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Section B

Existence and convergence of Neimark–Sacker bifurcation for delay differential equations using Runge–Kutta methods

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Pages 97-109 | Received 05 Oct 2008, Accepted 16 Jul 2009, Published online: 02 Nov 2010

References

  • Agliaria , A. , Gardinib , L. and Puuc , T. 2005 . Some global bifurcations related to the appearance of closed invariant curves . Math. Comput. Simulation , 68 : 201 – 219 .
  • Baker , C. T.H. , Bocharov , G. A. and Rihan , F. A. July 1999 . “ A report on the use of delay differential equations in numerical modelling in the biosciences ” . July , Mathematics Department, University of Manchester . NA Report No. 343 (ISSN: 1360-1725)
  • Bellen , A. and Zennaro , M. 2003 . Numerical Methods for Delay Differential Equations , Oxford : Oxford University Press .
  • Bellman , R. and Cooke , K. L. 1963 . Differential-difference Equations , New York : Academic Press .
  • Chen , G. R. , Moiola , J. L. and Wang , H. O. 2000 . Bifurcation control: theory, methods, and applications . Internat. J. Bifurc. Chaos , 10 ( 3 ) : 511 – 548 .
  • Cheng , Z. S. and Cao , J. D. 2006 . Bifurcation and stability analysis of a neural network model with distributed delays . Nonlinear Dyn. , 46 : 363 – 373 .
  • De la Sen , M. 2008 . On Minimal realizations and minimal partial realizations of linear time-invariant systems subject to point incommensurate delays . Math. Problems Eng. , 2008 doi: 10.1155/2008/790530
  • Ding , X. H. and Su , H. 2007 . Dynamics of a discretization physiological control system . Discrete Dyn. Nature Soc. , 2007 doi: 10.1155/2007/51406
  • Ding , X. H. , Su , H. and Liu , M. 2008 . Stability and bifurcation of numerical discretization of a second-order delay differential equation with negative feedback . Chaos, Solitons Fractals , 35 ( 4 ) : 795 – 807 .
  • Engelborghs , K. , Lemaire , V. , Bélair , J. and Roose , D. 2001 . Numerical bifurcation analysis of delay differential equations arising from physiological modeling . J. Math. Biol. , 42 : 361 – 385 .
  • Engelborghs , K. , Luzyanina , T. and Roose , D. 2002 . Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL . ACM Trans. Math. Software , 28 ( 1 ) : 1 – 21 .
  • Ford , N. J. and Wulf , V. 1999 . The use of boundary locus plots in the identification of bifurcation points in numerical approximation of delay differential equations . J. Comput. Appl. Math. , 111 ( 1–2 ) : 153 – 162 .
  • Gumela , A. B. , Twizell , E. H. and Yuc , P. 2000 . Numerical and bifurcation analyses for a population model of HIV chemotherapy . Math. Comput. Simulation , 54 : 169 – 181 .
  • Hairer , E. , Nørett , S. P. and Wanner , G. 1987 . Solving Ordinary Differential Equations, Vol. I Nonstiff Problems , Berlin, Heidelberg : Springer .
  • Hale , J. K. 1977 . Theory of Functional Differential Equations , Berlin : Springer .
  • Hassard , B. D. , Kazarinoff , N. D. and Wan , Y. H. 1981 . Theory and Applications of Hopf Bifurcation , Cambridge : Cambridge University Press .
  • He , J. H. 2005 . Perturbation method for bifurcation of nonlinear problems . Internat. J. Nonlinear Sci. Numer. Simulation , 6 ( 2 ) : 207 – 208 .
  • He , J. H. 2005 . Limit cycle and bifurcation of nonlinear problems . Chaos, Solitons Fractals, , 26 ( 3 ) : 827 – 833 .
  • In't Hout , K. J. and Lubich , C. 1998 . Periodic orbits of delay differential equations under discretization . BIT , 38 ( 1 ) : 72 – 91 .
  • Koto , T. 1999 . Neimark–Sacker bifurcations in the Euler method for a delay differential equation . BIT , 39 ( 1 ) : 110 – 115 .
  • Kuznetsov , Y. A. 1995 . Elements of Applied Bifurcation Theory , New York : Springer .
  • Lambert , J. D. 1991 . Numerical Methods for Ordinary Differential Systems , Chichester : Wiley .
  • Luzyanina , T. , Roose , D. and Bocharov , G. 2005 . Numerical bifurcation analysis of immunological models with time delays . J. Comput. Appl. Math. , 184 ( 1 ) : 165 – 176 .
  • Peng , M. S. and Uçar , A. 2004 . The use of the Euler method in identification of multiple bifurcations and chaotic behavior in numerical approximation of delay-differential equations . Chaos Solitons Fractals , 21 : 883 – 891 .
  • Wiggins , S. 1990 . Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics , Vol. 2 , Berlin : Springer .
  • Wulf , V. 1999 . “ Numerical analysis of delay differential equations undergoing a Hopf bifurcation ” . Liverpool : University of Liverpool . Ph.D. Thesis
  • Wulf , V. and Ford , N. J. 2000 . Numerical Hopf bifurcation for a class of delay differential equations . J. Comput. Appl. Math. , 115 ( 1–2 ) : 601 – 616 .
  • Xiao , M. and Cao , J. D. 2007 . Delayed feedback-based bifurcation control in an Internet congestion model . J. Comput. Appl. Math. , 332 : 1010 – 1027 .
  • Zhang , C. , Liu , M. and Zheng , B. 2004 . Hopf bifurcation in numerical approximation for delay differential equations . J. Appl. Math. Comput. , 14 : 319 – 328 .

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