Bifurcation delay, the postponement of instability during slow parameter variation, is a generic feature of nonautonomous dynamical systems. We study how diffusive coupling, unequal time scales, and communication lags alter this delay in pairs of autonomous and nonautonomous oscillators, using an ecological canard model and the FitzHugh-Nagumo neuron model. The delay depends on the dynamical state of the autonomous oscillator: coupling to a stable autonomous oscillator increases it linearly with the coupling strength, while coupling to an oscillatory autonomous oscillator reduces it. An internal time-scale ratio controls how fast the nonautonomous unit grows beyond threshold and yields a closed-form asymptotic law for the delay. A finite communication lag produces a periodic modulation of the delay whose period is set solely by the Hopf frequency of the nonautonomous oscillator and is independent of the sweep rate. Perturbation analysis of the delayed characteristic equation gives a quadratic amplitude law, which we confirm numerically. These results extend the static amplitude-death condition of autonomous coupled oscillators to the slow-passage setting and give analytical expressions for delayed onset in coupled multiscale systems.

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• Coupled oscillators
• Multiple time scale dynamics
• Nonlinear time-delay systems