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Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles pp 271–314Cite as

Limit Cycle Bifurcations Near a Center

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

Abstract

In Chap. 7, particular attention is given to bifurcation of limit cycles near a center. After normalizing the Hamiltonian function, detailed steps for computing the Melnikov function are described and formulas are given. Maple programs for computing the coefficients of the Melnikov function are developed and illustrative examples are presented.

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References

  1. 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 

  2. Caubergh, M., Dumortier, F.: Hopf–Takens bifurcations and centers. J. Differ. Equ. 202(1), 1–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. 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 

  6. 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 

  7. 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 

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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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  2. Pei Yu
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Correspondence to Maoan Han .

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Han, M., Yu, P. (2012). Limit Cycle Bifurcations Near a Center. 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_7

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