Higher-order phase reduction for delay-coupled oscillators beyond the phase-shift approximation

TL;DR

The paper develops a higher-order phase reduction for delay-coupled oscillators by recasting delay differential equations as an ODE with a transport equation for history, then solving conjugacy and homological equations order by order in the coupling strength ε. At first order, the delay enters as a phase shift, recovering the conventional phase-shift approximation; at second order, delays influence the phase dynamics in a more intricate way, enabling phenomena such as bistability between synchronization patterns. The authors provide a constructive algorithm to compute f^{(ε)}, e^{(ε)}, and E^{(ε)} to arbitrary order, and demonstrate the method on a two-oscillator Stuart–Landau system, showing bistability and close agreement with full DDE simulations. The framework yields finite-dimensional phase equations on the invariant torus that capture global phase dynamics and facilitate analysis of synchronization, basins of attraction, and nontrivial phase-locked states arising from delay. This approach offers a principled route to higher-order delay effects and can be extended to larger networks and other oscillatory systems, with potential computational implementations for motif-based scaling.

Abstract

Network interactions between dynamical units are often subject to time delay. We develop a phase reduction method for delay-coupled oscillator networks. The method is based on rewriting the delay-differential equation as an ordinary differential equation coupled with a transport equation, expanding in the coupling strength, and solving the resulting equations order-by-order. This approach yields an approximation of the finite-dimensional phase dynamics to arbitrary order. While in the first-order approximation the time delay acts as a phase shift as expected, the higher-order phase reduction generally displays a less trivial dependence on the delay. In particular, exploiting second-order phase reduction, we prove the existence of a region of bistability in the synchronization dynamics of two delay-coupled Stuart-Landau oscillators.

Figures (1)

• Figure 1: The bifurcation curves of the one-dimensional phase reduced equations \ref{['eq:psi_ord2']}---computed both analytically (light colored) and numerically (dark colored)---reflect the infinite-dimensional dynamics of the delay-coupled Stuart--Landau oscillators \ref{['eq:sl_coupled']} for $\varepsilon=0.1$. The background color corresponds to $\Psi = \arg(z_1(T))-\arg(z_2(T))$ for numerical solutions $z(t)$ of the DDE after $T=1000$ time units, for initial conditions chosen with a uniformly random phase difference $\Psi(0)$; blue corresponds to in-phase synchrony ($\Psi=0$), red corresponds to anti-phase synchrony ($\Psi=\pi$). The top panel shows the analytic approximation of the bifurcation lines of in-phase synchrony $\psi=0$ (dark blue line) and anti-phase synchrony $\psi=\pi$ (dark red line) predicted from the phase-reduced dynamics \ref{['eq:psi_ord2']} in Proposition \ref{['prop:expansions']}. For comparison, the bifurcation lines obtained by numerically solving the bifurcation equations \ref{['eq:lambdasync']}--\ref{['eq:lambdasplay']} for in-phase synchrony (blue line) and anti-phase synchrony (red line) only deviate slightly from the analytical approximations. The bottom panel shows the numerically computed bifurcation lines for a larger range of time delays. These indicate that the second-order phase reduction gives a reasonable approximation of the infinite-dimensional delay equation even for larger delays.

Theorems & Definitions (29)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 1
• Lemma 2.5
• proof : Outline of the proof