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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, 45 (1990). Google Scholar 
[2] 
J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems,, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1. doi: 10.1016/00457825(72)900187. Google Scholar 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGrawHill Book Company, (1955). Google Scholar 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,", SpringerVerlag, (1995). Google Scholar 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays,", Ph.D thesis, (2006). Google Scholar 
[6] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDEBIFTOOL,, ACM Transactions on Mathematical Software, 28 (2002), 1. doi: 10.1145/513001.513002. Google Scholar 
[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), 277. Google Scholar 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69,, SpringerVerlag, (1988). Google Scholar 
[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, (1990). Google Scholar 
[10] 
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type,, Applied Mathmeatics Letters, 24 (2011), 724. doi: 10.1016/j.aml.2010.12.017. Google Scholar 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99,, SpringerVerlag, (1993). Google Scholar 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. Google Scholar 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay,, Journal of Differential Equations, 248 (2010), 2801. Google Scholar 
[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), 2701. Google Scholar 
[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), 140. doi: 10.1016/j.ijnonlinmec.2007.11.002. Google Scholar 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent delay model for regenerative cutting processes,, in, (2005). Google Scholar 
[17] 
D. Jackson, "The Theory of Approximation," Vol. XI,, AMS Colloquium Publication, (1930). Google Scholar 
[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, (1997). Google Scholar 
[19] 
T. Krisztin, A local unstable manifold for differential equations with statedependent delay,, Discrete Contin. Dynam. Systems, 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. Google Scholar 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112,, SpringerVerlag, (2004). Google Scholar 
[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), 101. Google Scholar 
[22] 
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions,", Pitman Research Notes in Mathematics Series, 210 (1989). Google Scholar 
[23] 
H.O. Walther, Stable periodic motion of a system with state dependent delay,, Differential Integral Equations, 15 (2002), 923. Google Scholar 
[24] 
H.O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution,, Journal of Mathematical Sciences, 124 (2004), 5193. doi: 10.1023/B:JOTH.0000047253.23098.12. Google Scholar 
[25] 
E. Winston, Uniqueness of the zero solution for delay differential equations with state dependence,, J. Diff. Eqs., 7 (1970), 395. doi: 10.1016/00220396(70)90118X. Google Scholar 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799. doi: 10.1090/S0002994798020832. Google Scholar 
show all references
References:
[1] 
E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods,", Reprint of the 1990 edition [SpringerVerlag, 45 (1990). Google Scholar 
[2] 
J. Baumgarte, Stabilization of constraints and integrals of motion in dynamical systems,, Computer Methods in Applied Mechanics and Engineering, 1 (1972), 1. doi: 10.1016/00457825(72)900187. Google Scholar 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGrawHill Book Company, (1955). Google Scholar 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,", SpringerVerlag, (1995). Google Scholar 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays,", Ph.D thesis, (2006). Google Scholar 
[6] 
K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDEBIFTOOL,, ACM Transactions on Mathematical Software, 28 (2002), 1. doi: 10.1145/513001.513002. Google Scholar 
[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), 277. Google Scholar 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69,, SpringerVerlag, (1988). Google Scholar 
[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, (1990). Google Scholar 
[10] 
S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type,, Applied Mathmeatics Letters, 24 (2011), 724. doi: 10.1016/j.aml.2010.12.017. Google Scholar 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99,, SpringerVerlag, (1993). Google Scholar 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. Google Scholar 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay,, Journal of Differential Equations, 248 (2010), 2801. Google Scholar 
[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), 2701. Google Scholar 
[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), 140. doi: 10.1016/j.ijnonlinmec.2007.11.002. Google Scholar 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent delay model for regenerative cutting processes,, in, (2005). Google Scholar 
[17] 
D. Jackson, "The Theory of Approximation," Vol. XI,, AMS Colloquium Publication, (1930). Google Scholar 
[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, (1997). Google Scholar 
[19] 
T. Krisztin, A local unstable manifold for differential equations with statedependent delay,, Discrete Contin. Dynam. Systems, 9 (2003), 993. doi: 10.3934/dcds.2003.9.993. Google Scholar 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112,, SpringerVerlag, (2004). Google Scholar 
[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), 101. Google Scholar 
[22] 
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions,", Pitman Research Notes in Mathematics Series, 210 (1989). Google Scholar 
[23] 
H.O. Walther, Stable periodic motion of a system with state dependent delay,, Differential Integral Equations, 15 (2002), 923. Google Scholar 
[24] 
H.O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution,, Journal of Mathematical Sciences, 124 (2004), 5193. doi: 10.1023/B:JOTH.0000047253.23098.12. Google Scholar 
[25] 
E. Winston, Uniqueness of the zero solution for delay differential equations with state dependence,, J. Diff. Eqs., 7 (1970), 395. doi: 10.1016/00220396(70)90118X. Google Scholar 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799. doi: 10.1090/S0002994798020832. Google Scholar 
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