Synchronization of four coupled van der Pol oscillators

Abstract

It is possible that self-excited vibrations in turbomachine blades synchronize due to elastic coupling through the shaft. The synchronization of four coupled van der Pol oscillators is presented here as a simplified model. For quasilinear oscillations, a stability condition is derived from an analysis based on linearizing the original equation around an unperturbed limit cycle and transforming it into Hill's equation. For the nonlinear case, numerical simulations show the existence of two well-defined regions of phase relationships in parameter space in which a multiplicity of periodic attractors is embedded. The size of these regions strongly depends on the values of the oscillator and coupling constants. For the coupling constant below a critical value, there exists a region in which a diversity of phase-shift attractors is present, whereas for values above the critical value an in-phase attractor is predominant. It is observed that the presence of an anti-phase attractor in the subcritical region is associated with sudden changes in the period of the coupled oscillators. The convergence of the coupled system to a particular periodic attractor is explored using several initial conditions. The study is extended to non-identical oscillators, and it is found that there is synchronization even over a wide range of difference among the oscillator constants.

Synchronization patterns in geometrically frustrated rings of relaxation oscillators

“…The PLA approximation has been used in previous studies to analyze strong relaxation oscillators, such as the Van der Pol oscillator [12] and the Tyson-Fife model [22]. This has proven to be an effective method for studying relaxation oscillators with different coupling mechanisms including diffusive coupling [12] and delay coupling [23], and in systems with two [12], three [24] and four oscillators [25].…”

Theoretical Study of Synchronous Behavior in a Dual‐Pendulum‐Rotor System

“…The so-called synchronization phenomenon refers to self-adjusting different frequencies of oscillating objects to a unified frequency relying on their internal weak couplings [1,2]. In recent years, the study of synchronization phenomenon is involved in the fields of physics, chemistry, and biology, such as the practical application on complex dynamic network systems [3][4][5], nonlinear coupled chaotic systems [6,7], coupled pendulum system [8][9][10][11][12], and rotor system [1,[13][14][15][16][17][18][19]. Nevertheless, the latter two systems can be categorized as synchronous problem of mechanics system in detail.…”

Anti-phase synchronization and symmetry-breaking bifurcation of impulsively coupled oscillators

“…In particular, a special phenomenon in Since the transition of synchronization can be revealed through dynamical analysis, the synchronization of coupled oscillators was always investigated by studying its dynamics, e.g. [14][15][16]. In [14], Rene investigated different dynamical states of synchronization for a ring of four mutually inertia coupled self-sustained electrical systems which were described by the coupled Rayleigh-Duffing equations.…”

“…He also studied the stability properties of periodic solutions and the transition boundaries between different dynamical states by using the Floquet theory. The dynamical behaviors and synchronization of a ring of mutually coupled Van der Pol oscillators were studied by Barrón and Sen [15]. Later on, Perlikowski et al [16] investigated the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators which indicated that although the individual uncoupled oscillator has one globally stable equilibrium, the response of the coupled oscillators could evolve into periodic, quasi-periodic, and chaotic motions when the coupling strength increases.…”