Bifurcation analysis of multistability of synchronous states in the system of two delay-coupled oscillators

Abstract

Systems of mutually coupled oscillators with delay coupling are of great interest for various applications in electronics, laser physics, biophysics, etc. Time delay usually originates from the finite speed of propagation of the coupling signal. In this paper, we present the results of detailed bifurcation analysis of two delay-coupled limit-cycle (Landau–Stuart) oscillators. First, we study the simplified case when the delay time is much smaller than the oscillation build-up time. When the coupling signal propagates between the two counterparts, it acquires a phase shift, which strongly affects the synchronization pattern. Depending on this phase shift, the system may demonstrate the behavior typical for either dissipative or conservative (reactive) coupling. We examine stability of the in-phase and anti-phase synchronous states and reveal the complicated pattern of the synchronization domains on the frequency mismatch—coupling strength parameter plane paying a special attention to the mechanisms of appearance and disappearance of the phase multistability. We demonstrate that taking into account reactive phase nonlinearity the coupling signal acquires an additional phase shift, which depends on the signal intensity. We also examine the more complicated case of finite delay time. The increase of the reactive nonlinearity parameter and the delay time leads to transformations of synchronization domains similar to those that occur when the phase shift increases. For the bifurcation analysis, we employ XPPAUT and DDEBifTool package and verify the results by direct numerical integration.

Theory of Peer-to-Peer Locking of High-Power Gyrotron Oscillators Coupled with Delay

“…Following [25][26][27][28], consider the situation when the normalized delay time is small, i.e., τ d << 1. In that case, we can neglect the delay in the right-hand sides of (10), i.e., A 1,2 (τ − τ d ) ≈ A 1,2 (τ), and obtain the system of ordinary differential equations (ODEs):…”

“…For a more rigorous analysis, we investigate the third-order ODE system (12) employing the XPPAUT public-domain software [33] that is a powerful tool for bifurcation analysis of ODEs. In [26], we applied XPPAUT for a detailed analysis of peer-to-peer locking of two coupled Landau-Stuart oscillators with a weak cubic nonlinearity. We revealed a complicated structure of locking domains on the ∆, ρ-plane and intriguing scenarios of transition to the phase-locked modes.…”

“…The results obtained by the simulation of the DDE model (10) are in a rather good agreement with the results obtained in the small-delay approximation, as shown in Figure 4b. However, in the case of finite delay, there appears an additional phase shift ωτ d , where ω is the oscillation frequency [25,26]. Thus, the phase advance ψ should be adjusted to obtain a good agreement.…”

“…In [27,28], we proposed a simple model, which extends the analysis presented in [25,26] to the peer-to-peer locking of gyrotrons. This model allows for a comprehensive theoretical analysis of peer-to-peer locking regimes, including the use of modern software tools for the bifurcation analysis of nonlinear dynamical systems.…”

“…Since at microwave frequencies, the distance between coupled oscillators significantly exceeds the operating wavelength, the delay of the signal propagating between the oscillators should be taken into account, which greatly increases the complexity of the system. In [25,26], we studied a simple model of two delay-coupled oscillators in the form of two coupled delay-differential equations (DDEs). Due to the delay, the coupling signal acquires a phase shift that strongly affects the locking process.…”