Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

Ivana Kovacic, Richard H. Rand, Si Mohamed Sah

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This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.
Original languageEnglish
Article number020802
JournalApplied Mechanics Reviews
Issue number2
Number of pages22
Publication statusPublished - 2018


  • Parametric excitation
  • Stability chart
  • Transition curves
  • Perturbation method
  • Floquet theory
  • Harmonic balancing
  • Geometric nonlinearity;
  • Damping nonlinearity
  • Fractional derivative
  • Delay
  • Quasiperiodic excitation
  • Elliptic-type excitation


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