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 & Continuous Dynamical Systems - A, 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. doi: 10.1137/040603425. Google Scholar

[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. doi: 10.1137/0152048. Google Scholar

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,'', Numerical Mathematics and Scientific Computation, (2003). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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 (1995). Google Scholar

[7]

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

[8]

M. Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'', Ph.D thesis, (2006). Google Scholar

[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. doi: 10.1145/513001.513002. Google Scholar

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

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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. doi: 10.1016/j.jtbi.2005.06.021. Google Scholar

[12]

R. Gambell, Birds and mammals: Antarctic whales,, in, (1985), 223. Google Scholar

[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. doi: 10.1016/S0167-2789(02)00656-5. Google Scholar

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J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', Applied Mathematical Sciences, 42 (1983). Google Scholar

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

[16]

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

[17]

J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,'', Applied Mathematical Sciences, 99 (1993). Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'', 3rd edition, 112 (2004). Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[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. doi: 10.1007/BF01790539. Google Scholar

[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. doi: 10.1007/BF00418497. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

MATLAB R2011a, The MathWorks Inc.,, Natick, (2011). Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0006-3495(04)74085-0. Google Scholar

[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. Google Scholar

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'', Texts in Applied Mathematics, 57 (2011). Google Scholar

[37]

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

[38]

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

[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. doi: 10.1016/S0006-3495(03)70013-7. Google Scholar

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. doi: 10.1137/040603425. Google Scholar

[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. doi: 10.1137/0152048. Google Scholar

[3]

A. Bellen and M. Zennaro, "Numerical Methods for Delay Differential Equations,'', Numerical Mathematics and Scientific Computation, (2003). Google Scholar

[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). Google Scholar

[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. Google Scholar

[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 (1995). Google Scholar

[7]

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

[8]

M. Eichmann, "A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays,'', Ph.D thesis, (2006). Google Scholar

[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. doi: 10.1145/513001.513002. Google Scholar

[10]

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

[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. doi: 10.1016/j.jtbi.2005.06.021. Google Scholar

[12]

R. Gambell, Birds and mammals: Antarctic whales,, in, (1985), 223. Google Scholar

[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. doi: 10.1016/S0167-2789(02)00656-5. Google Scholar

[14]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', Applied Mathematical Sciences, 42 (1983). Google Scholar

[15]

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

[16]

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

[17]

J. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,'', Applied Mathematical Sciences, 99 (1993). Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,'', 3rd edition, 112 (2004). Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[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. doi: 10.1007/BF01790539. Google Scholar

[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. doi: 10.1007/BF00418497. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

MATLAB R2011a, The MathWorks Inc.,, Natick, (2011). Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0006-3495(04)74085-0. Google Scholar

[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. Google Scholar

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,'', Texts in Applied Mathematics, 57 (2011). Google Scholar

[37]

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

[38]

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

[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. doi: 10.1016/S0006-3495(03)70013-7. Google Scholar

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