oscillators described in [14, 17]. We hope that such a theory will ultimately lead to novel sensing mechanisms using MEMS devices; for simplicity, here we will consider a model system which represents only a caricature of such devices. Specifically, in this paper we describe interesting synchronization phenomena that are possible for coupled parametrically forced oscillators. For example, consider two uncoupled oscillators whose response is at half the frequency of the driving voltage, as is common for MEMS devices [25]. Both oscillators could identically lock to the forcing, or they could lock one forcing period apart both situations are allowable due to a discrete time-translation symmetry for the problem. We will show that different combinations of these states will persist if the oscillators are now weakly coupled, with stability inherited from the stability properties of the periodic orbits which exist for the uncoupled system. We first consider the dynamics of a specific single parametrically forced oscillator. We then consider two uncoupled parametrically forced oscillators, identifying different periodic states for such systems. Next, we show that provided the periodic orbits for the uncoupled system are hyperbolic, there will be periodic orbits for the coupled system close to the periodic states identified for the uncoupled system. This is then demonstrated through numerical bifurcation analysis. We finally describe how these results can be generalized to N coupled parametrically forced oscillators. A PARAMETRICALLY FORCED OSCILLATOR Consider the equation for a damped, parametrically forced oscillator ẍ+bẋ+ x+ x = xF cos(ω f t). (1) Here the term bẋ represents damping (we assume b > 0), the term x + x3 represents a nonlinear restoring force, and the term xF cos(ω f t) represents parametric excitation which can be viewed as a time-periodic modulation of the linear part of the restoring force. For this system, if F = 0 then x → 0 as t → ∞, as follows. Letting V (x, ẋ) = 1 2 x2+ 1 4 x4+ 1 2 ẋ2, (2) 1.50 1.75 2.00 2.25 2.50 2.75 -0.25 0.00 0.25 0.50 0.75 1.00