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

• Published in: Physica. D Vol. 386-387; no. 1er janvier 2019; pp. 1 - 7
• Main Author: Lazarus, Arnaud
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2019; Elsevier
• Subjects: Classical Physics; Dynamical systems; Floquet theory; General Physics; Mechanics; Modal analysis; Physics; Stability analysis; Standing waves; Vibrations; Standing waves; Floquet theory; Stability analysis; Modal analysis; Dynamical systems
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2018.08.001

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 Schrödinger 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. •Possible to stabilize, in theory, a Kapitza oscillator with slow forcing excitations.•In this case, stabilization is discrete in the modulation parameter space.•The stable vibrational modes have a compact support on each period.•Those modes can be analytically predicted by a stationary wave equation.