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<p>A recent study employing Fourier transform techniques presents an incomplete treatment of the Hyers–Ulam stability of linear differential equations with constant coefficients, particularly in the context of the second-order undamped harmonic oscillator (i.e., the spring-mass system). That paper asserts that such systems are universally Hyers–Ulam stable. This paper aims to clarify that the Hyers–Ulam stability of these equations is, in fact, more nuanced and critically dependent on system parameters such as the mass, damping coefficient, and spring constant. Specifically, the undamped spring–mass system is Hyers–Ulam unstable when the mass and spring constant are both positive or both negative. This instability arises due to resonance phenomena in the perturbed system. We also discuss and illustrate additional instability cases to provide a more complete stability analysis.</p>

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Resonance-Induced Hyers–Ulam Instability in Undamped Spring-Mass Systems

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Douglas R. Anderson,
  • Collin Smolke

摘要

A recent study employing Fourier transform techniques presents an incomplete treatment of the Hyers–Ulam stability of linear differential equations with constant coefficients, particularly in the context of the second-order undamped harmonic oscillator (i.e., the spring-mass system). That paper asserts that such systems are universally Hyers–Ulam stable. This paper aims to clarify that the Hyers–Ulam stability of these equations is, in fact, more nuanced and critically dependent on system parameters such as the mass, damping coefficient, and spring constant. Specifically, the undamped spring–mass system is Hyers–Ulam unstable when the mass and spring constant are both positive or both negative. This instability arises due to resonance phenomena in the perturbed system. We also discuss and illustrate additional instability cases to provide a more complete stability analysis.