References
Z. Afsharnejad. Bifurcation geometry of Mathieu's equation. Indian J. Pure Appl. Math. 17 (1986), 1284–1308.
V. I. Arnold. Lectures on bifurcations in versal families. Russ. Math. Surv. 27 (1972), 54–123.
V. I. Arnold. Loss of stability of self-oscillation close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11 (1977), 85–92.
V. I. Arnold. Mathematical Methods of Classical Mechanics, Springer-Verlag, 1980.
V. I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, 1983.
V. I. Arnold. Remarks on the perturbation theory for problems of Mathieu type. Russ. Math. Surv. 38 (1983), 215–233.
H. W. Broer & G. Vegter. Bifurcational aspects of parametric resonance. Dynamics Reported, New Series 1 (1992), 1–53.
I. M. Gelfand & V. B. Lidskii. On the structure of stability of linear canonical systems of differential equations with periodic coefficients. Amer. Math. Soc. Transl. (2) 8 (1958), 143–181.
J. Pöschel & E. Trubowitz. Inverse Spectral Theory. Academic Press, 1986.
M. Levi. Stability of the inverted pendulum — a topological explanation. SIAM Review 30 (1988), 639–644.
D. M. Levy & J. B. Keller. Instability intervals of Hill's equation. Comm. Pure Appl. Math. 16 (1963), 469–479.
J. Meixner & F. W. Schäfke. Mathieusche Funktionen und Sphäroidfunktionen. Springer-Verlag, 1954.
J. J. Stoker. Nonlinear Vibrations. Interscience, 1950.
Th. Bröcker & L. Lander. Differentiable Germs and Catastrophes. Cambridge University Press, 1976.
B. van der Pol & M. J. O. Strutt. On the stability of the solutions of Mathieu's equation. The London, Edinburgh and Dublin Phil. Mag. 7th Series 5 (1928), 18–38.
M. I. Weinstein & J. B. Keller. Hill's equation with a large potential. SIAM J. Appl. Math. 45 (1985), 954–958.
M. I. Weinstein & J. B. Keller. Asymptotic behavior of stability regions for Hill's equation. SIAM J. Appl. Math. 47 (1987), 941–958.
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Communicated by R. Mcgehee
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Broer, H., Levi, M. Geometrical aspects of stability theory for Hill's equations. Arch. Rational Mech. Anal. 131, 225–240 (1995). https://doi.org/10.1007/BF00382887
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DOI: https://doi.org/10.1007/BF00382887