Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

2018/01/27 by Ivana Kovacic, Ivana Kovačić, Richard Rand +2 · 292 citations
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Computer science #Control Systems and Identification #Differential equation #First-order partial differential equation #Hill differential equation #Mathematical analysis #Mathematics #Mathieu function #Quantum chaos and dynamical systems #Scientific Measurement and Uncertainty Evaluation #Stability (learning theory)

paper · open access · doi:10.1115/1.4039144

published in Applied Mechanics Reviews 70(2) (American Society of Mechanical Engineers)

openalex publication_date 2018/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

This work is concerned with Mathieu's equation—a classical differential equation, which has the form of a linear second-order ordinary differential equation (ODE) 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.

Citations

• Fractional delayed damped Mathieu equation
• On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon

Cited by

• Mixing induced by Faraday surface waves
• Full-band reciprocal parametric amplification in coupled oscillator arrays
• Steady-states and response functions of the periodically driven O(N) scalar field theory
• Pattern formation in ring condensates subjected to bichromatic driving
• Parametric resonance, chaos and spatial structure in the Lotka-Volterra model
• Parametric resonant enhancement of motional entanglement under optimal control: an analytical study
• Free oscillations of a standing surface wave and its mechanical analogue
• Periodic Cosmic String Formation and Dynamics
• Detecting the π-axiverse through parametric resonance
• Decay and lifetime of oscillons coupled to an external scalar field: Insights from instability band analysis
• Kibble-Zurek mechanism for dissipative discrete time crystals
• Geometric Floquet theory
• On the Prediction of the Subharmonic Threshold of an Ultrasound Contrast Agent
• Stability analysis of time-periodic shear flow generated by an oscillating density interface
• Particle decay as asymptotic narrow parametric resonance
• Parametric Instability in Discrete Models of Spatiotemporally Modulated Materials
• Parametric excitations in a harmonically trapped binary Bose-Einstein condensate
• Parametric Stability Analysis for Circuit Quantum Electrodynamical Hardwares
• Rotational Locking of Charged Microparticles in Quadrupole Ion Traps
• Dirac materials under linear polarized light: quantum wave function evolution and topological Berry phases as classical charged particles trajectories under electromagnetic fields
• Parametric Resonance and Backreaction Effects in Magnetogenesis from Ultralight Dark Matter
• A Magnetic Torsional Pendulum for Exploring Forced Resonance, Parametric Resonance, and Parametric Amplification
• Nonlinear stabilization of chiral modes in space-time modulated parametric oscillators
• Faraday pattern formation in dipolar superfluid and supersolid quantum gases
• Generalized parametric resonance in a spin-1 Bose-Einstein condensate
• Constraining the gravitational coupling of axion dark matter at LIGO
• Stability of axion dark matter-photon conversion
• Floquet Phases of Matter via Classical Prethermalization
• Controlled bit-flip of period-doubling and discrete time crystalline states in open systems
• Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy. [europepmc]
• Optomechanical synchronization across multi-octave frequency spans. [europepmc]
• Comprehensive Overview of Bottom-Up Proteomics Using Mass Spectrometry. [europepmc]
• Galloping Bubbles. [europepmc]
• Meta-frequency modulation in LQR vibration control with chirp excitation. [europepmc]
• Stability and control mechanism of nonlinear horizontal vibration for rolling system with gyroscope precession effect. [europepmc]
• Parametric anti-phase excitation of resonant MEMS mirrors for fast start-up. [europepmc]
• Effects of strong parametric excitation on cantilever beam: non-perturbative approach. [europepmc]
• Chaotic and dynamic vibration analysis of a time-delayed nonlinear mathieu oscillator via non-perturbative approach. [europepmc]

Related

• Computation and applications of Mathieu functions: A historical perspective
• The Mathieu Differential Equation and Generalizations to Infinite Fractafolds
• The sinusoid and the phasor
• Fractional delayed damped Mathieu equation
• Combined homotopy and Galerkin stability analysis of Mathieu-like equations