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The critical points of the period function of \(x''-x^ 2(x-\alpha)(x- 1)=0(0\leq \alpha <1)\). (English) Zbl 0641.34029

The author proves that the number of critical points of the period function \(P_{\alpha}(c)\) of equation \(x''-x^ 2(x-\alpha)(x-1)=0\) \((0\leq \alpha <1)\) is bounded for \(\alpha\in [0,1)\). The proof is based on the analyticity of function \(P_{\alpha}(c)\) in \(\alpha\) and c, which is proved by using Picard-Fuchs equation.
Reviewer: Duo Wang

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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[1] Arnold, V. I., Geometric Methods in the Theory of Ordinary Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0507.34003
[2] Brunovsky, P.; Chow, S. N., Generic properties of stationary state solutions of reaction-diffusion equations, J. diff. Eqns, 53, 1, 1-23 (1984) · Zbl 0544.34019
[3] Carr, J.; Chow, S. N.; Hale, J. K., Abelian integrals and bifurcation theory, J. diff. Eqns, 59, 413-436 (1985) · Zbl 0587.34033
[4] Chow, S. N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York
[5] Chow, S. N.; Sanders, J. A., On the number of critical points of the period, J. diff. Eqns, 64, 51-66 (1986) · Zbl 0594.34028
[6] Chow S. N. & Wang D., On the monotonicity of the period function of some second order equations, to appear.; Chow S. N. & Wang D., On the monotonicity of the period function of some second order equations, to appear. · Zbl 0603.34034
[7] Loud, W. S., Periodic solution of \(x″+cx′+g(x)=∈ƒ(t)\), Mem. Am. Math. Soc., 31, 1-57 (1959) · Zbl 0085.30701
[8] Obi, C., Analytical theory of nonlinear oscillation, VII. The periods of the periodic solution of the equation \(ẍ+g(x)=0\), J. math. Anal. Appl., 55, 295-301 (1976) · Zbl 0359.34038
[9] Opial, Z., Sur les periodes des solutions de l’equation differentielle \(ẍ+g(x)=0\), Annls Soc. pol. Math., 10, 49-72 (1961) · Zbl 0096.29604
[10] Schaaf, R., Global behavior of solution branches for some Neumann problems depending on one or several parameters, J. reine angew. Math, 346, 1-31 (1984) · Zbl 0513.34033
[11] Struble, R. A., Nonlinear Differential Equations (1972), McGraw-Hill: McGraw-Hill New York, Toronto, London · Zbl 0124.04904
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