Skip to main content

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

  • 1970 Accesses

Abstract

Chapter 1 is an introduction, presenting the background for nonlinear dynamics, bifurcation and stability, normal form method, Melnikov function and Hilbert’s 16th problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Doedel, E.J.: AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, Montreal, Canada (August 2007)

    Google Scholar 

  2. Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972)

    Article  Google Scholar 

  3. 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 

  4. Arnold, V.I.: Geometric Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)

    Book  Google Scholar 

  5. Ashkenazi, M., Chow, S.N.: Normal forms near critical points for differential equations and maps. IEEE Trans. Circuits Syst. 35, 850–862 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atherton, D.P.: Stability of Nonlinear Systems. Research Studies Press, New York (1981)

    MATH  Google Scholar 

  7. Baider, A.: Unique normal forms for vector fields and Hamiltonians. J. Differ. Equ. 78, 33–52 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baider, A., Churchill, R.: Unique normal forms for planar vector fields. Math. Z. 199, 303–310 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Baider, A., Sanders, J.A.: Unique normal forms: the nilpotent Hamiltonian case. J. Differ. Equ. 92, 282–304 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baider, A., Sanders, J.A.: Further reduction of the Takens–Bogdanov normal forms. J. Differ. Equ. 99, 205–244 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. 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 

  12. Belitskii, G.: Invariant normal forms of formal series. Funct. Anal. Appl. 13, 46–47 (1979)

    Article  MathSciNet  Google Scholar 

  13. Belitskii, G.: Normal forms relative to a filtering action of a group. Trans. Mosc. Math. Soc. 2, 1–39 (1981)

    MathSciNet  Google Scholar 

  14. Bhatia, N.P., Szegö, G.P.: Stability Theory of Dynamical Systems. Springer, Berlin (1970)

    MATH  Google Scholar 

  15. Birkhoff, G.D.: Dynamical Systems. AMS, Providence (1927)

    MATH  Google Scholar 

  16. Campbell, S.A., Stone, E.: Analysis of the chatter instability in a nonlinear model for drilling. ASME J. Comput. Nonlinear Dyn. 1, 294–306 (2006)

    Article  Google Scholar 

  17. Chen, G.R., Dong, X.: From Chaos to Order. World Scientific, Singapore (1998)

    MATH  Google Scholar 

  18. 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 

  19. Chow, S.N., Drachman, B., Wang, D.: Computation of normal forms. J. Comput. Appl. Math. 29, 1290–1430 (1990)

    Article  MathSciNet  Google Scholar 

  20. Chow, S.-N., Li, C.-C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  21. Chow, S.-N., Li, C., Yi, Y.: The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment. Ergod. Theory Dyn. Syst. 22(4), 1233–1261 (2002)

    MathSciNet  Google Scholar 

  22. Christopher, C.J., Lloyd, N.G.: Polynomial systems: a lower bound for the Hilbert numbers. Proc. R. Soc. Lond. A 450, 219–224 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Christopher, C.J., Lloyd, N.G.: Small-amplitude limit cycles in polynomial Liénard systems. Nonlinear Differ. Equ. Appl. 3(2), 183–190 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Christopher, C.J., Lynch, S.: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12(4), 1099–1112 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields—Part I: Theory and algorithm. IEEE Trans. Circuits Syst. 35, 863–880 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chua, L.O., Kokubu, H.: Normal forms for nonlinear vector fields—Part II: Applications. IEEE Trans. Circuits Syst. 36, 51–70 (1988)

    Article  MathSciNet  Google Scholar 

  27. 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 

  28. 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 

  29. Gasull, A., Torregrosa, J.: Small-amplitude limit cycles in Liénard systems via multiplicity. J. Differ. Equ. 159(1), 186–211 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gazor, M., Yu, P.: Infinite order parametric normal form of Hopf singularity. Int. J. Bifurc. Chaos 18(11), 3393–3408 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Golubitsky, M.S., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory. Springer, New York (1985)

    MATH  Google Scholar 

  32. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 4th edn. Springer, New York (1993)

    Google Scholar 

  33. Han, M.: On the number of limit cycles bifurcating from a homoclinic or heteroclinic loop. Sci. China Ser. A 36(2), 113–132 (1993)

    Google Scholar 

  34. Han, M.: Bifurcations of invariant tori and subharmonic solutions for periodic perturbed systems. Sci. China Ser. A 37(11), 1325–1336 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Han, M.: Cyclicity of planar homoclinic loops and quadratic integrable systems. Sci. China Ser. A 40(12), 1247–1258 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Han, M.: Bifurcations of limit cycles from a heteroclinic cycle of Hamiltonian systems. Chin. Ann. Math., Ser. B 19(2), 189–196 (1998)

    MATH  Google Scholar 

  37. Han, M.: Liapunov constants and Hopf cyclicity of Liénard systems. Ann. Differ. Equ. 15(2), 113–126 (1999)

    MATH  Google Scholar 

  38. Han, M.: On Hopf cyclicity of planar systems. J. Math. Anal. Appl. 245, 404–422 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Han, M.: Periodic Solution and Bifurcation Theory of Dynamical Systems. Science Publication, Beijing (2002) (in Chinese)

    Google Scholar 

  40. Han, M.: Bifurcation theory of limit cycles of planar systems. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, Ordinary Differential Equations, vol. 3. Elsevier, Amsterdam (2006). Chapter 4

    Chapter  Google Scholar 

  41. Han, M., Hu, S., Liu, X.: On the stability of double homoclinic and heteroclinic cycles. Nonlinear Anal. 53, 701–713 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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 

  43. Han, M., Luo, D., Zhu, D.: Uniqueness of limit cycles bifurcating from a singular closed orbit. II. Acta Math. Sin. 35(4), 541–548 (1992) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  44. Han, M., Shang, D., Wang, Z., Yu, P.: Bifurcation of limit cycles in a 4th-order near-Hamiltonian systems. Int. J. Bifurc. Chaos 17(11) (2007)

    Google Scholar 

  45. Han, M., Wang, Z., Zang, H.: Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems. Chin. J. Contemp. Math. 28(4), 423–434 (2007)

    MathSciNet  Google Scholar 

  46. Han, M., Wu, Y., Bi, P.: Bifurcation of limit cycles near polycycles with n vertices. Chaos Solitons Fractals 22(2), 383–394 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Han, M., Wu, Y., Bi, P.: A new cubic system having eleven limit cycles. Discrete Contin. Dyn. Syst. 12(4), 675–686 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Han, M., Yang, C.: On the cyclicity of a 2-polycycle for quadratic systems. Chaos Solitons Fractals 23, 1787–1794 (2005)

    MathSciNet  MATH  Google Scholar 

  49. Han, M., Yang, J., Tarta, A., Yang, G.: Limit cycles near homoclinic and heteroclinic loops. J. Dyn. Differ. Equ. 20, 923–944 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Han, M., Yang, J., Yu, P.: Hopf bifurcations for near-Hamiltonian systems. Int. J. Bifurc. Chaos 19(12), 4117–4130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Han, M., Ye, Y.: On the coefficients appearing in the expansion of Melnikov function in homoclinic bifurcations. Ann. Differ. Equ. 14(2), 156–162 (1998)

    MathSciNet  MATH  Google Scholar 

  52. Han, M., Ye, Y., Zhu, D.: Cyclicity of homoclinic loops and degenerate cubic Hamiltonians. Sci. China Ser. A 42(6), 605–617 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  53. Han, M., Zhang, T., Zang, H.: On the number and distribution of limit cycles in a cubic system. Int. J. Bifurc. Chaos 14(12), 4285–4292 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  54. Han, M., Zhang, T., Zang, H.: Bifurcation of limit cycles near equivariant compound cycles. Sci. China Ser. A 50(4), 503–514 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Han, M., Zhu, H.: The loop quantities and bifurcations of homoclinic loops. J. Differ. Equ. 234(2), 339–359 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  57. 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 

  58. Hopf, E.: Abzweigung einer periodischen Losung von stationaren Losung einers differential-systems. Ber. Math. Phys. Kl. Sachs Acad. Wiss. Leipzig 94, 1–22 (1942); and Ber. Math. Phys. Kl. Sachs Acad. Wiss. Leipzig Math.-Nat. Kl. 95, 3–22 (1942)

    Google Scholar 

  59. Hou, Y., Han, M.: Melnikov functions for planar near-Hamiltonian systems and Hopf bifurcations. J. Shanghai Norm. Univ. (Nat. Sci.) 35(1), 1–10 (2006)

    Google Scholar 

  60. Iooss, G.: Bifurcation of Maps and Applications. North-Holland, New York (1979)

    MATH  Google Scholar 

  61. Ilyashenko, Yu.: Centennial history of Hilbert’s 16th problem. Bull., New Ser., Am. Math. Soc. 39(3), 301–354 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  62. 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 

  63. Jiang, Q., Han, M.: Melnikov functions and perturbation of a planar Hamiltonian system. Chin. Ann. Math., Ser. B 20(2), 233–246 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  64. Jiang, X., Yuan, Z., Yu, P., Zou, X.: Dynamics of an HIV-1 therapy model of fighting a virus with another virus. J. Biol. Dyn. 3(4), 387–409 (2009)

    Article  MathSciNet  Google Scholar 

  65. Joyal, P., Rousseau, C.: Saddle quantities and applications. J. Differ. Equ. 78, 374–399 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  66. Langford, W.F., Zhan, K.: Strongly resonant Hopf bifurcation and vortex induced vibrations. In: IUTAM Symposium, Nonlinearity and Chaos in Engineering Dynamics. Springer, New York (1994)

    Google Scholar 

  67. Leblanc, V.G., Langford, W.F.: Classification and unfoldings of 1:2 resonant Hopf bifurcation. Arch. Ration. Mech. Anal. 136, 305–357 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  68. 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 

  69. 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 

  70. Li, J., Huang, Q.: Bifurcations of limit cycles forming compound eyes in the cubic system. Chin. Ann. Math., Ser. B 8, 391–403 (1987)

    MATH  Google Scholar 

  71. 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 

  72. Li, J., Lin, Y.: Global bifurcations in a perturbed cubic system with Z 2-symmetry. Acta Math. Appl. Sin. 8(2), 131–143 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  73. Li, J., Liu, Z.: Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publ. Math. 35, 487–506 (1991)

    MATH  Google Scholar 

  74. Li, J., Liu, Z.: Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system, in ordinary and delay differential equations. In: Pitman Research Notes in Mathematics Ser., vol. 272, pp. 116–128. Longman, Harlow (1991)

    Google Scholar 

  75. 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 

  76. 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 

  77. Li, J., Zhang, M., Li, S.: Bifurcations of limit cycles in a Z 2-equivariant planar polynomial vector field of degree 7. Int. J. Bifurc. Chaos 16(4), 925–943 (2006)

    Article  MATH  Google Scholar 

  78. Li, J., Zhao, X.: Rotation symmetry groups of planar Hamiltonian systems. Ann. Differ. Equ. 5, 25–33 (1989)

    MATH  Google Scholar 

  79. Liao, X.X., Wang, L., Yu, P.: Stability of Dynamical Systems. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  80. Liao, X.X., Yu, P.: Absolute Stability of Nonlinear Control Systems. Springer, New York (2008)

    Book  MATH  Google Scholar 

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

    Google Scholar 

  82. 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 

  83. Llibre, J.: Integrability of polynomial differential systems. In: Canada, A., Drabek, P., Fonda, A. (eds.) Handbook of Differential Equations, Ordinary Differential Equations, vol. 1. Elsevier, Amsterdam (2004). Chapter 5

    Google Scholar 

  84. Llibre, J., Rodríguez, G.: Configurations of limit cycles and planar polynomial vector fields. J. Differ. Equ. 198(2), 374–380 (2004)

    Article  MATH  Google Scholar 

  85. Lloyd, N.G.: Limit cycles of polynomial systems. In: Bedford, T., Swift, J. (eds.) New Directions in Dynamical Systems. London Mathematical Society Lecture Notes, vol. 40, pp. 192–234 (1988)

    Chapter  Google Scholar 

  86. Luo, D., Wang, X., Zhu, D., Han, M.: Bifurcation Theory and Methods of Dynamical Systems. Advanced Series in Dynamical Systems, vol. 15. World Scientific, Singapore (1997)

    MATH  Google Scholar 

  87. Ma, H., Han, M.: Limit cycles of a Z 3-equivariant near-Hamiltonian system. Nonlinear Anal. 71(9), 3853–3871 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  88. Mebatsion, T., Finke, S., Weiland, F., Conzelmann, K.: A CXCR4/CD4 pseudotype rhabdovirus that selectively infects HIV-1 envelope protein-expressing cells. Cell 90, 841–847 (1997)

    Article  Google Scholar 

  89. Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Mosc. Math. Soc. 12, 1–57 (1963)

    Google Scholar 

  90. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste (1892–1899)

    Google Scholar 

  91. Roussarie, R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat. 17(2), 57–101 (1986)

    Article  MathSciNet  Google Scholar 

  92. Sanders, J.A.: Normal form theory and spectral sequences. J. Differ. Equ. 192, 536–552 (2003)

    Article  MATH  Google Scholar 

  93. Schlomiuk, D.: Algebraic and geometric aspects of the theory of polynomial vector fields. In: Schlomiuk, D. (ed.) Bifurcations and Periodic Orbits of Vector Fields. NATO ASI Series C, vol. 408, pp. 429–467. Kluwer Academic, London (1993)

    Google Scholar 

  94. 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 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  97. Takens, F.: Unfoldings of certain singularities of vector fields: generalized Hopf bifurcations. J. Differ. Equ. 14, 476–493 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  98. Ushiki, S.: Normal forms for singularities of vector fields. Jpn. J. Appl. Math. 1, 1–37 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  100. 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 

  101. 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 

  102. 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 

  103. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)

    MATH  Google Scholar 

  104. Wu, J., Faria, T., Huang, Y.S.: Synchronization and stable phase-locking in a network of neurons with memory. Math. Comput. Model. 30, 117–138 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  105. Wu, Y., Han, M.: On the study of limit cycles of the generalized Rayleigh–Liénard oscillator. Int. J. Bifurc. Chaos 14(8), 2905–2914 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  106. Wu, Y., Han, M., Liu, X.: On the study of limit cycles of a cubic polynomials system under Z 4-equivariant quintic perturbation. Chaos Solitons Fractals 24(4), 999–1012 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  107. Xia, H., Wolkowicz, G.S.K., Wang, L.: Transient oscillations induced by delayed growth response in the chemostat. J. Math. Biol. 50, 489–530 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  108. Xu, Z., Hangan, H., Yu, P.: Analytical solutions for invisicid Gaussian impinging jets. ASME J. Appl. Mech. 75, 021019 (2008)

    Article  Google Scholar 

  109. Xu, F., Yu, P., Liao, X.: Global analysis on n-scroll-chaotic attractors of modified Chua’s circuit. Int. J. Bifurc. Chaos 19(1), 135–157 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  110. Yang, J., Han, M.: Limit cycles near a double homoclinic loop. Ann. Differ. Equ. 23(4), 536–545 (2007)

    MathSciNet  MATH  Google Scholar 

  111. 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 

  112. 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 

  113. Ye, Y.Q.: Theory of Limit Cycles. Transl. Math. Monographs, vol. 66. AMS, Providence (1986)

    MATH  Google Scholar 

  114. Yu, P.: Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales. Ann. Differ. Equ. 27, 19–53 (2002)

    MATH  Google Scholar 

  115. Yu, P.: Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling. J. Comput. Appl. Math. 144(1–2), 359–373 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  116. Yu, P.: Bifurcation dynamics in control systems. In: Chen, R., Hill, D.J., Yu, X. (eds.) Chaos and Bifurcation Control: Theory and Applications, vol. 293, pp. 99–126. Springer, New York (2003)

    Google Scholar 

  117. Yu, P.: A simple and efficient method for computing center manifold and normal forms associated with semi-simple cases. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 10(1–3), 273–286 (2003)

    MathSciNet  MATH  Google Scholar 

  118. Yu, P.: Closed-form conditions of bifurcation points for general differential equations. Int. J. Bifurc. Chaos 15(4), 1467–1483 (2005)

    Article  MATH  Google Scholar 

  119. Yu, P.: Local and global bifurcations to limit cycles in a class of Liénard equation. Int. J. Bifurc. Chaos 17(1), 183–198 (2007)

    Article  MATH  Google Scholar 

  120. Yu, P., Chen, G.: Hopf bifurcation control using nonlinear feedback with polynomial functions. Int. J. Bifurc. Chaos 14(5), 1683–1704 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  121. Yu, P., Chen, R.: The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations. Ann. Differ. Equ. 50(1–2), 297–313 (2007)

    MathSciNet  MATH  Google Scholar 

  122. Yu, P., Chen, R.: Computation of focus values with applications. Ann. Differ. Equ. 51(3), 409–427 (2008)

    MathSciNet  MATH  Google Scholar 

  123. 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 

  124. 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 

  125. 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 

  126. 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 

  127. Yu, P., Leung, A.Y.T.: The simplest normal form of Hopf bifurcation. Nonlinearity 16(1), 277–300 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  128. Yu, P., Leung, A.Y.T.: A perturbation method for computing the simplest normal forms of dynamical systems. J. Sound Vib. 261(1), 123–151 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  129. Yu, P., Leung, A.: Normal forms of vector fields with perturbation parameters. Chaos Solitons Fractals 34(2), 564–579 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  130. Yu, P., Leung, A.: The simplest normal form and its application to bifurcation control. Chaos Solitons Fractals 33(3), 845–863 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  131. Yu, P., Yuan, Y., Xu, J.: Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback. Commun. Nonlinear Sci. Numer. Simul. 7(1–2), 69–91 (2002)

    Article  MathSciNet  Google Scholar 

  132. Yu, P., Yuan, Y.: A matching pursuit technique for computing the simplest normal forms of vector fields. J. Symb. Comput. 35(5), 591–615 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  133. Yu, P., Yuan, Y.: An efficient method for computing the simplest normal forms of vector fields. Int. J. Bifurc. Chaos 13(1), 19–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  134. Yu, P., Zhu, S.: Computation of the normal forms for general M-DOF systems using multiple time scales. Part I: Autonomous systems. Commun. Nonlinear Sci. Numer. Simul. 10(8), 869–905 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  135. 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 

  136. Zhang, T., Zang, H., Han, M.: Bifurcations of limit cycles in a cubic system. Chaos Solitons Fractals 20, 629–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Maoan Han .

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag London Limited

About this chapter

Cite this chapter

Han, M., Yu, P. (2012). Introduction. 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_1

Download citation

Publish with us

Policies and ethics