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Phase models and oscillators with time delayed coupling
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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