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 | References | Similar Articles | Additional Information

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 is studied. In particular, for such a family, the period function is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by ) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with as bifurcation parameter.

Generally, if the function , given by , vanishes to order at the origin, then it is shown that the period function, after a perturbation of , has at most critical points near the origin. If vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of for perturbations of depends on the number of generators of the ideal of all Taylor coefficients of , where the coefficients are considered elements of the ring of convergent power series in . Specifically, if the ideal is generated by the first Taylor coefficients, then a perturbation of produces at most critical points of 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.

**[1]**A. A. Andronov,*Theory of bifurcations of dynamical systems on a plane*, Wiley, New York, 1973.**[2]**V. I. Arnol′d,*Ordinary differential equations*, MIT Press, Cambridge, Mass.-London, 1978. Translated from the Russian and edited by Richard A. Silverman. MR**0508209****[3]**V. I. Arnol′d,*Geometrical methods in the theory of ordinary differential equations*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR**695786****[4]**Alberto Baider and Richard Churchill,*Unique normal forms for planar vector fields*, Math. Z.**199**(1988), no. 3, 303–310. MR**961812**, https://doi.org/10.1007/BF01159780**[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*, American Math. Soc. Translation**1954**(1954), no. 100, 19. MR**0059426****[6]**Piotr Biler,*On the stationary solutions of Burgers’ equation*, Colloq. Math.**52**(1987), no. 2, 305–312. MR**893547**, https://doi.org/10.4064/cm-52-2-305-312**[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), no. 3-4, 215–239. MR**768345**, https://doi.org/10.1017/S030821050001341X**[8]**N. Bourbaki,*Commutative algebra*, Addison-Wesley, Reading, Mass., 1969.**[9]**Egbert Brieskorn and Horst Knörrer,*Plane algebraic curves*, Birkhäuser Verlag, Basel, 1986. Translated from the German by John Stillwell. MR**886476****[10]**B. Buchberger,*Gröbner bases*:*An algorithmic method in polynomial ideal theory*, Multidimensional Systems Theory (N. K. Bose, ed.), Reidel, Boston, Mass., 1985.**[11]**Carmen Chicone,*The monotonicity of the period function for planar Hamiltonian vector fields*, J. Differential Equations**69**(1987), no. 3, 310–321. MR**903390**, https://doi.org/10.1016/0022-0396(87)90122-7**[12]**-,*Geometric methods for nonlinear two point boundary value problems*, J. Differential Equations (to appear).**[13]**Carmen Chicone and Freddy Dumortier,*A quadratic system with a nonmonotonic period function*, Proc. Amer. Math. Soc.**102**(1988), no. 3, 706–710. MR**929007**, https://doi.org/10.1090/S0002-9939-1988-0929007-7**[14]**Shui Nee Chow and Jack K. Hale,*Methods of bifurcation theory*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR**660633****[15]**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**, https://doi.org/10.1016/0022-0396(86)90071-9**[16]**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****[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**, https://doi.org/10.1016/0022-0396(66)90070-2**[20]**J.-P. Françoise,*Cycles limites études locale*, Report /83/M/13, Inst. Hautes Études Sci., 1983.**[21]**J.-P. Françoise and C. C. Pugh,*Keeping track of limit cycles*, J. Differential Equations**65**(1986), no. 2, 139–157. MR**861513**, https://doi.org/10.1016/0022-0396(86)90030-6**[22]**William Fulton,*Algebraic curves. An introduction to algebraic geometry*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss; Mathematics Lecture Notes Series. MR**0313252****[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]**Peter Henrici,*Applied and computational complex analysis*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Volume 1: Power series—integration—conformal mapping—location of zeros; Pure and Applied Mathematics. MR**0372162****[26]**Donald E. Knuth,*The art of computer programming*, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR**0378456****[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]**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****[29]**A. Lyapunov,*Problème général de la stabilité du mouvement*, Ann. of Math. Studies, No. 17, Princeton Univ. Press, Princeton, N. J., 1949.**[30]**Francis J. Murray and Kenneth S. Miller,*Existence theorems for ordinary differential equations*, New York University Press, New York, 1954. MR**0064934****[31]**L. M. Perko,*On the accumulation of limit cycles*, Proc. Amer. Math. Soc.**99**(1987), no. 3, 515–526. MR**875391**, https://doi.org/10.1090/S0002-9939-1987-0875391-1**[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,*The number of zeros of complete elliptic integrals*, Funktsional. Anal. i Prilozhen.**18**(1984), no. 2, 73–74 (Russian). MR**745710****[34]**G. S. Petrov,*Elliptic integrals and their nonoscillation*, Funktsional. Anal. i Prilozhen.**20**(1986), no. 1, 46–49, 96 (Russian). MR**831048****[35]**Tim Poston and Ian Stewart,*Catastrophe theory and its applications*, Pitman, London-San Francisco, Calif.-Melbourne: distributed by Fearon-Pitman Publishers, Inc., Belmont, Calif., 1978. With an appendix by D. R. Olsen, S. R. Carter and A. Rockwood; Surveys and Reference Works in Mathematics, No. 2. MR**0501079****[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*, Revised edition. Translated from the Italian by Ainsley H. Diamond. International Series of Monographs in Pure and Applied Mathematics, Vol. 67, A Pergamon Press Book. The Macmillan Co., New York, 1964. MR**0177153****[39]**K. S. Sibirskiĭ,*On the number of limit cycles in the neighborhood of a singular point*, Differencial′nye Uravnenija**1**(1965), 53–66 (Russian). MR**0188542****[40]**Carl Ludwig Siegel and Jürgen K. Moser,*Lectures on celestial mechanics*, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme; Die Grundlehren der mathematischen Wissenschaften, Band 187. MR**0502448****[41]**Renate Schaaf,*A class of Hamiltonian systems with increasing periods*, J. Reine Angew. Math.**363**(1985), 96–109. MR**814016**, https://doi.org/10.1515/crll.1985.363.96**[42]**A. Seidenberg,*Elements of the theory of algebraic curves*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR**0248139****[43]**J. Smoller and A. Wasserman,*Global bifurcation of steady-state solutions*, J. Differential Equations**39**(1981), no. 2, 269–290. MR**607786**, https://doi.org/10.1016/0022-0396(81)90077-2**[44]**J. Sotomayor and R. Paterlini,*Quadratic vector fields with finitely many periodic orbits*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 753–766. MR**730297**, https://doi.org/10.1007/BFb0061444**[45]**Minoru Urabe,*Potential forces which yield periodic motions of a fixed period*, J. Math. Mech.**10**(1961), 569–578. MR**0123060****[46]**Minoru Urabe,*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**, https://doi.org/10.1007/BF00253926**[47]**A. N. Varchenko,*Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles*, Funktsional. Anal. i Prilozhen.**18**(1984), no. 2, 14–25 (Russian). MR**745696****[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]**Jörg Waldvogel,*The period in the Lotka-Volterra system is monotonic*, J. Math. Anal. Appl.**114**(1986), no. 1, 178–184. MR**829122**, https://doi.org/10.1016/0022-247X(86)90076-4**[52]**Yan-Qian Ye, et al.*Theory of limit cycles*, Transl. Math. Monographs, Vol. 66, Amer. Math. Soc., Providence, R.I., 1984.**[53]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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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