×

zbMATH — the first resource for mathematics

Bifurcation of critical periods for plane vector fields. (English) Zbl 0678.58027
A bifurcation problem in families of plane vector fields which have a nondegenerate center at the origin for all values of a parameter \(\lambda \in {\mathbb{R}}^ N\) is studied. In particular, for such a family, the period function \((\xi,\lambda)\to P(\xi,\lambda)\) is defined; it assigns the minimum period to each member of the continuous band of periodic orbits surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with \(\lambda\) as bifurcation parameter. The obtained results are applied to the quadratic systems with Bautin centers and to one degree of freedom \(``kinetic+potential''\) Hamiltonian systems with polynomial potentials.
Reviewer: J.Šiška

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. A. Andronov, Theory of bifurcations of dynamical systems on a plane, Wiley, New York, 1973.
[2] V. I. Arnol\(^{\prime}\)d, Ordinary differential equations, MIT Press, Cambridge, Mass.-London, 1978. Translated from the Russian and edited by Richard A. Silverman.
[3] V. I. Arnol\(^{\prime}\)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.
[4] Alberto Baider and Richard Churchill, Unique normal forms for planar vector fields, Math. Z. 199 (1988), no. 3, 303 – 310. · Zbl 0691.58012
[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. · Zbl 0059.08201
[6] Piotr Biler, On the stationary solutions of Burgers’ equation, Colloq. Math. 52 (1987), no. 2, 305 – 312. · Zbl 0635.35008
[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. · Zbl 0603.34020
[8] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1969. · Zbl 0205.06001
[9] Egbert Brieskorn and Horst Knörrer, Plane algebraic curves, Birkhäuser Verlag, Basel, 1986. Translated from the German by John Stillwell. · Zbl 0588.14019
[10] B. Buchberger, Gröbner bases: An algorithmic method in polynomial ideal theory, Multidimensional Systems Theory , Reidel, Boston, Mass., 1985. · Zbl 0587.13009
[11] Carmen Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987), no. 3, 310 – 321. · Zbl 0622.34033
[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. · Zbl 0651.34043
[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. · Zbl 0487.47039
[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. · Zbl 0594.34028
[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). · Zbl 0603.34034
[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. · Zbl 0143.11903
[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. · Zbl 0602.34019
[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. · Zbl 0681.14011
[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. · Zbl 0113.29003
[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. · Zbl 0313.30001
[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. · Zbl 0357.68001
[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. · Zbl 0139.04301
[28] V. A. Lunkevich and K. S. Sibirskiĭ, Integrals of a general quadratic differential system in cases of the center, Differentsial\(^{\prime}\)nye Uravneniya 18 (1982), no. 5, 786 – 792, 915 (Russian).
[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. · Zbl 0347.34001
[31] L. M. Perko, On the accumulation of limit cycles, Proc. Amer. Math. Soc. 99 (1987), no. 3, 515 – 526. · Zbl 0626.34022
[32] I. Pleshkan, A new method of investigating the isochronicity of a system of two differential equations, Differential Equations 5 (1969), 796-802. · Zbl 0252.34034
[33] G. S. Petrov, The number of zeros of complete elliptic integrals, Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 73 – 74 (Russian). · Zbl 0547.14003
[34] G. S. Petrov, Elliptic integrals and their nonoscillation, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 46 – 49, 96 (Russian). · Zbl 0656.34017
[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. · Zbl 0382.58006
[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. · Zbl 0128.08403
[39] K. S. Sibirskiĭ, On the number of limit cycles in the neighborhood of a singular point, Differencial\(^{\prime}\)nye Uravnenija 1 (1965), 53 – 66 (Russian). · Zbl 0196.35702
[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. · Zbl 0817.70001
[41] Renate Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math. 363 (1985), 96 – 109. · Zbl 0565.34037
[42] A. Seidenberg, Elements of the theory of algebraic curves, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. · Zbl 0159.33303
[43] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981), no. 2, 269 – 290. · Zbl 0425.34028
[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. · Zbl 0557.34038
[45] Minoru Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961), 569 – 578. · Zbl 0100.29901
[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. · Zbl 0134.07205
[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). · Zbl 0545.58038
[48] W. Vasconcelos, private communication, 1987.
[49] B. L. van der Waerden, Algebra, Vol. II, Ungar, New York, 1950. · Zbl 0037.01903
[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. · Zbl 0588.92018
[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. · Zbl 0322.13001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.