Discrete dynamical stabilization of a naturally diverging mass in a harmonically time-varying potential

TL;DR

This paper explores how a diverging mass in a harmonically modulated potential can be stabilized in discrete regimes, revealing new stability patterns and analytical solutions related to Floquet modes and Schrödinger equations.

Contribution

It demonstrates that dynamical stabilization occurs in discrete regimes beyond the Kapitza limit, with analytical predictions of stability stripes and modal shapes using a Schrödinger equation analogy.

Findings

Stability stripes are predicted analytically and confirmed numerically.

Almost-periodic Floquet modes are the only stable vibrational solutions.

Transformation to a Schrödinger equation provides insight into modal shapes.

Abstract

We numerically investigate the stability and linear oscillatory behavior of a naturally diverging mass whose potential energy is harmonically modulated. It is known that in the Kapitza limit, i.e. when the period of modulation is much smaller than the diverging time, the collapsing mass can be dynamically stabilized and behave like an effective classic harmonic oscillator. We find that in the regime where the period of modulation is larger than the collapsing time of the mass, dynamical stabilization is still possible but in a discrete fashion. Only almost-periodic vibrational modes, or Floquet forms (FFs), are allowed that are located in independent stability stripes in the modulation parameter space. Reducing the FFs to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schr\"odinger stationary wave equation with a harmonic potential. This transformation allows for an analytical prediction of the stability stripes and the modal shapes of the vibrating mass. These results shed new light on the stability of linear dynamical systems, analytical solutions of Mathieu equations and on the relations between Initial and Boundary Value Problems.