# American Institute of Mathematical Sciences

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

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

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:
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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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##### 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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