Pais-Uhlenbeck Oscillator with a Benign Friction Force

TL;DR

This paper explores a generalized Pais-Uhlenbeck oscillator with combined first and third order damping, identifying conditions for stability and demonstrating how specific damping parameters influence system behavior.

Contribution

It introduces a more general damping model for the Pais-Uhlenbeck oscillator, extending previous work and analyzing stability conditions based on damping constants.

Findings

System stability depends on positive damping constants below critical thresholds.

Equal damping constants with opposite signs lead to instability.

Generalized damping terms can stabilize or destabilize the oscillator.

Abstract

It is shown that the Pais-Uhlenbeck oscillator with damping, considered by Nesterenko, is a special case of a more general oscillator that has not only a first order, but also a third order friction term. If the corresponding damping constants, \alpha\ and \beta, are both positive and below certain critical values, then the system is stable. In particular, if \alpha = - \beta, then we have the unstable Nesterenko's oscillator