Asymptotic phase revisited (Q1881136)

It is well known that if the condition of the Andronov-Witt theorem holds, i.e., the second characteristic multiplier of the variational system with respect to the periodic orbit of a planar system is less than 1, then the periodic orbit is orbitally asymptotically stable with asymptotic phase. If a periodic orbit has a neighbourhood such that each solution of the system starting in that neighbourhood has an asymptotic phase (as \(t\) tends to plus or minus infinity), then the orbit is said to be isochronous. Conditions are given here under which a nonhyperbolic orbit, i.e., one with the second characteristic multiplier equal to 1, is isochronous. It is also shown that if the conditions do not hold, then the orbit is not isochronous. Examples are provided for either case.