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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). 
[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. 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGrawHill Book Company, (1955). 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,", SpringerVerlag, (1995). 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays,", Ph.D thesis, (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), 1. 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), 277. 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69,, SpringerVerlag, (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, (1990). 
[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. 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99,, SpringerVerlag, (1993). 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay,, Journal of Differential Equations, 248 (2010), 2801. 
[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. 
[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. 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent delay model for regenerative cutting processes,, in, (2005). 
[17] 
D. Jackson, "The Theory of Approximation," Vol. XI,, AMS Colloquium Publication, (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, (1997). 
[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. 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112,, SpringerVerlag, (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), 101. 
[22] 
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions,", Pitman Research Notes in Mathematics Series, 210 (1989). 
[23] 
H.O. Walther, Stable periodic motion of a system with state dependent delay,, Differential Integral Equations, 15 (2002), 923. 
[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. 
[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. 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799. 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, 45 (1990). 
[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. 
[3] 
E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGrawHill Book Company, (1955). 
[4] 
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,", SpringerVerlag, (1995). 
[5] 
Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with StateDependent Delays,", Ph.D thesis, (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), 1. 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), 277. 
[8] 
M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69,, SpringerVerlag, (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, (1990). 
[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. 
[11] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to FunctionalDifferential Equations," Applied Mathematical Sciences, 99,, SpringerVerlag, (1993). 
[12] 
F. Hartung, T. Krisztin, H.O. Walther and J. Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. 
[13] 
Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with statedependent delay,, Journal of Differential Equations, 248 (2010), 2801. 
[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. 
[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. 
[16] 
T. Insperger, G. Stépán and J. Turi, Statedependent delay model for regenerative cutting processes,, in, (2005). 
[17] 
D. Jackson, "The Theory of Approximation," Vol. XI,, AMS Colloquium Publication, (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, (1997). 
[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. 
[20] 
Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112,, SpringerVerlag, (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), 101. 
[22] 
G Stépán, "Retarded Dynamical Systems: Stability and Characteristic Functions,", Pitman Research Notes in Mathematics Series, 210 (1989). 
[23] 
H.O. Walther, Stable periodic motion of a system with state dependent delay,, Differential Integral Equations, 15 (2002), 923. 
[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. 
[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. 
[26] 
J. Wu, Symmetric functionaldifferential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799. doi: 10.1090/S0002994798020832. 
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