On the number of critical points of the period. (English) Zbl 0594.34028

It is shown that for the equation \(\ddot u+f(u)=0\), where \(f\) is a cubic polynomial, the period map can have at most three critical points. This is done by deriving the Picard-Fuchs equation for the period and analyzing the behavior of its solutions. One could make the conjecture that the maximal number is one. This would be the sharpest possible result without conditions on the coefficients of f. Applications are mentioned in the bifurcation theory of steady-state solutions.


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 New York · Zbl 0507.34003
[2] Bogdanov, R.I, Bifurcation of limit cycle of a family of plane vector fields, Selecta math. soviet, 1, 373-387, (1981) · Zbl 0518.58029
[3] Brunovsky, P; Chow, S.N, Generic properties of stationary state solutions of reaction-diffusion equation, J. differential equations, 53, 1-23, (1984) · Zbl 0544.34019
[4] Carr, J, Application of center manifold theory, (1981), Springer-Verlag New York
[5] Chow, S.-N; Hale, J.K, Methods of bifurcation theory, (1982), Springer-Verlag New York
[6] Cushman, R.H; Sanders, J.A, A codimension two bifurcation with a third-order Picard-Fuchs equation, 1983, J. differential equations, 59, 243-256, (1985) · Zbl 0571.34021
[7] Cushman, R; Sanders, J.A, Limit cycles in the Josephson equation, (1984)
[8] Hale, J.K; Taboas, P.Z, Interaction of damping and forcing in a second-order equation, Nonlinear anal., 2, 77-84, (1978) · Zbl 0369.34014
[9] Hsu, S.B, A remark on the period of the periodic solution in the Lotka-Volterra systems, J. math. anal. appl., 95, 428-436, (1983) · Zbl 0515.92020
[10] Iłyashenko, Yu, The multiplicity of limit cycles arising from perturbations of the form \(w′ = P2Q1\)of a Hamiltonian equation in the real and complex domain, Amer. math. soc. transl., 118, 191-202, (1982)
[11] Opial, Z, Sur LES périodes des solutions de l’équation différentielle x″ + g(x) = 0, Ann. polon. math., 10, 49-72, (1961) · Zbl 0096.29604
[12] {\scR. Schaaf}, Global behavior of solution branches for some Neumann problems depending on one or several parameters. · Zbl 0513.34033
[13] Simon, C, Bounded orbits in mechanical systems with two degrees of freedom and symmetry, (), 515-525
[14] Smoller, J; Wasserman, A, Global bifurcation of steady-state solutions, J. differential equations, 39, 269-290, (1981) · Zbl 0425.34028
[15] {\scJ. Waldvogel}, The period in the Lotka-Volterra system is monotonic, preprint. · Zbl 0588.92018
[16] Waldvogel, J, The period in the Lotka-Volterra predator-prey model, SIAM J. numer. anal., 20, (1983) · Zbl 0533.65051
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