Comment on "Asymptotic Phase for Stochastic Oscillators".

In his comment [1] on our paper [2], Pikovsky contrasts two definitions for the phase of a stochastic oscillator by way of an analytically solvable model system. In [3] the phase is defined in terms of a system of isochrons Σθ, analogous to Poincaré sections, with the property that for any initial condition on one isochron Σa, the mean first passage time (MFPT) to a second isochron Σb, b > a, will depend only on the phase difference b−a. In our approach [2] the phase is defined as the complex argument of the slowest decaying eigenfunction of the backward Kolmogorov operator, provided the first nontrivial eigenvalue is complex and is well separated from the next slowest decaying eigenvalue. Pikovsky argues that our eigenfunction approach does not properly work in all situations and proposes the following example to demonstrate this. Consider two independent phase-like variables, each taking values in [0, 2π), that obey