Nonlinear Dynamics

, Volume 52, Issue 3, pp 199–206 | Cite as

Normal forms for NFDEs with parameters and application to the lossless transmission line

  • Chuncheng Wang
  • Junjie Wei
Original Paper


A method for the computation of normal forms for neutral functional differential equations (NFDEs) with parameters is developed by considering an extension of phase space, based on the method of computing normal forms for FDEs with parameters previously introduced by Faria. The Hopf bifurcation of the differential difference equation is considered as an example of a circuit involving a lossless transmission line. The direction and stability of the bifurcating periodic solutions are also determined. Finally, numerical simulations are carried out to support the analytic results.


Normal form NFDE Lossless transmission line 


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  1. 1.
    Brayton, R.K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type. Quart. Appl. Math. 24, 215–224 (1966) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brayton, R.K.: Nonlinear oscillations in a distributed network. Quart. J. Appl. Math. 24, 289–301 (1967) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Brumley, W.E.: On the asymptotic behavior of solutions of differential-difference equations of neutral type. J. Differ. Equ. 7, 175–188 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Carr, J.: Applications of Centre Manifold Theory. Springer, New York (1981) zbMATHGoogle Scholar
  5. 5.
    Cao, J., He, G.: Periodic solutions for higher order neutral differential equations with several delays. Comput. Math. Appl. 48, 1491–1503 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chow, S.-N., Lu, K.: C k center unstable manifolds. Proc. Roy. Soc. Edinb. A 108, 303–320 (1988) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chow, S.-N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equ. 26, 112–159 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cruz, M.A., Hale, J.K.: Stability of functional differential equations of neutral type. J. Differ. Equ. 7, 334–355 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Faria, T., Magalhaes, L.: Normal forms for retarded functional differential equation and applications to Bogdanov–Takens singularity. J. Differ. Equ. 122, 201–224 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Faria, T., Magalhaes, L.: Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation. J. Differ. Equ. 122, 181–200 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) zbMATHGoogle Scholar
  12. 12.
    Hale, J.K., Weedermann, M.: On perturbations of delay-differential equations with periodic orbits. J. Differ. Equ. 197, 219–246 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) zbMATHGoogle Scholar
  14. 14.
    Hausrath, R.: Stability in the critical case of pure imaginary roots for neutral functional differential equations. J. Differ. Equ. 13, 329–397 (1973) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Krawcewicz, W., Ma, S., Wu, J.: Multiple slowly oscillating periodic solutions in coupled lossless transmission lines. Nonlinear Anal. 5, 309–354 (2004) zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Krawcewicz, W., Wu, J., Xia, H.: Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems. Can. Appl. Math. Quart. 1(2), 167–219 (1993) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Lopes, O.: Stability and forced oscillations. J. Math. Anal. Appl. 55, 686–698 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Qu, Y., Wei, J.: Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure. Nonlinear Dyn. 49, 285–294 (2007) CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Weedermann, M.: Normal forms for neutral functional differential equations. In: Topics in Functional Differential and Difference Equations, vol. 29, pp. 361–368. American Mathematical Society, Providence (2001) Google Scholar
  20. 20.
    Weedermann, M.: Hopf bifurcation calculations for scalar delay differential equations. Nonlinearity 19, 2091–2102 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Wei, J., Ruan, S.: Stability and global Hopf bifurcation for neutral differential quations. Acta Math. Sin. 45(1), 94–104 (2002) MathSciNetGoogle Scholar
  22. 22.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990) zbMATHGoogle Scholar
  23. 23.
    Yu, W., Cao, J.: Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay. Nonlinear Anal. 62, 141–165 (2005) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina

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