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On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay
1. | Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan |
References:
[1] |
K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.
doi: 10.1016/0022-247X(82)90243-8. |
[2] |
R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function, Lecture Notes in Comput. Sci., 2385 (2002), Springer, Berlin, 76-89.
doi: 10.1007/3-540-45470-5_10. |
[3] |
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359.
doi: 10.1007/BF02124750. |
[4] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[5] |
M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phy. Rev. E, 67 (2003), 036204.
doi: 10.1103/PhysRevE.67.036204. |
[6] |
B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett., 98 (2007), 114101.
doi: 10.1103/PhysRevLett.98.114101. |
[7] |
N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation, J. London Math. Soc., 25 (1950), 226-232. |
[8] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
doi: 10.1017/CBO9780511810817. |
[9] |
P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods, Phy. Rev. E, 72 (2005), 046203. |
[10] |
H. Kokame, K. Hirata, K. Konishi and T. Mori, State difference feedback for stabilizing uncertain steady states of non-linear systems, Internat. J. Control, 74 (2001), 537-546.
doi: 10.1080/00207170010017275. |
[11] |
H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system, Proc. Amer. Math. Soc., 136 (2008), 4305-4312.
doi: 10.1090/S0002-9939-08-09396-9. |
[12] |
K. Pyragas, Continuous control of chaos by self-controlling feedback, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118-123.
doi: 10.1016/B978-012396840-1/50038-2. |
[13] |
H. Shinozaki and T. Mori, Robust stability analysis of linear time-delay systems by Lambert $W$ function: Some extreme point results, Automatica, 42 (2006), 1791-1799.
doi: 10.1016/j.automatica.2006.05.008. |
[14] |
G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Harlow, 1989. |
[15] |
J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay, Nonlinear Anal., 59 (2004), 657-671.
doi: 10.1016/j.na.2004.07.027. |
[16] |
E. M. Wright, Solution of the equation $z e^z = a$, Bull. Amer. Math. Soc., 65 (1959), 89-93.
doi: 10.1090/S0002-9904-1959-10290-1. |
show all references
References:
[1] |
K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627.
doi: 10.1016/0022-247X(82)90243-8. |
[2] |
R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function, Lecture Notes in Comput. Sci., 2385 (2002), Springer, Berlin, 76-89.
doi: 10.1007/3-540-45470-5_10. |
[3] |
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359.
doi: 10.1007/BF02124750. |
[4] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[5] |
M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phy. Rev. E, 67 (2003), 036204.
doi: 10.1103/PhysRevE.67.036204. |
[6] |
B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett., 98 (2007), 114101.
doi: 10.1103/PhysRevLett.98.114101. |
[7] |
N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation, J. London Math. Soc., 25 (1950), 226-232. |
[8] |
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
doi: 10.1017/CBO9780511810817. |
[9] |
P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods, Phy. Rev. E, 72 (2005), 046203. |
[10] |
H. Kokame, K. Hirata, K. Konishi and T. Mori, State difference feedback for stabilizing uncertain steady states of non-linear systems, Internat. J. Control, 74 (2001), 537-546.
doi: 10.1080/00207170010017275. |
[11] |
H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system, Proc. Amer. Math. Soc., 136 (2008), 4305-4312.
doi: 10.1090/S0002-9939-08-09396-9. |
[12] |
K. Pyragas, Continuous control of chaos by self-controlling feedback, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118-123.
doi: 10.1016/B978-012396840-1/50038-2. |
[13] |
H. Shinozaki and T. Mori, Robust stability analysis of linear time-delay systems by Lambert $W$ function: Some extreme point results, Automatica, 42 (2006), 1791-1799.
doi: 10.1016/j.automatica.2006.05.008. |
[14] |
G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Harlow, 1989. |
[15] |
J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay, Nonlinear Anal., 59 (2004), 657-671.
doi: 10.1016/j.na.2004.07.027. |
[16] |
E. M. Wright, Solution of the equation $z e^z = a$, Bull. Amer. Math. Soc., 65 (1959), 89-93.
doi: 10.1090/S0002-9904-1959-10290-1. |
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