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Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems
On the stability of periodic orbits in delay equations with large delay
1. | Harrison Building, North Park Road, CEMPS, University of Exeter, Exeter, EX4 4QF, United Kingdom |
2. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany |
3. | Institute of Mathematics, Humboldt University of Berlin, Rudower Chaussee 25, 12489, Berlin |
References:
[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. |
[2] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. |
[3] |
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. |
[4] |
R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection properties, IEEE J. of Quant. El., 16 (1980), 347-355. |
[5] |
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. |
[6] |
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. |
[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. |
[8] |
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. |
[9] |
E. Schöll and H. Schuster, "Handbook of Chaos Control," Wiley, New York, 2 edition, 2008. |
[10] |
J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522
doi: 10.1137/100796455. |
[11] |
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. |
[12] |
G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Longman Scientific and Technical, Harlow, Essex, 1989. |
[13] |
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. |
[14] |
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. |
[15] |
M Wolfrum and S Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006), 220201.
doi: 10.1103/PhysRevLett.96.220201. |
[16] |
S Yanchuk and P Perlikowski, Delay and periodicity, Physical Review E., 79 (2009), 46221.
doi: 10.1103/PhysRevE.79.046221. |
[17] |
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. |
show all references
References:
[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. |
[2] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993. |
[3] |
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. |
[4] |
R. Lang and K. Kobayashi, External optical feedback effects on semiconductor injection properties, IEEE J. of Quant. El., 16 (1980), 347-355. |
[5] |
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. |
[6] |
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. |
[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. |
[8] |
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. |
[9] |
E. Schöll and H. Schuster, "Handbook of Chaos Control," Wiley, New York, 2 edition, 2008. |
[10] |
J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM Journal on Applied Dynamical Systems, 10 (2011), 129-147. arXiv:1005.4522
doi: 10.1137/100796455. |
[11] |
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. |
[12] |
G. Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Longman Scientific and Technical, Harlow, Essex, 1989. |
[13] |
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. |
[14] |
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. |
[15] |
M Wolfrum and S Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett., 96 (2006), 220201.
doi: 10.1103/PhysRevLett.96.220201. |
[16] |
S Yanchuk and P Perlikowski, Delay and periodicity, Physical Review E., 79 (2009), 46221.
doi: 10.1103/PhysRevE.79.046221. |
[17] |
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. |
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