Nonlinear Dynamics

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

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

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) MATHMathSciNetGoogle Scholar
  2. 2.
    Brayton, R.K.: Nonlinear oscillations in a distributed network. Quart. J. Appl. Math. 24, 289–301 (1967) MATHMathSciNetGoogle 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) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Carr, J.: Applications of Centre Manifold Theory. Springer, New York (1981) MATHGoogle 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) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chow, S.-N., Lu, K.: C k center unstable manifolds. Proc. Roy. Soc. Edinb. A 108, 303–320 (1988) MathSciNetMATHGoogle Scholar
  7. 7.
    Chow, S.-N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equ. 26, 112–159 (1977) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cruz, M.A., Hale, J.K.: Stability of functional differential equations of neutral type. J. Differ. Equ. 7, 334–355 (1970) MATHCrossRefMathSciNetGoogle 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) MATHCrossRefMathSciNetGoogle 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) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATHGoogle Scholar
  12. 12.
    Hale, J.K., Weedermann, M.: On perturbations of delay-differential equations with periodic orbits. J. Differ. Equ. 197, 219–246 (2004) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981) MATHGoogle 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) MATHCrossRefMathSciNetGoogle 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) MATHMathSciNetCrossRefGoogle 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) MATHMathSciNetGoogle Scholar
  17. 17.
    Lopes, O.: Stability and forced oscillations. J. Math. Anal. Appl. 55, 686–698 (1976) MATHCrossRefMathSciNetGoogle 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) CrossRefMathSciNetMATHGoogle 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) MATHCrossRefMathSciNetGoogle 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) MATHGoogle 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) MATHCrossRefMathSciNetGoogle 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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