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Application (I)—Hilbert’s 16th Problem

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Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles

Part of the book series: Applied Mathematical Sciences ((AMS,volume 181))

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Abstract

In Chap. 4, Hopf bifurcation and computation of normal forms are applied to consider planar vector fields and focus on the well-known Hilbert’s 16th problem. Attention is given to general cubic order and higher order systems are considered to find the maximal number of limit cycles possible for such systems i.e., to find the lower bound of the Hilbert number for certain vector fields. The Liénard system is also investigated and critical periods of bifurcating periodic solutions from two special type of planar systems are studied.

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Notes

  1. 1.

    The command realroot, which uses dyadic rationals and binary splitting after a method of Collins, provides reliable intervals guaranteed to contain roots. However, fsolve has evolved to be equally reliable.

References

  1. Arnold, V.I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl. 11, 85–92 (1977)

    Article  Google Scholar 

  2. Bautin, N.N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. (N. S.) 30(72), 181–196 (1952)

    MathSciNet  Google Scholar 

  3. Beyn, W.J., Champneys, A., Doedel, E., Govaerts, W., Kuznetsov, Yu.A., Sandstede, B.: Numerical continuation, and computation of normal forms. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 149–219. North-Holland, Amsterdam (2002)

    Chapter  Google Scholar 

  4. Caubergh, M., Dumortier, F.: Hilbert’s 16th problem for classical Liénard equations of even degree. J. Differ. Equ. 244(6), 1359–1394 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caubergh, M., Dumortier, F., Luca, S.: Cyclicity of unbounded semihyperbolic 2-saddle cycles in polynomial Liénard systems. Discrete Contin. Dyn. Syst. 27(3), 963–980 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caubergh, M., Françoise, J.P.: Generalized Liénard equations, cyclicity and Hopf–Takens bifurcations. Qual. Theory Dyn. Syst. 5(2), 195–222 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, L.S., Wang, M.S.: The relative position, and the number, of limit cycles of a quadratic differential system. Acta Math. Sin. 22, 751–758 (1979)

    Article  MATH  Google Scholar 

  8. Chen, G., Wu, Y., Yang, X.: The number of limit cycles for a class of quintic Hamiltonian systems under quintic perturbations. J. Aust. Math. Soc. 73, 37–53 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312, 433–486 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    Book  MATH  Google Scholar 

  11. Dhooge, A., Govaerts, W., Kuznetsov, Yu.A.: MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dhooge, A., Govaerts, W., Kuznetsov, Yu.A., Meijer, H.G.E., Sautois, B.: New features of the software MatCont for bifurcation analysis of dynamical systems. Math. Comput. Model. Dyn. Syst. 14, 145–175 (2008)

    Article  MathSciNet  Google Scholar 

  13. Dumortier, F.: Compactification and desingularization of spaces of polynomial Liénard equations. J. Differ. Equ. 224(2), 296–313 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dumortier, F., Li, C.: Quadratic Liénard equations with quadratic damping. J. Differ. Equ. 139(1), 41–59 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dumortier, F., Panazzolo, D., Roussarie, R.: More limit cycles than expected in Liénard equations. Proc. Am. Math. Soc. 135, 1895–1904 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gonzalez-Vega, L., Rouillier, F., Roy, M.-F., Trujillo, G.: Symbolic Recipes for Real Solutions, in Some Tapas of Computer Algebra. Springer, Heidelberg (1999)

    Google Scholar 

  17. Han, M., Lin, Y., Yu, P.: A study on the existence of limit cycles of a planar system with 3rd-degree polynomials. Int. J. Bifurc. Chaos 14(1), 41–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hilbert, D.: Mathematical problems (M. Newton, transl.). Bull. Am. Math. Soc. 8, 437–479 (1902); reprinted in Bull. Am. Math. Soc. (N. S.), 37, 407–436 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ilyashenko, Yu., Panov, A.: Some upper estimations of the number of limit cycles of planar vector fields with application to the Liénard equation. Mosc. Math. J. 1(4), 583–599 (2001)

    MathSciNet  MATH  Google Scholar 

  20. James, E.M., Lloyd, N.G.: A cubic system with eight small-amplitude limit cycles. IMA J. Appl. Math. 47, 163–171 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jiang, J., Han, M., Yu, P., Lynch, S.: Limit cycles in two types of symmetric Liénard systems. Int. J. Bifurc. Chaos 17(6), 2169–2174 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolutsky, G.: One upper estimate on the number of limit cycles in even degree Liénard equations in the focus case. arXiv:0911.3516v1 [math.DS], November 18 (2009)

  23. Kuznetsov, Yu.A.: Numerical normalization techniques for all codimension 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36, 1104–1124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuznetsov, Yu.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)

    MATH  Google Scholar 

  25. Kuznetsov, Yu.A.: Practical computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcations. Int. J. Bifurc. Chaos 15(11), 3535–3546 (2005)

    Article  MATH  Google Scholar 

  26. Li, J.: Chaos and Melnikov Method. Chongqing University Press, Chongqing (1989)

    Google Scholar 

  27. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos 13, 47–106 (2003)

    Article  MATH  Google Scholar 

  28. Li, J., Bai, J.X.: The cyclicity of multiple Hopf bifurcation in planar cubic differential systems: M(3)≥7. Preprint, Kunming Institute of Technology (1989)

    Google Scholar 

  29. Li, J., Chan, H.S.Y., Chung, K.W.: Bifurcations of limit cycles in a Z 3-equivariant planar vector field of degree 5. Int. J. Bifurc. Chaos 11, 2287–2298 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, J., Chan, H.S.Y., Chung, K.W.: Investigations of bifurcations of limit cycles in Z 2-equivariant planar vector fields of degree 5. Int. J. Bifurc. Chaos 12(10), 2137–2157 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, J., Li, C.F.: Planar cubic Hamiltonian systems and distribution of limit cycles of (E 3). Acta Math. Sin. 28, 509–521 (1985)

    MATH  Google Scholar 

  32. Li, C., Liu, L., Yang, J.: A cubic system with thirteen limit cycles. J. Differ. Equ. 246, 3609–3619 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, J., Zhang, M.Q.: Bifurcations of limit cycles in a Z 8-equivariant planar vector field of degree 7. J. Differ. Equ. Dyn. Syst. 16(4), 1123–1139 (2004)

    Article  MATH  Google Scholar 

  34. Li, J., Zhou, H.: On the control of parameters of distributions of limit cycles for a Z 2-equivariant perturbed planar Hamiltonian polynomial vector field. Int. J. Bifurc. Chaos 15, 137–155 (2005)

    Article  MATH  Google Scholar 

  35. Liénard, A.: Etude des oscillations entretenues. Rev. Gén. électr. 23, 901–912 (1928)

    Google Scholar 

  36. Lins, A., De Melo, W., Pugh, C.C.: On Liénard equation. Lect. Notes Math. 597, 335–357 (1977)

    Article  Google Scholar 

  37. Liu, Y., Li, J.: New results on the study of Z q -equivariant planar polynomial vector fields. Qual. Theory Dyn. Syst. 9, 167–219 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, Y., Li, J., Huang, W.: Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous Systems. Science Press, Beijing (2008)

    Google Scholar 

  39. Lloyd, N.G., Blows, T.R., Kalenge, M.C.: Some cubic systems with several limit cycles. Nonlinearity 1, 653–669 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lloyd, N., Pearson, J.: Symmetry in planar dynamical systems. J. Symb. Comput. 33, 357–366 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lynch, S.: Generalized cubic Liénard equations. Appl. Math. Lett. 12, 1–6 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lynch, S., Christopher, C.J.: Limit cycles in highly non-linear differential equations. J. Sound Vib. 224(3), 505–517 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mañosas, F., Villadelprat, J.: A note on the critical periods of potential systems. Int. J. Bifurc. Chaos 16, 765–774 (2006)

    Article  MATH  Google Scholar 

  44. Rayleigh, J.: The Theory of Sound. Dover, New York (1945)

    MATH  Google Scholar 

  45. Roussarie, R.: Putting a boundary to the space of Liénard equations. Discrete Contin. Dyn. Syst. 17(2), 441–448 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Rousseau, C., Toni, B.: Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third degree. Can. Math. Bull. 36, 473–484 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  47. Shi, S.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sin. 11, 1051–1056 (1979) (in Chinese); 23, 153–158 (1980) (in English)

    Google Scholar 

  48. Smale, S.: Dynamics retrospective: great problem, attempts that failed. Physica D 51, 261–273 (1991)

    Article  MathSciNet  Google Scholar 

  49. Smale, S.: Mathematical problems for the next century. Math. Intell. 20(2), 7–15 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. Van der Pol, B.: On relaxation-oscillations. Philos. Mag. 2(7), 978–992 (1926)

    Google Scholar 

  51. Wang, D.M.: A class of cubic differential systems with 6-tuple focus. J. Differ. Equ. 87, 305–315 (1990)

    Article  MATH  Google Scholar 

  52. Wang, S.: Hilbert’s 16th problem and computation of limit cycles. PhD Thesis, The University of Western Ontario, Canada (2004)

    Google Scholar 

  53. Wang, S., Yu, P.: Bifurcation of limit cycles in a quintic Hamiltonian system under sixth-order perturbation. Chaos Solitons Fractals 26(5), 1317–1335 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wang, S., Yu, P.: Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11. Chaos Solitons Fractals 30(3), 606–621 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  55. Wang, S., Yu, P., Li, J.: Bifurcation of limit cycles in Z 10-equivariant vector fields of degree 9. Int. J. Bifurc. Chaos 16(8), 2309–2324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yang, J., Han, M., Li, J., Yu, P.: Existence conditions of thirteen limit cycles in a cubic system. Int. J. Bifurc. Chaos 20(8), 2569–2577 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Yao, W., Yu, P.: Bifurcation of small limit cycles in Z 5-equivariant planar vector fields of order 5. J. Math. Anal. Appl. 328(1), 400–413 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yu, P.: Computation of normal forms via a perturbation technique. J. Sound Vib. 211(1), 19–38 (1998)

    Article  MATH  Google Scholar 

  59. Yu, P., Han, M.: Limit cycles in 3rd-order planar system. In: International Congress of Mathematicians, Beijing, China, August 20–28, 2002

    Google Scholar 

  60. Yu, P.: Computation of limit cycles—the second part of Hilbert’s 16th problem. Fields Inst. Commun. 49, 151–177 (2006)

    Google Scholar 

  61. Yu, P., Chen, R.: Bifurcation of limit cycles in a 5th-order Z 6-equivariant planar vector field. Preprint

    Google Scholar 

  62. Yu, P., Corless, R.M.: Symbolic computation of limit cycles associated with Hilbert’s 16th problem. Commun. Nonlinear Sci. Numer. Simul. 14(12), 4041–4056 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  63. Yu, P., Han, M.: Twelve limit cycles in a 3rd-order planar system with Z 2 symmetry. Commun. Pure Appl. Anal. 3(3), 515–526 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. Yu, P., Han, M.: Twelve limit cycles in a cubic case of the 16th Hilbert problem. Int. J. Bifurc. Chaos 15(7), 2191–2205 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  65. Yu, P., Han, M.: Small limit cycles from fine focus points in cubic order Z 2-equivariant vector fields. Chaos Solitons Fractals 24(1), 329–348 (2005)

    MathSciNet  MATH  Google Scholar 

  66. Yu, P., Han, M.: Limit cycles in generalized Liénard systems. Chaos Solitons Fractals 30(5), 1048–1068 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Yu, P., Han, M.: On limit cycles of the Liénard equations with Z 2 symmetry. Chaos Solitons Fractals 31(3), 617–630 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  68. Yu, P., Han, M.: Critical periods of planar reversible vector field with 3rd-degree polynomial functions. Int. J. Bifurc. Chaos 19(1), 419–433 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  69. Yu, P., Han, M., Yuan, Y.: Analysis on limit cycles of Z q -equivariant polynomial vector fields with degree 3 or 4. J. Math. Anal. Appl. 322(1), 51–65 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  70. Zhang, T.H., Han, M., Zang, H., Meng, X.Z.: Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos Solitons Fractals 22, 1127–1138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  71. Zhang, W.N., Hou, X.R., Zeng, Z.B.: Weak centers and bifurcation of critical periods in reversible cubic systems. Comput. Math. Appl. 40, 771–782 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  72. Zoladek, H.: Eleven small limit cycles in a cubic vector field. Nonlinearity 8, 843–860 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, China, People’s Republic

    Maoan Han

  2. Dept. Applied Mathematics, University of Western Ontario, London, Ontario, Canada

    Pei Yu

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  1. Maoan Han
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Han, M., Yu, P. (2012). Application (I)—Hilbert’s 16th Problem. In: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Applied Mathematical Sciences, vol 181. Springer, London. https://doi.org/10.1007/978-1-4471-2918-9_4

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