The monotonicity and convexity of the period function for a class of symmetric Newtonian systems of degree 8

Document Type : Research Paper


1 Department of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, Iran.

2 Department of Mathematics, Yazd University, Yazd, 89195-741 Iran.


In this paper, we study the monotonicity and convexity of the period function associated with centers of a specific class of symmetric Newtonian systems of degree 8. In this regard, we prove that if the period annulus surrounds only one elementary center, then the corresponding period function is monotone; but, for the other cases, the period function has exactly one critical point. We also prove that in all cases, the period function is convex.


  • [1]          L. Bonorino, E. Brietzke, J. Lukaszczyk, and C. Taschetto, Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation, J. Differential Equations, 214 (2005), 156– 175.
  • [2]          F. Chen, C. Li, J. Llibre, and Z. Zhang, A unified proof on the weak Hilbert 16th problem for n = 2, J. Differential Equations, 221 (2006), no. 2, 309–342.
  • [3]          C. Chicone, The monotonicity of the period function for planar Hamiltonian vector field, J. Differential Equations, 69 (1987), 310–321.
  • [4]          F. Dumortier, J. Llibre, and J. C. Art´es, Qualitative theory of planar differential systems, Universitext, Springer- Verlag, Berlin, 2006.
  • [5]          C. Li and K. Lu, The period function of hyperelliptic Hamiltonians of degree 5 with real critical points, Nonlin- earity, 21 (2008), 465–483.
  • [6]          F. Mann˜osas and J. Villadelprat, Criteria to bound the number of critical periods,   J. Differential Equations, 246 (2009), 2415–2433.
  • [7]          M. Sabatini, Period function’s convexity for Hamiltonian centers with separable variables, Ann. Polon. Math., 85 (2005), 153–163.
  • [8]          R. Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math., 363 (1985), 96–109.
  • [9]          R. Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several pa- rameters, J. Reine Angew. Math., 346 (1984), 1–31.
  • [10]        X. Sun, H. Xi, H.R. Zangeneh, and R. Kazemi, Bifurcation of limit cycles in small perturbation of  a  class  of Linard systems, Internat. J. Bifur. Chaos, 24(1) (2014), 1450004, 23.
  • [11]        A. Zevin and M. Pinsky, Monotonicity criteria for an energy-period function in planar Hamiltonian systems, Nonlinearity, 14 (2001), 1425–1432.