Nonlinear and divergent responses of fluctuation-driven systems

Abstract

Most network dynamical systems are out of equilibrium and externally driven by fluctuations. Linear response theory generically characterizes systems responses to such fluctuations for small driving amplitudes yet cannot capture response properties that are either due to strong driving or intrinsically nonlinear. For oscillation-driven systems, we here report average response offsets that scale quadratically with asymptotically small amplitudes. At some critical driving amplitude, responses cease to stay close to a given operating point and may diverge. Standard response theory fails to predict these amplitudes even at arbitrarily high orders. We propose an integral self-consistency condition that captures the full nonlinear system dynamics. We illustrate our approach for a minimal one-dimensional model and capture the nonlinear shift of voltages in the phase, frequency and voltage dynamics of AC power grid networks. Our approach may help to quantitatively predict intrinsically nonlinear response dynamics as well as bifurcations emerging at large driving amplitudes in non-autonomous dynamical systems.