Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A quadratic system with a nonmonotonic period function


Authors: Carmen Chicone and Freddy Dumortier
Journal: Proc. Amer. Math. Soc. 102 (1988), 706-710
MSC: Primary 58F22; Secondary 34C25
MathSciNet review: 929007
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a certain $ {c_ * } > 1.4$ and $ c \in \left( {1.4,{c_ * }} \right)$ the quadratic system $ \dot x = - y + xy,\dot y = x + 2{y^2} - c{x^2}$ has a center at the origin surrounded by a one-parameter family of periodic trajectories. We show the period is not a monotone function of the parameter.


References [Enhancements On Off] (What's this?)

  • [1] Carmen Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), no. 3, 310–321. MR 903390, 10.1016/0022-0396(87)90122-7
  • [2] -, Geometric methods for two point nonlinear boundary value problems, preprint, 1986.
  • [3] S.-N. Chow and J. A. Sanders, On the number of critical points of the period, J. Differential Equations 64 (1986), no. 1, 51–66. MR 849664, 10.1016/0022-0396(86)90071-9
  • [4] Shui-Nee Chow and Duo Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat. 111 (1986), no. 1, 14–25, 89 (English, with Russian and Czech summaries). MR 833153
  • [5] Roberto Conti, About centers of quadratic planar systems, preprint, Instituto Matematico, Universita Degli Studi di Firenze, 1986.
  • [6] W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations 3 (1964), 21–36. MR 0159985
  • [7] V. A. Lunkevich and K. S. Sibirskiĭ, Integrals of a general quadratic differential system in cases of the center, Differentsial′nye Uravneniya 18 (1982), no. 5, 786–792, 915 (Russian). MR 661356
  • [8] Franz Rothe, Thermodynamics, real and complex periods of the Volterra model, Z. Angew. Math. Phys. 36 (1985), no. 3, 395–421 (English, with German summary). MR 797236, 10.1007/BF00944632
  • [9] -, Periods of oscillation, nondegeneracy and specific heat of Hamiltonian systems in the plane, Proc. Internat. Conf. Differential Equations and Math. Phys. (Birmingham, Alabama, 1986) (to appear).
  • [10] Renate Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math. 363 (1985), 96–109. MR 814016, 10.1515/crll.1985.363.96
  • [11] Renate Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984), 1–31. MR 727393, 10.1515/crll.1984.346.1
  • [12] Duo Wang, On the existence of 2𝜋-periodic solutions of the differential equation 𝑥+𝑔(𝑥)=𝑝(𝑡), Chinese Ann. Math. Ser. A 5 (1984), no. 1, 61–72 (Chinese). An English summary appears in Chinese Ann. Math Ser. B 5 (1984), no. 1, 133. MR 743783
  • [13] J. Waldvogel, The period in the Lotka-Volterra predator-prey model, SIAM J. Numer. Anal. 20 (1983).
  • [14] Jörg Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986), no. 1, 178–184. MR 829122, 10.1016/0022-247X(86)90076-4

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F22, 34C25

Retrieve articles in all journals with MSC: 58F22, 34C25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0929007-7
Article copyright: © Copyright 1988 American Mathematical Society