Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl

PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników

Czasopismo

2015 | 13 | 1 |

Tytuł artykułu

Computation of double Hopf points for delay differential equations

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

Wydawca

Czasopismo

Rocznik

Tom

13

Numer

1

Opis fizyczny

Daty

otrzymano
2015-08-06
zaakceptowano
2015-11-06
online
2015-11-24

Twórcy

autor
  • School of Mathematics and Statistics, Northeast Normal University,
autor
  • School of Mathematics and Statistics, Northeast Normal University,

Bibliografia

  • [1] Hale J.K., Lunel S.M.V., Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
  • [2] Driver R.D., Ordinary and Delay Differential Equations, Springer, Berlin, 1977.
  • [3] Bellman R., Cooke K.L., Differential-Difference Equations, Academic Press, New York, 1963.
  • [4] Cooke K.L., Forced periodic solutions of a stable non-linear differential-difference equation, Ann. Math., 1955, (61), 381–387.
  • [5] Hassard B.D., Kazarinoff N.D., Wan Y.-H., Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981.
  • [6] Govaerts W., Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM, Philadelphia, 2000.
  • [7] Griewand A., Reddien G., The calculation of Hopf points by a direct method, IMA J. Numer. Anal. 1983, (3), 295–303.
  • [8] Yang Z.H., Nonlinear bifurcation: Theory and Computation (in Chinese), Science Press, Beijing, 2007.
  • [9] Luzyanina T., Roose D., Numerical stabilty analysis and computation of Hopf bifurcation points for delay differential equations, J. Comput. Appl. Math., 1996, (72), 379–392. [WoS]
  • [10] Khibnik A.I., Kuznetsov Yu.A., Levitin V.V., Nikolaev E.N., Continuation techniques and iterative software for bifurcation analysis of ODEs and iterated maps, Physica D, 1993, (62), 360–371.
  • [11] Allgower E.L., Georg K., Numerical path following, Handb. Numer. Anal., 1997, (5), 3–207.
  • [12] Buonoa P.-L., Bélair J., Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations, 2003, (189), 234–266.
  • [13] Ding Y., Jiang W., Yu P., Double Hopf bifurcation in a container crane model with delayed position feedback, Appl. Math. Comput., 2013, (219), 9270–9281. [WoS]
  • [14] Li Y., Non-resonant double Hopf bifurcation of a class-B laser system, Appl. Math. Comput., 2014, (226), 564–574. [WoS]
  • [15] Li Y., Jiang W., Wang H., Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators, J. Math. Anal. Appl., 2012, (387), 1114–1126. [WoS]
  • [16] Ma S., Lu Q., Feng Z., Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control, J. Math. Anal. Appl., 2008, (338), 993–1007. [WoS]
  • [17] Qesmi R., Babram M.A., Double Hopf bifurcation in delay differential equations, Arab J. Math. Sci., 2014, (20), 280–301.
  • [18] Shen Z., Zhang C., Double Hopf bifurcation of coupled dissipative Stuart-Landau oscillators with delay, Appl. Math. Comput., 2014, (227), 553–566. [WoS]
  • [19] Engelborghs K., Luzyanina T., Roose D., Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Softw., 2002, (280), 1-21.
  • [20] Wage B., Normal Form Computations for Delay Differential Equations in DDE-BIFTOOL, Mater thesis, Universiteit Utrecht, 2014.
  • [21] Xu Y., Huang M., Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differential Equations, 2008, (244), 582–598. [WoS]
  • [22] Xu Y., Mabonzo V.D., Analysis on Takens-Bogdanov points for delay differential equations, Appl. Math. Comput., 2012, (218), 11891–11899.
  • [23] Diekmann O., van Gils S.A., Lunel S.M.V., Walther H.-O., Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer, New York, 1995.
  • [24] Otsuka K., Chen J.L., High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback, Optics Letters, 1991, (16), 1759–1761.

Typ dokumentu

Bibliografia

Identyfikatory

Identyfikator YADDA

bwmeta1.element.doi-10_1515_math-2015-0076
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.