Magnus,
W.
, and
Winkler,
S.
, 1961, “
Hill's Equation—Part II: Transformations, Approximation, Examples,” New York University, New York, Report No. BR-38.

https://archive.org/details/hillsequationiit00magn
McLachlan,
N. W.
, 1947, Theory and Applications of Mathieu Functions,
Clarendon Press,
Oxford, UK.

Erdelyi,
A.
, 1955, Higher Transcendental Functions, Vol.
III,
McGraw-Hill Book Company,
New York.

Stoker,
J. J.
, 1950, Nonlinear Vibrations in Mechanical and Electrical Systems,
Interscience Publishers,
New York.

Cartmell,
M.
, 1990, Introduction to Linear, Parametric and Nonlinear Vibrations,
Chapman and Hall,
London.

Ruby,
L.
, 1996, “
Applications of the Mathieu Equation,” Am. J. Phys.,
64(1), pp. 39–44.

[CrossRef]
Mathieu,
E.
, 1868, “
Mémoire sur Le Mouvement Vibratoire d'une Membrane de forme Elliptique,” J. Math. Pures Appl.,
13, pp. 137–203.

Heine,
E.
, 1878, Hanbuch Der Kugelfunktionen, Vol.
2, Georg Reimer, Berlin, p. 81.

Floquet,
G.
, 1883, “
Sur les equations differetielles lineaires,” Ann. de L' Cole Normale Super.,
12, pp. 47–88.

[CrossRef]
Hill,
G. W.
, 1886, “
Mean Motion of the Lunar Perigee,” Acta Math.,
8, pp. 1–36.

[CrossRef]
Sieger,
B.
, 1908, “
Die Beugung einer ebened elektrischen Weilen an einem Schirm von elliptischen Querschnitt,” A. der P.,
27(13), p. 626.

Whittaker,
E. T.
, 1912, “
Elliptic Cylinder Functions in Harmonic Analysis,” Paediatric Intensive Care Medicine, Vol.
1, Springer, London, p. 366.

Ince,
E.
, 1927, “
Research Into the Characteristic Numbers of Mathieu Equation,” Proc. R. Soc. Edinburgh,
46, pp. 20–29.

[CrossRef]
Strutt,
M. J. O.
, 1928, “
Zur Wellenmechanik des Atomgitters,” Ann. Phys.,
391(10), pp. 319–324.

[CrossRef]
Stephenson,
A.
, 1908, “
On a New Type of Dynamic Stability,” Memoirs and Proceedings of the Manchester Literary and Philosophical Society, Vol.
52, Manchester Literary and Philosophical Society, Manchester, UK, pp. 1–10.

Stephenson,
A.
, 1908, “
On Induced Stability,” Philos. Mag.,
15(86), pp. 233–236.

[CrossRef]
Thomsen,
J. J.
, 2003, Vibrations and Stability: Advanced Theory, Analysis, and Tools,
Springer-Verlag,
Berlin.

Seyranian,
A. P.
, and
Mailybaev,
A. A.
, 2003, Multiparameter Stability Theory with Mechanical Applications, Vol.
13,
World Scientific, Singapore.

[CrossRef]
Verhulst,
F.
, 2009, “
Perturbation Analysis of Parametric Resonance,” Encyclopedia of Complexity and Systems Science, R. Meyers, ed., Springer, New York.

Vlajic,
N.
,
Liu,
X.
,
Karki,
H.
, and
Balachandran,
B.
, 2014, “
Torsional Oscillations of a Rotor With Continuous Stator Contact,” Int. J. Mech. Sci.,
83, pp. 65–75.

[CrossRef]
Yang,
T. L.
, and
Rosenberg,
R. M.
, 1967, “
On the Vibrations of a Particle in the Plane,” Int. J. Non-Linear Mech.,
2(1), pp. 1–25.

[CrossRef]
Yang,
T. L.
, and
Rosenberg,
R. M.
, 1968, “
On the Forced Vibrations of a Particle in the Plane,” Int. J. Non-Linear Mech.,
3(1), pp. 47–63.

[CrossRef]
Cole,
J. D.
, 1968, Perturbation Methods in Applied Mathematics,
Blaisdell, Waltham, MA.

Nayfeh,
A.
, 1973, Perturbation Methods,
Wiley, New York.

Insperger,
T.
, and
Stépán,
G.
, 2011, Semi Discretization for Time Delay Systems: Stability and Engineering Applications (Applied Mathematical Science, Vol. 178),
Springer Science+Business Media, New York.

[CrossRef]
Kovacic,
I. I.
, and
Rand,
R.
, 2014, “
Duffing-Type Oscillators With Amplitude-Independent Period,” Applied Nonlinear Dynamical Systems, Vol.
93,
J. Awrejcewicz
, ed.,
Springer, Berlin, pp. 1–10.

[CrossRef]
Kovacic,
I.
, and
Brennan,
M. J.
, 2011, The Duffing Equation: Nonlinear Oscillators and Their Behaviour,
Wiley,
Chichester, UK.

[CrossRef]
Rand,
R. H.
,
Ramani,
D. V.
,
Keith,
W. L.
, and
Cipolla,
K. M.
, 2000, “
The Quadratically Damped Mathieu Equation and Its Application to Submarine Dynamics,” Control of Noise and Vibration: New Millenium, AD-Vol. 61,
ASME,
New York, pp. 39–50.

Ramani,
D. V.
,
Keith,
W. L.
, and
Rand,
R. H.
, 2004, “
Perturbation Solution for Secondary Bifurcation in the Quadratically-Damped Mathieu Equation,” Int. J. Non-Linear Mech.,
39(3), pp. 491–502.

[CrossRef]
Morrison,
T. M.
, and
Rand,
R. H.
, 2007, “
2:1 Resonance in the Delayed Nonlinear Mathieu Equation,” Nonlinear Dyn.,
50(1–2), pp. 341–352.

[CrossRef]
Insperger,
T.
, and
Stepan,
G.
, 2002, “Stability Chart for the Delayed Mathieu Equation,” Proc. R. Soc. A,
458(2024), pp. 1989–1998.

Butcher,
E. A.
, and
Mann,
B. P.
, 2009, “
Stability Analysis and Control of Linear Periodic Delayed Systems Using Chebyshev and Temporal Finite Element Methods,” Delay Differential Equations: Recent Advances and New Directions,
B. Balachandran
,
D. Gilsinn
, and
T. Kalmar-Nagy
, eds.,
Springer,
New York.

[CrossRef]
Atay,
F. M.
, 1998, “
Van Der Pol's Oscillator Under Delayed Feedback,” J. Sound Vib.,
218(2), pp. 333–339.

[CrossRef]
Wirkus,
S.
, and
Rand,
R. H.
, 2002, “
The Dynamics of Two Coupled Van Der Pol Oscillators With Delay Coupling,” Nonlinear Dyn.,
30(3), pp. 205–221.

[CrossRef]
Sah,
S. M.
, and
Rand,
R.
, 2016, “
Delay Terms in the Slow Flow,” J. Appl. Nonlinear Dyn.,
5(4), pp. 471–484.

[CrossRef]
Bernstein,
A.
, and
Rand,
R.
, 2016, “
Delay-Coupled Mathieu Equations in Synchrotron Dynamics,” J. Appl. Nonlinear Dyn.,
5(3), pp. 337–348.

[CrossRef]
Rand,
R. H.
,
Sah,
S. M.
, and
Suchorsky,
M. K.
, 2010, “
Fractional Mathieu Equation,” Commun. Nonlinear Sci. Numer. Simul.,
15(11), pp. 3254–3262.

[CrossRef]
Ross,
B.
, 1975, “
A Brief History and Exposition of the Fundamental Theory of Fractional Calculus,” Fractional Calculus and Its Applications (Springer Lecture Notes in Mathematics, Vol. 57), Springer, New York, pp. 1–36.

[CrossRef]
Mesbahi,
A.
,
Haeri,
M.
,
Nazari,
M.
, and
Butcher,
E. A.
, 2015, “
Fractional Delayed Damped Mathieu Equation,” Int. J. Control,
88(3), pp. 622–630.

[CrossRef]
Rand,
R.
,
Zounes,
R.
, and
Hastings,
R.
, 1997, “
Dynamics of a Quasiperiodically Forced Mathieu Oscillator,” Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume,
A. Guran
, ed.,
World Scientific, Singapore, pp. 203–221.

[CrossRef]
Zounes,
R. S.
, and
Rand,
R. H.
, 1998, “
Transition Curves in the Quasiperiodic Mathieu Equation,” SIAM J. Appl. Math.,
58(4), pp. 1094–1115.

[CrossRef]
Abouhazim,
N.
,
Rand,
R. H.
, and
Belhaq,
M.
, 2006, “
The Damped Nonlinear Quasiperiodic Mathieu Equation Near 2:2:1 Resonance,” Nonlinear Dyn.,
45(3–4), pp. 237–247.

[CrossRef]
Rand,
R.
,
Guennoun,
K.
, and
Belhaq,
M.
, 2003, “
2:2:1 Resonance in the Quasiperiodic Mathieu Equation,” Nonlinear Dyn.,
31(4), pp. 367–374.

[CrossRef]
Rand,
R.
, and
Morrison,
T.
, 2005, “
2:1:1 Resonance in the Quasi-Periodic Mathieu Equation,” Nonlinear Dyn.,
40(2), pp. 195–203.

[CrossRef]
Sharma,
A.
, and
Sinha,
S. C.
, 2017, “
An Approximate Analysis of Quasi-Periodic Systems Via Floquet Theory,” ASME J. Comput. Nonlinear Dyn.,
13(2), p. 021008.

[CrossRef]
Zounes,
R. S.
, and
Rand,
R. H.
, 2002, “
Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation,” Nonlinear Dyn.,
27(1), pp. 87–105.

[CrossRef]
Abramowitz,
M.
, and
Stegun,
I.
, 1965, Handbook of Mathematical Functions,
Dover Publications,
Mineola, NY.

Kovacic,
I.
,
Cveticanin,
L.
,
Zukovic,
M.
, and
Rakaric,
Z.
, 2016, “
Jacobi Elliptic Functions: A Review of Nonlinear Oscillatory Application Problems,” J. Sound Vib.,
380, pp. 1–36.

[CrossRef]
Byrd,
P.
, and
Friedman,
M.
, 1954, Handbook of Elliptic Integrals for Engineers and Scientists,
Springer,
Berlin.

[CrossRef]
Gradshteyn,
I. S.
, and
Ryzhik,
I. M.
, 2000, Tables of Integrals, Series and Products,
Academic Press,
New York.

Kovacic,
I.
, and
Zukovic,
M.
, 2014, “
A Pendulum With an Elliptic-Type Parametric Excitation: Stability Charts for a Damped and Undamped System,” Commun. Nonlinear Sci. Numer. Simul.,
19(4), pp. 1185–1202.

[CrossRef]
Sah,
S. M.
, and
Mann,
B.
, 2012, “
Transition Curves in a Parametrically Excited Pendulum With a Force of Elliptic Type,” Proc. R. Soc. A,
468(2148), pp. 3995–4007.

[CrossRef]
Dingle,
R. B.
, and
Müller-Kirsten,
H. J. W.
, 1962, “
Asymptotic Expansions of Mathieu Functions and Their Characteristic Numbers,” J. Die Reine Angew. Math.,
1962(211), pp. 11–32.