Explanation for Oscillating Backbone Curves Based on Fractional Spectral Submanifolds

The instantaneous relationship between frequency and amplitude (backbone curve) of a decaying trajectory represents an important feature of a system, as it provides information about the forced response, if forcing and damping are small. The backbone curve of a system can be directly approximated by the reduced dynamics on spectral submanifolds (SSMs) constructed either from governing equations or from data, without the necessity for extensive numerical computations (numerical continuation or time integration). In this chapter, we study oscillating backbone curves observed in experiments and numerical simulations. Oscillations become more apparent, as the initial condition of decaying trajectory moves farther away from the primary SSM, which is indeed incapable of reproducing this phenomenon. Conversely, a new class of manifolds, fractional (or secondary) SSMs, offer a clear explanation for this observation.