Local Exponents of Nonlinear Compression in Periodically Driven Noisy Oscillators

Nonlinear compression of periodic signals is a key feature of the active amplifier in inner ear organs of all vertebrates. Different exponents α₀∈[-0.88,-0.5] of the sensitivity vs forcing amplitude |χ|∼f^α₀ have been observed. Here we calculate analytically the local exponent for a generic oscillator, the normal form of a Hopf bifurcation driven by noise and a periodic signal. For weak noise and sufficient distance from the bifurcation on the unstable side, the exponent may be close to -1 for moderate forcing amplitudes beyond linear response. Such strong compression is also found in a model of hair bundle motility.