Novel methods to distinguish between quasiperiodicity and a high periodicity

Distinguishing between a quasiperiodic orbit and a high-periodic mode-locked orbit in a nonlinear system remains a challenge. In this paper, we present two approaches to address this issue. The first method analyzes how the distance between two nearby initial conditions located on the invariant closed curve evolves with time. For a high-periodic orbit, the distance asymptotically dies down, while for a quasiperiodic orbit, it remains of the same order of magnitude as the initial distance. We present an algorithm to utilize this property fruitfully to distinguish between the two types of orbits. The second method uses the spectral bifurcation diagram that plots the evolution of the spectral components as a parameter is varied. The gradual convergence of the spectral components allows one to identify the periodic windows. These methods provide convenient tools for distinguishing between high-periodicity and quasi-periodicity, subject to the limitations of machine precision.