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Preface
Finding periodic orbits in statedependent 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 [SpringerVerlag, 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), 116. doi: 10.1016/00457825(72)900187. 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGrawHill Book Company, Inc., New YorkTorontoLondon, 1955. 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," SpringerVerlag, New York, 1995. 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays," Ph.D thesis, University of Giessen, 2006. 
[6] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDEBIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 121. 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 functionaldifferential equations, J. Differ. Eq., 98 (1992), 277298. 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, SpringerVerlag, 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, SpringerVerlag, New York, 1990. 
[10] 
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724730. doi: 10.1016/j.aml.2010.12.017. 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99, SpringerVerlag, New York, 1993. 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent 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/NorthHolland, Amsterdam, (2006), 435545. 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay, Journal of Differential Equations, 248 (2010), 28012840. 
[14] 
A. R. Humphries, O. DeMasi, F. M. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple statedependent delays, Discrete and Continuous Dynamical Systems  Series A, 32 (2012), 27012727. 
[15] 
T. Insperger, D. A. W. Barton and G. Stépán, Criticality of Hopf bifurcation in statedependent delay model of turning processes, International Journal of NonLinear Mechanics, 43 (2008), 140149. doi: 10.1016/j.ijnonlinmec.2007.11.002. 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent 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 statedependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 9931028. doi: 10.3934/dcds.2003.9.993. 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, SpringerVerlag, New York, 2004. 
[21] 
J. MalletParet, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functionaldifferential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. 
[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), 923944. 
[24] 
H.O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 51935207. 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), 395405. doi: 10.1016/00220396(70)90118X. 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 47994838. doi: 10.1090/S0002994798020832. 
show all references
References:
[1] 
E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [SpringerVerlag, 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), 116. doi: 10.1016/00457825(72)900187. 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGrawHill Book Company, Inc., New YorkTorontoLondon, 1955. 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," SpringerVerlag, New York, 1995. 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays," Ph.D thesis, University of Giessen, 2006. 
[6] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDEBIFTOOL, ACM Transactions on Mathematical Software, 28 (2002), 121. 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 functionaldifferential equations, J. Differ. Eq., 98 (1992), 277298. 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, SpringerVerlag, 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, SpringerVerlag, New York, 1990. 
[10] 
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724730. doi: 10.1016/j.aml.2010.12.017. 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99, SpringerVerlag, New York, 1993. 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent 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/NorthHolland, Amsterdam, (2006), 435545. 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay, Journal of Differential Equations, 248 (2010), 28012840. 
[14] 
A. R. Humphries, O. DeMasi, F. M. Magpantay and F. Upham, Dynamics of a delay differential equation with multiple statedependent delays, Discrete and Continuous Dynamical Systems  Series A, 32 (2012), 27012727. 
[15] 
T. Insperger, D. A. W. Barton and G. Stépán, Criticality of Hopf bifurcation in statedependent delay model of turning processes, International Journal of NonLinear Mechanics, 43 (2008), 140149. doi: 10.1016/j.ijnonlinmec.2007.11.002. 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent 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 statedependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 9931028. doi: 10.3934/dcds.2003.9.993. 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, SpringerVerlag, New York, 2004. 
[21] 
J. MalletParet, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functionaldifferential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. 
[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), 923944. 
[24] 
H.O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 51935207. 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), 395405. doi: 10.1016/00220396(70)90118X. 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 47994838. doi: 10.1090/S0002994798020832. 
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