Advanced Search
Article Contents
Article Contents

On the stability of periodic orbits in delay equations with large delay

Abstract Related Papers Cited by
  • We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.
    Mathematics Subject Classification: 34K08, 34K13, 34K20, 37G15, 37L10.


    \begin{equation} \\ \end{equation}
  • [1]

    K. Engelborghs, T. Luzyanina and G. Samaey, "DDE-BIFTOOL v.2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations," Report TW 330, Katholieke Universiteit Leuven, 2001.


    J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.


    M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517.doi: 10.2307/2154470.


    R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection properties, IEEE J. of Quant. El., 16 (1980), 347-355.


    M. Lichtner, M. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802.doi: 10.1137/090766796.


    J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, "Topics in Time Delay Systems: Analysis, Algorithms and Control," 388 of Lecture Notes in Control and Information Sciences. Springer, 2009.doi: 10.1007/978-3-642-02897-7.


    D. Roose and R. Szalai, Continuation and bifurcation analysis of delay differential equations, in " Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems" (eds. B Krauskopf, H M Osinga and J Galán-Vioque), Springer-Verlag, Dordrecht (2007), 51-75.doi: 10.1007/978-1-4020-6356-5_12.


    G. Samaey, K. Engelborghs and D. Roose, Numerical computation of connecting orbits in delay differential equations, Numer. Algorithms, 30 (2002), 335-352.doi: 10.1023/A:1020102317544.


    E. Schöll and H. Schuster, "Handbook of Chaos Control," Wiley, New York, 2 edition, 2008.


    J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522doi: 10.1137/100796455.


    A. L. Skubachevskii and H.-O. Walther, On the Floquet multipliers of periodic solutions to nonlinear functional differential equations, J. Dynam. Diff. Eq., 18 (2006), 257-355.doi: 10.1007/s10884-006-9006-5.


    G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Longman Scientific and Technical, Harlow, Essex, 1989.


    R. Szalai, G. Stépán and S. J. Hogan, Continuation of bifurcations in periodic delay differential equations using characteristic matrices, SIAM Journal on Scientific Computing, 28 (2006), 1301-1317.doi: 10.1137/040618709.


    H.-O. Walther, Density of slowly oscillating solutions of $\dot x(t)=-f(x(t-1))$, Journal of Mathematical Analysis and Applications, 79 (1981), 127-140.doi: 10.1016/0022-247X(81)90014-7.


    M Wolfrum and S Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006), 220201.doi: 10.1103/PhysRevLett.96.220201.


    S Yanchuk and P Perlikowski, Delay and periodicity, Physical Review E., 79 (2009), 46221.doi: 10.1103/PhysRevE.79.046221.


    S Yanchuk and M Wolfrum, Stability of external cavity modes in the Lang-Kobayashi system with large delay, SIAM J. Appl. Dyn. Sys., 9 (2010), 519-535.doi: 10.1137/090751335.

  • 加载中

Article Metrics

HTML views() PDF downloads(240) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint