August  2012, 32(8): 2701-2727. doi: 10.3934/dcds.2012.32.2701

Dynamics of a delay differential equation with multiple state-dependent delays

1. 

Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada, Canada, Canada, Canada

Received  June 2011 Revised  October 2011 Published  March 2012

We study the dynamics of a linear scalar delay differential equation $$\epsilon \dot{u}(t)=-\gamma u(t)-\sum_{i=1}^N\kappa_i u(t-a_i-c_iu(t)),$$ which has trivial dynamics with fixed delays ($c_i=0$). We show that if the delays are allowed to be linearly state-dependent ($c_i\ne0$) then very complex dynamics can arise, when there are two or more delays. We present a numerical study of the bifurcation structures that arise in the dynamics, in the non-singularly perturbed case, $\epsilon=1$. We concentrate on the case $N=2$ and $c_1=c_2=c$ and show the existence of bistability of periodic orbits, stable invariant tori, isola of periodic orbits arising as locked orbits on the torus, and period doubling bifurcations.
Citation: A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701
References:
[1]

K. A. Abell, C. E. Elmer, A. R. Humphries and E. S. Van Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Sys., 4 (2005), 755-781. doi: 10.1137/040603425.

[2]

W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048.

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,'' Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003.

[4]

J. De Luca, N. Guglielmi, A. R. Humphries and A. Politi, Electromagnetic two-body problem: Recurrent dynamics in the presence of state-dependent delay, J. Phys. A, 43 (2010), 205103, 20 pp.

[5]

J. De Luca, A. R. Humphries and S. B. Rodrigues, Finite element boundary value integration of Wheeler-Feynman electrodynamics, J. Comput. Appl. Math., (2012). doi: 10.1016/j.cam.2012.02.039.

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,'' Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995.

[7]

R. Driver, Existence theory for a delay-differential system, Contrib. Diff. Eq., 1 (1963), 317-366.

[8]

M. Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'' Ph.D thesis, Universität Gieß en, Germany, 2006.

[9]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Soft., 28 (2002), 1-21. doi: 10.1145/513001.513002.

[10]

J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback, and bistability, Curr. Opin. Chem. Biol., 6 (2002), 140-148.

[11]

C. Foley, S. Bernard and M. C. Mackey, Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses, J. Theor. Biol., 238 (2006), 754-763. doi: 10.1016/j.jtbi.2005.06.021.

[12]

R. Gambell, Birds and mammals: Antarctic whales, in "Key Environments Antarctica'' (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, New York, (1985), 223-241.

[13]

K. Green, B. Krauskopf and K. Engelborghs, Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback, Physica D, 173 (2002), 114-129. doi: 10.1016/S0167-2789(02)00656-5.

[14]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.

[15]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[16]

I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst., 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.

[17]

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

[18]

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,'' Vol 3 (eds. A Cañada, P. Drábek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545.

[19]

G. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[20]

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

[21]

T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.

[22]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.

[23]

J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Diff. Eq., 248 (2010), 992-1016.

[24]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[25]

M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors, J. Econ. Theory, 48 (1989), 497-509. doi: 10.1016/0022-0531(89)90039-2.

[26]

J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura. Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.

[27]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rat. Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497.

[28]

J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Top. Meth. Nonlin. Anal., 3 (1994), 101-162.

[29]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. II, J. Reine Angew. Math., 477 (1996), 129-197.

[30]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. III, J. Diff. Eq., 189 (2003), 640-692.

[31]

J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Diff. Eq., 250 (2011), 4037-4084.

[32]

MATLAB R2011a, The MathWorks Inc., Natick, MA, 2011.

[33]

T. H. Price, G. S. Chatta and D. C. Dale, Effect of recombinant granulocyte colony stimulating factor on neutrophil kinetics in normal young and elderly humans, Blood, 88 (1996), 335-340.

[34]

M. Santillán and M. C. Mackey, Why the lysogenic state of phage $\lambda$ is so stable: A mathematical modeling approach, Biophysical J., 86 (2004), 75-84. doi: 10.1016/S0006-3495(04)74085-0.

[35]

J. Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2607-2651.

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Texts in Applied Mathematics, 57, Springer, New York, 2011.

[37]

H.-O. Walther, On a model for soft landing with state dependent delay, J. Dyn. Diff. Eqns., 19 (2003), 593-622. doi: 10.1007/s10884-006-9064-8.

[38]

E. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87.

[39]

N. Yildirim and M. C. Mackey, Feedback regulation in the lactose operon: A mathematical modeling study and comparison with experimental data, Biophysical J., 84 (2003), 2841-2851. doi: 10.1016/S0006-3495(03)70013-7.

show all references

References:
[1]

K. A. Abell, C. E. Elmer, A. R. Humphries and E. S. Van Vleck, Computation of mixed type functional differential boundary value problems, SIAM J. Appl. Dyn. Sys., 4 (2005), 755-781. doi: 10.1137/040603425.

[2]

W. G. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869. doi: 10.1137/0152048.

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,'' Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003.

[4]

J. De Luca, N. Guglielmi, A. R. Humphries and A. Politi, Electromagnetic two-body problem: Recurrent dynamics in the presence of state-dependent delay, J. Phys. A, 43 (2010), 205103, 20 pp.

[5]

J. De Luca, A. R. Humphries and S. B. Rodrigues, Finite element boundary value integration of Wheeler-Feynman electrodynamics, J. Comput. Appl. Math., (2012). doi: 10.1016/j.cam.2012.02.039.

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis,'' Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995.

[7]

R. Driver, Existence theory for a delay-differential system, Contrib. Diff. Eq., 1 (1963), 317-366.

[8]

M. Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'' Ph.D thesis, Universität Gieß en, Germany, 2006.

[9]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Soft., 28 (2002), 1-21. doi: 10.1145/513001.513002.

[10]

J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback, and bistability, Curr. Opin. Chem. Biol., 6 (2002), 140-148.

[11]

C. Foley, S. Bernard and M. C. Mackey, Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses, J. Theor. Biol., 238 (2006), 754-763. doi: 10.1016/j.jtbi.2005.06.021.

[12]

R. Gambell, Birds and mammals: Antarctic whales, in "Key Environments Antarctica'' (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, New York, (1985), 223-241.

[13]

K. Green, B. Krauskopf and K. Engelborghs, Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback, Physica D, 173 (2002), 114-129. doi: 10.1016/S0167-2789(02)00656-5.

[14]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983.

[15]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[16]

I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst., 18 (2007), 773-791. doi: 10.3934/dcds.2007.18.773.

[17]

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

[18]

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,'' Vol 3 (eds. A Cañada, P. Drábek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545.

[19]

G. Hutchinson, Circular causal systems in ecology, Ann. N.Y. Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[20]

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

[21]

T. Insperger, G. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.

[22]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'' 3rd edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 2004.

[23]

J.-P. Lessard, Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation, J. Diff. Eq., 248 (2010), 992-1016.

[24]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[25]

M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors, J. Econ. Theory, 48 (1989), 497-509. doi: 10.1016/0022-0531(89)90039-2.

[26]

J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura. Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.

[27]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I, Arch. Rat. Mech. Anal., 120 (1992), 99-146. doi: 10.1007/BF00418497.

[28]

J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Top. Meth. Nonlin. Anal., 3 (1994), 101-162.

[29]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. II, J. Reine Angew. Math., 477 (1996), 129-197.

[30]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags. III, J. Diff. Eq., 189 (2003), 640-692.

[31]

J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Diff. Eq., 250 (2011), 4037-4084.

[32]

MATLAB R2011a, The MathWorks Inc., Natick, MA, 2011.

[33]

T. H. Price, G. S. Chatta and D. C. Dale, Effect of recombinant granulocyte colony stimulating factor on neutrophil kinetics in normal young and elderly humans, Blood, 88 (1996), 335-340.

[34]

M. Santillán and M. C. Mackey, Why the lysogenic state of phage $\lambda$ is so stable: A mathematical modeling approach, Biophysical J., 86 (2004), 75-84. doi: 10.1016/S0006-3495(04)74085-0.

[35]

J. Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete and Continuous Dynamical Systems - Series A, 32 (2012), 2607-2651.

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'' Texts in Applied Mathematics, 57, Springer, New York, 2011.

[37]

H.-O. Walther, On a model for soft landing with state dependent delay, J. Dyn. Diff. Eqns., 19 (2003), 593-622. doi: 10.1007/s10884-006-9064-8.

[38]

E. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66-87.

[39]

N. Yildirim and M. C. Mackey, Feedback regulation in the lactose operon: A mathematical modeling study and comparison with experimental data, Biophysical J., 84 (2003), 2841-2851. doi: 10.1016/S0006-3495(03)70013-7.

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