## The monotonicity of the period function for planar Hamiltonian vector fields.(English)Zbl 0622.34033

Classical Hamiltonian systems on the plane with Hamiltonians of the form $$H(x.y)=(1/2)y^ 2+V(x)$$ are considered. Here V(x) is a smooth potential function with a nondegenerate relative minimum at the origin. The phase portrait of the Hamiltonian vector field $$X=y(\partial /\partial x)-V^ 1(\partial /\partial y)$$ has a center at the origin surrounded by periodic orbits each of which is an energy curve of H with energy $$E\in (0,E_*)$$. Periodic function T which assigns to each period orbit its minimum period is studied.
Let $$K(E)=\{x\in R:V(x)\leq E\}$$ and define $$N(x)=(V'(x))^ 4(V(x)/V'(x)^ 2)''.$$ The following theorem is proved. If N(x)$$\geq 0$$ for all $$x\in K(E)$$, then T is monotone increasing on (0,E). If N(x)$$\leq 0$$ for all $$x\in K(E)$$ then T is monotone decreasing on (0,E). Several examples are shown.
Reviewer: B.Cheshankov

### MSC:

 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 70H05 Hamilton’s equations
Full Text:

### References:

 [1] Arnold, V. I., Geometric Methods in the Theory of Ordinary Differential Equations (1983), Springer-Verlag: 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 [4] Chow, S. N.; Hale, J. K., Methods of bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0487.47039 [7] Cushman, R. H.; Sanders, J. A., A codimension two bifurcation with a third order Picard-Fuchs equations, Differential Equations, 59, 243-256 (1985) · Zbl 0571.34021 [8] 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 [9] Il’yashenko, 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) · Zbl 0494.34018 [10] Lasry, J. M.; Lions, P. L., Equations elliptiques non lineaires avec conditions aux limites infinies et contrôle stochastique avec contraintes d’état, C. R. Acad. Sci. Paris, Ser. I, 299, 7, 213-216 (1984) · Zbl 0568.35042 [11] Loud, W. S., Periodic solution of $$x″ + cx′ + g(x) = εƒ(t)$$, Mem. Amer. Math. Soc., 31, 1-57 (1959) · Zbl 0085.30701 [12] Obi, C., Analytical theory of nonlinear oscillation, VII, The periods of the periodic solutions of the equation $$x$$″ + $$g(x) = 0$$, J. Math. Anal. Appl., 55, 295-301 (1976) · Zbl 0359.34038 [13] 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 [14] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. Differential Equations, 39, 269-290 (1981) · Zbl 0425.34028 [15] Wang, D., On the existence of 2π-periodic solutions of differential equation $$x$$″ + $$g(x) = p(t)$$, Chinese Ann. Math. No. 1 A, 5, 61-72 (1984) · Zbl 0549.34043 [16] Waldvogel, J., The period in the Lotka-Volterra predator-prey model, SIAM J. Numer. Anal., 20 (1983) · Zbl 0533.65051 [18] Lunkevich, V.; Sibirskii, K., Integrals of a general quadratic differential system in cases of a center, Differential Equations, 18, 563-568 (1982) · Zbl 0499.34017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.