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Preface
Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations
1. | Department of Mathematics, Lion Gate Building, University of Portsmouth, Portsmouth, PO1 3HF, United Kingdom |
References:
[1] |
E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [Springer-Verlag, Berlin; MR1059455 (92a:65165)], Classics in Applied Mathematics, 45, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. |
[2] |
J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1-16.
doi: 10.1016/0045-7825(72)90018-7. |
[3] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
[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. |
[5] |
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays," Ph.D thesis, University of Giessen, 2006. |
[6] |
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 1-21.
doi: 10.1145/513001.513002. |
[7] |
L. H. Erbe, W. Krawcewicz, K. Geba and J. Wu, $S^1$-degree and global Hopf bifurcation theory of functional-differential equations, J. Differ. Eq., 98 (1992), 277-298. |
[8] |
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, Springer-Verlag, New York, 1988. |
[9] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990. |
[10] |
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724-730.
doi: 10.1016/j.aml.2010.12.017. |
[11] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[12] |
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations" (eds. P. Drábek, A. Cañada and A. Fonda), Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545. |
[13] |
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, Journal of Differential Equations, 248 (2010), 2801-2840. |
[14] |
A. R. Humphries, O. DeMasi, F. M. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2701-2727. |
[15] |
T. Insperger, D. A. W. Barton and G. Stépán, Criticality of Hopf bifurcation in state-dependent delay model of turning processes, International Journal of Non-Linear Mechanics, 43 (2008), 140-149.
doi: 10.1016/j.ijnonlinmec.2007.11.002. |
[16] |
T. Insperger, G. Stépán and J. Turi, State-dependent delay model for regenerative cutting processes, in "Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference," Eindhoven, Netherlands, 2005. |
[17] |
D. Jackson, "The Theory of Approximation," Vol. XI, AMS Colloquium Publication, New York, 1930. |
[18] |
W. Krawcewicz and J. Wu, "Theory of Degrees with Applications to Bifurcations and Differential Equations," Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. |
[19] |
T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 993-1028.
doi: 10.3934/dcds.2003.9.993. |
[20] |
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. |
[21] |
J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101-162. |
[22] |
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Pitman Research Notes in Mathematics Series, 210, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989. |
[23] |
H.-O. Walther, Stable periodic motion of a system with state dependent delay, Differential Integral Equations, 15 (2002), 923-944. |
[24] |
H.-O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 5193-5207.
doi: 10.1023/B:JOTH.0000047253.23098.12. |
[25] |
E. Winston, Uniqueness of the zero solution for delay differential equations with state dependence, J. Diff. Eqs., 7 (1970), 395-405.
doi: 10.1016/0022-0396(70)90118-X. |
[26] |
J. Wu, Symmetric functional-differential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |
show all references
References:
[1] |
E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [Springer-Verlag, Berlin; MR1059455 (92a:65165)], Classics in Applied Mathematics, 45, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. |
[2] |
J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1-16.
doi: 10.1016/0045-7825(72)90018-7. |
[3] |
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. |
[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. |
[5] |
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays," Ph.D thesis, University of Giessen, 2006. |
[6] |
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 1-21.
doi: 10.1145/513001.513002. |
[7] |
L. H. Erbe, W. Krawcewicz, K. Geba and J. Wu, $S^1$-degree and global Hopf bifurcation theory of functional-differential equations, J. Differ. Eq., 98 (1992), 277-298. |
[8] |
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, Springer-Verlag, New York, 1988. |
[9] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Revised and corrected reprint of the 1983 original, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990. |
[10] |
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724-730.
doi: 10.1016/j.aml.2010.12.017. |
[11] |
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. |
[12] |
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations" (eds. P. Drábek, A. Cañada and A. Fonda), Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545. |
[13] |
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, Journal of Differential Equations, 248 (2010), 2801-2840. |
[14] |
A. R. Humphries, O. DeMasi, F. M. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2701-2727. |
[15] |
T. Insperger, D. A. W. Barton and G. Stépán, Criticality of Hopf bifurcation in state-dependent delay model of turning processes, International Journal of Non-Linear Mechanics, 43 (2008), 140-149.
doi: 10.1016/j.ijnonlinmec.2007.11.002. |
[16] |
T. Insperger, G. Stépán and J. Turi, State-dependent delay model for regenerative cutting processes, in "Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference," Eindhoven, Netherlands, 2005. |
[17] |
D. Jackson, "The Theory of Approximation," Vol. XI, AMS Colloquium Publication, New York, 1930. |
[18] |
W. Krawcewicz and J. Wu, "Theory of Degrees with Applications to Bifurcations and Differential Equations," Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. |
[19] |
T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 993-1028.
doi: 10.3934/dcds.2003.9.993. |
[20] |
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004. |
[21] |
J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101-162. |
[22] |
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions," Pitman Research Notes in Mathematics Series, 210, Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989. |
[23] |
H.-O. Walther, Stable periodic motion of a system with state dependent delay, Differential Integral Equations, 15 (2002), 923-944. |
[24] |
H.-O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 5193-5207.
doi: 10.1023/B:JOTH.0000047253.23098.12. |
[25] |
E. Winston, Uniqueness of the zero solution for delay differential equations with state dependence, J. Diff. Eqs., 7 (1970), 395-405.
doi: 10.1016/0022-0396(70)90118-X. |
[26] |
J. Wu, Symmetric functional-differential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 4799-4838.
doi: 10.1090/S0002-9947-98-02083-2. |
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