Review Article

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

[+] Author and Article Information
Ivana Kovacic

Centre of Excellence for Vibro-Acoustic
Systems and Signal Processing,
Faculty of Technical Sciences,
University of Novi Sad,
Novi Sad 21215, Serbia

Richard Rand

Department of Mathematics,
Cornell University,
Ithaca, NY 14853;
Department of Mechanical and
Aerospace Engineering,
Cornell University,
Ithaca, NY 14853

Si Mohamed Sah

Technical University of Denmark,
Department of Mechanical Engineering,
Section of Solid Mechanics,
Lyngby 2800, Denmark

Manuscript received August 7, 2017; final manuscript received January 18, 2018; published online February 14, 2018. Editor: Harry Dankowicz.

Appl. Mech. Rev 70(2), 020802 (Feb 14, 2018) (22 pages) Paper No: AMR-17-1054; doi: 10.1115/1.4039144 History: Received August 07, 2017; Revised January 18, 2018

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.

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Grahic Jump Location
Fig. 1

(a) Mathematical pendulum whose support moves periodically in a vertical direction and (b) “the particle in the plane” problem

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Fig. 2

(a) Two transition curves of Mathieu's equation for the first region of instability called a tongue for an undamped case (solid line), Eq. (25), (b) example of motion of point P1 located inside the tongue, and (c) example of motion of point P2 located outside the tongue

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Fig. 3

A stability chart of classical Mathieu's equation (1) obtained by using numerical integration in conjunction with Floquet theory: gray region = unstable (U), white region = stable (S). Thick solid line-transition curves obtained by harmonic balancing, Eqs. (C1)(C9).

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Fig. 4

(a) Analytically obtained transition curves of Mathieu's equation for the first tongue for an undamped case (solid line), Eq. (25) and the case with linear viscous damping (dashed line), Eq. (52) and (b) numerically obtained stability chart of damped of Mathieu's equation, Eq. (45)

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Fig. 5

The δ-ϵ plane of Mathieu's equation with cubic geometric nonlinearity, Eq. (56): (a) existence of different equilibria and (b) bifurcations

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Fig. 6

Phase portraits of the Poincaré map in the different regions of the parameter plane in a quadratically damped Mathieu equation, Eq. (66)

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Fig. 8

Hopf bifurcation curve, Eq. (106), for ϵ=0.05, β=3/5,α=1. LC = limit cycle, No LC = no limit cycle.

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Fig. 7

(a) First instability tongue in the parameter space (δ,τ,ϵ), Eq. (99), for β=3/5, α=1, μ=0 and (b) three-dimensional stability chart in the parameter space (δ,b,ϵ) of the delayed undamped Mathieu equation with τ=2π, redrawn based on Ref. [33] (adapted from Ref. [32] with permission from The Royal Society)

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Fig. 9

First transition curves, Eq. (119), in the fractional Mathieu equation, Eq. (109) for c = 0.1 and different values of α

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Fig. 10

Plot of the minimum quantity of forcing amplitude ϵmin/c, Eq. (120) necessary to produce instability as a function of fractional derivative order α

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Fig. 11

First transition curves, Eq. (123), for the Mathieu equation with two fractional terms and delay, Eq. (122) for b = c = 0.1, different values of τ, Λ∗ satisfying Eq. (124) and: (a) α = 0.25 and (b) α = 0.75

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Fig. 14

(a) Stability of Eq. (125) as determined directly from numerical integration. ϵ=0.1, (b) enlargement of (a) around δ=0.25,ω=1. ϵ=0.1, and (c) enlargement of (a) around δ=0.25,ω=0.1. ϵ=0.1. Black = stable and white = unstable.

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Fig. 15

Stability of Eq. (125) as determined directly from numerical integration: (a) ϵ=0.01, (b) ϵ=0.05, (c) ϵ=0.1, (d) ϵ=0.5 (Note that Figs. 15(a)15(c) lie in the region bounded by the thick straight lines in this part), and (e) ϵ = 1. Black = stable and white = unstable.

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Fig. 12

As a first approximation, we may think of the instabilities occurring in Eq. (125) as consisting of the union of the instabilities in Eqs. (126) and (127)

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Fig. 13

Same information as in Fig. 12 plotted for fixed ϵ with varying ω and δ

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Fig. 16

Period of the elliptic-type cn excitation as a function of the elliptic parameter m for cases I–III

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Fig. 17

Stability charts of Eq. (129), where gray regions indicate instability: (a) m=−0.5, (b) m = 0.5, and (c) m = 1.5. Results obtained by means of the approximated equation of motion are labeled by thick solid lines (in all cases Ñ=10).

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Fig. 18

Stability chart redrawn based on a figure from page 28 of Ince's paper [15]



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