# American Institute of Mathematical Sciences

August  2012, 32(8): 2607-2651. doi: 10.3934/dcds.2012.32.2607

## 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

Received  November 2010 Revised  September 2011 Published  March 2012

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays, transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an extension of the Hopf bifurcation theorem by Eichmann (2006), along with an alternative proof.
Citation: Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607
##### References:
 [1] E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods,", Reprint of the 1990 edition [Springer-Verlag, 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: 10.1016/0045-7825(72)90018-7.  Google Scholar [3] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill 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,", Springer-Verlag, (1995).   Google Scholar [5] Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,", Ph.D thesis, (2006).   Google Scholar [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.  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 functional-differential 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,, Springer-Verlag, (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,, Springer-Verlag, (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 Functional-Differential Equations," Applied Mathematical Sciences, 99,, Springer-Verlag, (1993).   Google Scholar [12] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435.   Google Scholar [13] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent 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 state-dependent 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 state-dependent delay model of turning processes,, International Journal of Non-Linear Mechanics, 43 (2008), 140.  doi: 10.1016/j.ijnonlinmec.2007.11.002.  Google Scholar [16] T. Insperger, G. Stépán and J. Turi, State-dependent 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 state-dependent 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,, Springer-Verlag, (2004).   Google Scholar [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.   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: 10.1016/0022-0396(70)90118-X.  Google Scholar [26] J. Wu, Symmetric functional-differential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar

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##### References:
 [1] E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods,", Reprint of the 1990 edition [Springer-Verlag, 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: 10.1016/0045-7825(72)90018-7.  Google Scholar [3] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations,", McGraw-Hill 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,", Springer-Verlag, (1995).   Google Scholar [5] Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,", Ph.D thesis, (2006).   Google Scholar [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.  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 functional-differential 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,, Springer-Verlag, (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,, Springer-Verlag, (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 Functional-Differential Equations," Applied Mathematical Sciences, 99,, Springer-Verlag, (1993).   Google Scholar [12] F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications,, in, (2006), 435.   Google Scholar [13] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent 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 state-dependent 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 state-dependent delay model of turning processes,, International Journal of Non-Linear Mechanics, 43 (2008), 140.  doi: 10.1016/j.ijnonlinmec.2007.11.002.  Google Scholar [16] T. Insperger, G. Stépán and J. Turi, State-dependent 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 state-dependent 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,, Springer-Verlag, (2004).   Google Scholar [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.   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: 10.1016/0022-0396(70)90118-X.  Google Scholar [26] J. Wu, Symmetric functional-differential equations and neural networks with memory,, Transactions of the AMS, 350 (1998), 4799.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar
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