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 and Continuous Dynamical Systems, 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, 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), 1-16. doi: 10.1016/0045-7825(72)90018-7.

[3]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," Springer-Verlag, New York, 1995.

[5]

Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays," Ph.D thesis, University of Giessen, 2006.

[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), 1-21. 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 functional-differential equations, J. Differ. Eq., 98 (1992), 277-298.

[8]

M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, Springer-Verlag, 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, Springer-Verlag, New York, 1990.

[10]

S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724-730. doi: 10.1016/j.aml.2010.12.017.

[11]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[12]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent 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/North-Holland, Amsterdam, (2006), 435-545.

[13]

Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, Journal of Differential Equations, 248 (2010), 2801-2840.

[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), 2701-2727.

[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), 140-149. doi: 10.1016/j.ijnonlinmec.2007.11.002.

[16]

T. Insperger, G. Stépán and J. Turi, State-dependent 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 state-dependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 993-1028. doi: 10.3934/dcds.2003.9.993.

[20]

Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.

[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), 101-162.

[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), 923-944.

[24]

H.-O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 5193-5207. 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), 395-405. doi: 10.1016/0022-0396(70)90118-X.

[26]

J. Wu, Symmetric functional-differential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2.

show all references

References:
[1]

E. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [Springer-Verlag, 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), 1-16. doi: 10.1016/0045-7825(72)90018-7.

[3]

E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," Springer-Verlag, New York, 1995.

[5]

Markus Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays," Ph.D thesis, University of Giessen, 2006.

[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), 1-21. 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 functional-differential equations, J. Differ. Eq., 98 (1992), 277-298.

[8]

M. Golubitsky, D. G. Schaeffer and I. Stewart, "Singularities and Groups in Bifurcation Theory," Vol. II, Applied Mathematical Sciences, 69, Springer-Verlag, 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, Springer-Verlag, New York, 1990.

[10]

S. Guo, Equivariant Hopf bifurcation for functional differential equations of mixed type, Applied Mathmeatics Letters, 24 (2011), 724-730. doi: 10.1016/j.aml.2010.12.017.

[11]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993.

[12]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent 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/North-Holland, Amsterdam, (2006), 435-545.

[13]

Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, Journal of Differential Equations, 248 (2010), 2801-2840.

[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), 2701-2727.

[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), 140-149. doi: 10.1016/j.ijnonlinmec.2007.11.002.

[16]

T. Insperger, G. Stépán and J. Turi, State-dependent 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 state-dependent delay, Discrete Contin. Dynam. Systems, 9 (2003), 993-1028. doi: 10.3934/dcds.2003.9.993.

[20]

Y. A Kuznetsov, "Elements of Applied Bifurcation Theory," Third edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.

[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), 101-162.

[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), 923-944.

[24]

H.-O. Walther, Smoothness of semiflows for differential equations with delay that depends on the solution, Journal of Mathematical Sciences, 124 (2004), 5193-5207. 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), 395-405. doi: 10.1016/0022-0396(70)90118-X.

[26]

J. Wu, Symmetric functional-differential equations and neural networks with memory, Transactions of the AMS, 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2.

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