275
Views
3
CrossRef citations to date
0
Altmetric
Section B

Stability analysis in a second-order differential equation with delays

Pages 1345-1354 | Received 27 Apr 2010, Accepted 06 Mar 2012, Published online: 30 Apr 2012

References

  • Berger , B. S. , Rokni , M. and Minis , I. 1993 . Complex dynamics in metal cutting . Appl. Math. , 51 : 601 – 602 .
  • Beuter , A. , Belair , J. and Labrie , C. 1993 . Feedback and delays in neurological diseases: A modeling study using dynamical systems . Bull. Math. Biol. , 55 : 525 – 541 .
  • Boe , E. and Chang , H. C. 1989 . Dynamics of delayed systems under feedback control . Chem. Engng. Sci. , 44 : 1281 – 1294 .
  • Boe , E. and Chang , H. C. 1991 . Transition to chaos from a two torus in a delayed feedback system . Int. J. Bifurc. Chaos , 1 : 67 – 82 .
  • Boese , F. G. and van den Driessche , P. 1994 . Stability with respect to the delay in a class of differential delay equations . Can. Appl. Math. Quart. , 2 ( 2 ) : 151 – 176 .
  • Campbell , S. A. and Belair , J. 1995 . Analytical and symbolically assisted investigation of Hopf bifurcation in delay differential equations . Can. Appl. Math. Quart. , 3 ( 2 ) : 137 – 154 .
  • Campbell , S. A. , Belair , J. , Ohira , T. and Milton , J. 1995 . Limit cycles, tori and complex dynamics in a second-order differential equation with delayed negative feedback . J. Dynam. Differ. Eqns , 7 ( 1 ) : 213 – 236 .
  • Ding , X. H. , Fan , D. and Liu , M. Z. 2007 . Stability and bifurcation of a numerical discretization Mackey-Glass system . Chaos, Solitons & Fractals , 34 : 383 – 393 .
  • Fan , D. and Wei , J. 2008 . Hopf bifurcation analysis in a tri-neuron network with time delay . Nonlinear Anal. Real World Appl. , 9 : 9 – 25 .
  • Fofana , M. S. 2002 . Asymptotic stability of a stochastic delay equation . Probabilist Eng Mech , 17 : 385 – 392 .
  • Ford , N. J. and Wulf , V. 1998 . “ Numerical Hopf bifurcation for the delay logistic equation ” . 1 – 19 . Tech. Rep., Number: MCCM-NAR-323, Manchester Center for Computational Mathematics
  • Guo , S. , Chen , Y. and Wu , J. 2008 . Two-parameter bifurcations in a network of two neurons with multiple delays . J. Diff. Eqns , 244 : 444 – 486 .
  • Hassard , B. D. , Kazarinoff , N. D. and Wa , Y. H. 1981 . Theory and Applications of Hopf Bifurcation , Cambidge : Cambidge University Press .
  • Liu , M. Z. and Wang , Q. 2009 . Numerical Hopf bifurcation of linear multi-step methods for a class of delay differential equations . Appl. Math. Comput. , 208 : 462 – 474 .
  • Wang , Q. and Liu , M. Z. 2009 . Stability and Neimark–Sacker bifurcation in Runge–Kutta methods for a predator-prey system . Int. J. Comput. Math. , 86 : 2218 – 2214 .
  • Wang , Q. , Li , D. and Liu , M. Z. 2009 . Numerical Hopf bifurcation of Runge–Kutta methods for a class of delay differential equations . Chaos, Solitons & Fractals , 42 : 3087 – 3099 .
  • Wei , J. and Zhang , C. 2008 . Bifurcation analysis of a class of neural networks with delays . Nonlinear Anal. Real World Appl , 9 : 2234 – 2252 .
  • Wulf , V. and Ford , N. J. 2000 . Numerical Hopf bifurcation for a class of delay differential equations . J. Comput. Appl. Math. , 115 : 601 – 616 .
  • Zhang , C. and Wei , J. 2004 . Stability and bifurcation analysis in a kind of business cycle model with delay . Chaos, Solitons & Fractals , 22 : 883 – 896 .
  • Zhang , J. , Jin , Z. , Yan , J. and Sun , G. 2009 . Stability and Hopf bifurcation in a delayed competition system . Nonlinear Anal. Theor. Meth. & Appl. , 70 : 658 – 670 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.