Computing quadratic approximations for the isochrons of oscillators: A general theory and advanced numerical methods

We first review the notion of isochrons for oscillators, which has been developed and heavily utilized in mathematical biology in studying biological oscillations. Isochrons were instrumental in introducing a notion of generalized phase for an oscillation and form the basis for oscillator perturbation analysis formulations. Calculating the isochrons of an oscillator is a very difficult task. Except for some very simple planar oscillators, isochrons can not be calculated analytically and one has to resort to numerical techniques. Previously proposed numerical methods for computing isochrons can be regarded as brute-force, which become totally impractical for non-planar oscillators with dimension more than two. In this paper, we present a precise and carefully developed theory and advanced numerical techniques for computing local but quadratic approximations for isochrons. Previous work offers the theory and the numerical methods needed for computing only linear approximations for isochrons. Our treatment is general and applicable to oscillators with large dimension. We present examples for isochron computations, verify our results against exact calculations in a simple case, and allude to several applications among many where quadratic approximations of isochrons will be of use.