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Transactions of the American Mathematical Society

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Bifurcation of critical periods for plane vector fields


Authors: Carmen Chicone and Marc Jacobs
Journal: Trans. Amer. Math. Soc. 312 (1989), 433-486
MSC: Primary 58F14; Secondary 34C25, 58F05, 58F30
DOI: https://doi.org/10.1090/S0002-9947-1989-0930075-2
MathSciNet review: 930075
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Abstract: A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter $ \lambda \in {{\mathbf{R}}^N}$ is studied. In particular, for such a family, the period function $ (\xi ,\lambda) \mapsto P(\xi ,\lambda)$ is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by $ \xi \in {\mathbf{R}}$) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with $ \lambda $ as bifurcation parameter.

Generally, if the function $ \rho $, given by $ \xi \mapsto P(\xi ,{\lambda_\ast}) - P(0,{\lambda_\ast})$, vanishes to order $ 2k$ at the origin, then it is shown that the period function, after a perturbation of $ {\lambda_\ast}$, has at most $ k$ critical points near the origin. If $ \rho $ vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of $ P$ for perturbations of $ {\lambda_\ast}$ depends on the number of generators of the ideal of all Taylor coefficients of $ \rho (\xi ,\lambda)$, where the coefficients are considered elements of the ring of convergent power series in $ \lambda $. Specifically, if the ideal is generated by the first $ 2k$ Taylor coefficients, then a perturbation of $ {\lambda_\ast}$ produces at most $ k$ critical points of $ P$ near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven.

Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin's treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hilbert's 16th problem first posed by V. Arnold.


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

  • [1] A. A. Andronov, Theory of bifurcations of dynamical systems on a plane, Wiley, New York, 1973.
  • [2] V. I. Arnold, Ordinary differential equations (R. A. Silverman, translator), MIT Press, 1978. MR 0508209 (58:22707)
  • [3] -, Geometrical methods in the theory of ordinary differential equations (J. Szücs, translator), Springer-Verlag, New York, 1983. MR 695786 (84d:58023)
  • [4] A. Baider and R. Churchill, Unique normal forms for planar vector fields, Preprint, Hunter College, 1987. MR 961812 (90k:58146)
  • [5] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl. 100 (1954), 1-19. MR 0059426 (15:527h)
  • [6] P. Biler, On the stationary solutions of Burger's equation, Colloq. Math. 52 (1987), 305-312. MR 893547 (88h:35098)
  • [7] T. R. Blows and N. G. Lloyd, The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 215-239. MR 768345 (86g:34030)
  • [8] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1969.
  • [9] E. Brieskorn and H. Knörrer, Plane algebraic curves (J. Stillwell, translator), Birkhäuser-Verlag, Boston, Mass., 1986. MR 886476 (88a:14001)
  • [10] B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, Multidimensional Systems Theory (N. K. Bose, ed.), Reidel, Boston, Mass., 1985.
  • [11] C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), 310-321. MR 903390 (88i:58050)
  • [12] -, Geometric methods for nonlinear two point boundary value problems, J. Differential Equations (to appear).
  • [13] C. Chicone and F. Dumortier, A quadratic system with a non monotonic period function, Proc. Amer. Math. Soc. (to appear). MR 929007 (89d:58106)
  • [14] S. N. Chow and J. K. Hale, Methods of bifurcation theory, Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
  • [15] S. N. Chow and J. A. Sanders, On the number of critical points of the period, J. Differential Equations 64 (1986), 51-66. MR 849664 (87j:34075)
  • [16] S. N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, Časopis Pěst. Mat. 111 (1986), 14-25. MR 833153 (87e:34069)
  • [17] R. Conti, About centers of quadratic planar systems, Universita Degli Studi di Firenze, 1986.
  • [18] -, About centers of planar cubic systems, Universita Degli Studi di Firenze, 1986.
  • [19] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304. MR 0196182 (33:4374)
  • [20] J.-P. Françoise, Cycles limites études locale, Report /83/M/13, Inst. Hautes Études Sci., 1983.
  • [21] J.-P. Françoise and C. Pugh, Keeping track of limit cycles, J. Differential Equations 65 (1986), 139-157. MR 861513 (88a:58162)
  • [22] W. Fulton, Algebraic curves, Benjamin, New York, 1969. MR 0313252 (47:1807)
  • [23] J. Guckenheimer, R. Rand, and D. Schlomink, Degenerate homoclinic cycles in perturbation of quadratic Hamiltonian systems, Preprint, 1987.
  • [24] M. Hervé, Several complex variables, Oxford Univ. Press, 1963.
  • [25] P. Henrici, Applied and computational complex analysis, Vol. 1, Wiley-Interscience, New York, 1974. MR 0372162 (51:8378)
  • [26] D. Knuth, The art of computer programming, Addison-Wesley, Reading, Mass., 1981. MR 0378456 (51:14624)
  • [27] 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 (28:3199)
  • [28] V. Lunkevich and K. Sibirskii, Integrals of a general quadratic differential system in cases of a center, Differential Equations 18 (1982), 563-568. MR 661356 (83i:34005)
  • [29] A. Lyapunov, Problème général de la stabilité du mouvement, Ann. of Math. Studies, No. 17, Princeton Univ. Press, Princeton, N. J., 1949.
  • [30] F. Murray and K. Miller, Existence theorems for ordinary differential equations, New York Univ. Press, New York, 1954. MR 0064934 (16:358b)
  • [31] L. M. Perko, On the accumulation of limit cycles, Proc. Amer. Math. Soc. 99 (1987), 515-526. MR 875391 (88b:34040)
  • [32] I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Differential Equations 5 (1969), 796-802.
  • [33] G. S. Petrov, Number of zeros of complete elliptic integrals, Functional Anal. Appl. 18 (1984), 73-74. MR 745710 (85j:33002)
  • [34] -, Elliptic integrals and their nonoscillation, Functional Anal. Appl. 20 (1986), 37-40. MR 831048 (87f:58031)
  • [35] T. Poston and I. Stewart, Catastrophe theory and its applications, Pitman, London, 1978. MR 0501079 (58:18535)
  • [36] R. Roussarie, private communication, 1987.
  • [37] F. Rothe, Periods of oscillation, nondegeneracy and specific heat of Hamiltonian systems in the plane, Proc. Internat. Conf. on Differential Equations and Math. Physics, Birmingham, Alabama, 1986.
  • [38] G. Sansone and R. Conti, Non-linear differential equations, Macmillan, New York, 1964. MR 0177153 (31:1417)
  • [39] K. Sibirskii, On the number of limit cycles in the neighborhood of a singular point, Differential Equations 1 (1965), 36-47. MR 0188542 (32:5980)
  • [40] C. K. Siegel and J. K. Moser, Lectures on celestial mechanics, Springer-Verlag, New York, 1971. MR 0502448 (58:19464)
  • [41] R. Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math. 363 (1985), 96-109. MR 814016 (87b:58029)
  • [42] A. Seidenberg, Elements of algebraic curves, Addison-Wesley, Reading, Mass., 1968. MR 0248139 (40:1393)
  • [43] J. Smoller and A. Wasserman, Global bifurcation of steady state solutions, J. Differential Equations 39 (1981), 269-290. MR 607786 (82d:58056)
  • [44] J. Sotomayor and R. Paterlini, Quadratic vector fields with finitely many periodic orbits, Internat. Sympos. on Dynamical Systems, I.M.P.A., Rio de Janeiro, 1983. MR 730297 (85b:58107)
  • [45] M. Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961), 569-578. MR 0123060 (23:A391)
  • [46] -, The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity, Arch. Rational Mech. Anal. 11 (1962), 27-33. MR 0141834 (25:5231)
  • [47] A. N. Varchenko, Estimation of the number of zeros of an Abelian integral depending on a parameter, and limit cycles, Functional Anal. Appl. 18 (1984), 98-108. MR 745696 (85g:32033)
  • [48] W. Vasconcelos, private communication, 1987.
  • [49] B. L. van der Waerden, Algebra, Vol. II, Ungar, New York, 1950.
  • [50] -, Algebra, Vol. II, Ungar, New York, 1970.
  • [51] J. Waldvogel, The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986), 178-184. MR 829122 (87j:92034)
  • [52] Yan-Qian Ye, et al. Theory of limit cycles, Transl. Math. Monographs, Vol. 66, Amer. Math. Soc., Providence, R.I., 1984.
  • [53] O. Zariski and P. Samuel, Commutative algebra, Vol. II, Van Nostrand, Princeton, N.J., 1960. MR 0120249 (22:11006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0930075-2
Keywords: Period function, center, bifurcation, quadratic system, Hamiltonian system, linearization
Article copyright: © Copyright 1989 American Mathematical Society

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