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Resonance phenomena in a scalar delay differential equation with two state-dependent delays. (English) Zbl 1371.34115


MSC:

34K18 Bifurcation theory of functional-differential equations
37M20 Computational methods for bifurcation problems in dynamical systems
34K06 Linear functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
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[1] 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), pp. 855–869, . · Zbl 0760.92018
[2] D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee, Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Commun. Math. Phys., 83 (1982), pp. 303–354, . · Zbl 0499.70034
[3] J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation, SIAM J. Appl. Math., 54 (1994), pp. 1402–1424, . · Zbl 0809.34077
[4] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford Science Publications, New York, 2003. · Zbl 1038.65058
[5] A. Bellen, M. Zennaro, S. Maset, and N. Guglielmi, Recent trends in the numerical solution of retarded functional differential equations, Acta Numer., 18 (2009), pp. 1–110, . · Zbl 1178.65078
[6] R. E. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963, . · Zbl 0105.06402
[7] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1–23, . · Zbl 1352.34101
[8] G. Brown, C. M. Postlethwaite, and M. Silber, Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation, Phys. D, 240 (2011), pp. 859–871, . · Zbl 1219.37062
[9] M. Craig, A. R. Humphries, and M. C. Mackey, A mathematical model of granulopoiesis incorporating the negative feedback dynamics and kinetics of G-CSF/neutrophil binding and internalization, Bull. Math. Biol., 78 (2016), pp. 2304–2357, . · Zbl 1361.92022
[10] 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, . · Zbl 1195.37031
[11] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka, and A. M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), pp. 277–318, . · Zbl 1208.92082
[12] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis, Appl. Math. Sci. 110, Springer-Verlag, New York, Berlin, 1995. · Zbl 0826.34002
[13] H. A. Dijkstra, Dynamical Oceanography, Springer-Verlag, New York, Berlin, 2008.
[14] R. D. Driver, Existence theory for a delay-differential system, Contrib. Differential Equations, 1 (1963), pp. 317–336. · Zbl 0126.10102
[15] R. D. Driver, Ordinary and Delay Differential Equations, Appl. Math. Sci. 20, Springer-Verlag, New York, Berlin, 1977. · Zbl 0374.34001
[16] M. Eichmann, A Local Hopf Bifurcation Theorem for Differential Equations with State-Dependent Delays, Ph.D. thesis, Mathematics, Universität Gieß en, Germany, 2006, .
[17] K. Engelborghs, T. Luzyanina, and D. Roose, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), pp. 1–21, . · Zbl 1070.65556
[18] G. Fan, S. A. Campbell, G. S. Wolkowicz, and H. Zhu, The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons, J. Dynam. Differential Equations, 25 (2013), pp. 193–216, . · Zbl 1279.34084
[19] B. Fiedler, V. Flunkert, M. Georgi, P. Hövel, and E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett., 98 (2007), 114101, .
[20] J. Foss, A. Longtin, B. Mensour, and J. Milton, Multistability and delayed recurrent loops, Phys. Rev. Lett., 76 (1996), pp. 708–711, .
[21] P. Getto and M. Waurick, A differential equation with state-dependent delay from cell population biology, J. Differential Equations, 260 (2016), pp. 6176–6200, . · Zbl 1335.34099
[22] K. Green, B. Krauskopf, and K. Engelborghs, Bistability and torus break-up in a semiconductor laser with phase-conjugate feedback, Phys. D, 173 (2002), pp. 114–129, . · Zbl 1023.34064
[23] S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Appl. Math. Sci. 184, Springer-Verlag, New York, 2013. · Zbl 1316.34003
[24] I. Györi and F. Hartung, On the exponential stability of a state-dependent delay equation, Acta Sci. Math. (Szeged), 66 (2000), pp. 71–84.
[25] I. Györi and F. Hartung, Exponential stability of a state-dependent delay system, Discrete Contin. Dyn. Syst. Ser. A, 18 (2007), pp. 773–791, .
[26] J. K. Hale, Theory of Functional Differential Equations, Appl. Math. Sci. 3, Springer-Verlag, New York, 1977. · Zbl 0352.34001
[27] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Appl. Math. Sci. 99, Springer-Verlag, New York, 1993. · Zbl 0787.34002
[28] F. Hartung, Nonlinear variation of constants formula for differential equations with state-dependent delays, J. Dynam. Differential Equations, 28 (2016), pp. 1187–1213, . · Zbl 1355.34098
[29] 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 3, A. Can͂ada, P. Drábek, and A. Fonda, eds., Elsevier - North-Holland, Amsterdam, 2006, pp. 435–545.
[30] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method I: Finitely differentiable, hyperbolic case, J. Dynam. Differential Equations, (2016), . · Zbl 1384.34080
[31] X. He and R. de la Llave, Construction of quasi-periodic solutions of state-dependent delay differential equations by the parameterization method II: Analytic case, J. Differential Equations, 261 (2016), pp. 2068–2108, . · Zbl 1407.34094
[32] Q. Hu and J. Wu, Global Hopf bifurcation for differential equations with state-dependent delay, J. Differential Equations, 248 (2010), pp. 2801–2840, . · Zbl 1214.34057
[33] A. R. Humphries, D. A. Bernucci, R. Calleja, N. Homayounfar, and M. Snarski, Periodic solutions of a singularly perturbed delay differential equation with two state-dependent delays, J. Dynam. Differential Equations, 28 (2016), pp. 1215–1263, . · Zbl 1353.34083
[34] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, and F. Upham, Dynamics of a delay differential equation with multiple state-dependent delays, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), pp. 2701–2727, . · Zbl 1246.34073
[35] T. Insperger, J. Milton, and G. Stépán, Acceleration feedback improves balancing against reflex delay, J. Roy. Soc. Interface, 10 (2012), 79, .
[36] T. Insperger, G. Stépán, and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), pp. 275–283, . · Zbl 1177.74197
[37] S. G. Janssens, On a normalization technique for codimension two bifurcations of equilibria of delay differential equations, Master’s thesis, Faculty of Science, Universiteit Utrecht, The Netherlands, 2010, .
[38] R. Jessop and S. A. Campbell, Approximating the stability region of a neural network with a general distribution of delays, Neural Netw., 23 (2010), pp. 1187–1201, .
[39] W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hövel, and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Phys. Rev. E, 76 (2007), 026210, .
[40] D. M. Kane and K. A. Shore, eds., Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers, Wiley, New York, 2005.
[41] H. Kaper and H. Engler, Mathematics and Climate, SIAM, Philadelphia, 2013. · Zbl 1285.86001
[42] M. Kloosterman, S. A. Campbell, and F. J. Poulin, A closed NPZ model with delayed nutrient recycling, J. Math. Biol., 68 (2014), pp. 815–850, . · Zbl 1351.37275
[43] G. Kozyreff and T. Erneux, Singular Hopf bifurcation in a differential equation with large state-dependent delay, Proc. R. Soc. A, 470 (2013), 0596, . · Zbl 1353.34086
[44] B. Krauskopf, Bifurcation sequences at 1:4 resonance: An inventory, Nonlinearity, 7 (1994), pp. 1073–1091, . · Zbl 0804.58041
[45] B. Krauskopf, The bifurcation set for the 1:4 resonance problem, Exper. Math., 3 (1994), pp. 107–128, . · Zbl 0828.34027
[46] B. Krauskopf and D. D. Lenstra, eds., Fundamental Issues of Nonlinear Laser Dynamics, AIP Conf. Proc. 548, American Institute of Physics, College Park, MD, 2000.
[47] B. Krauskopf and K. Green, Computing unstable manifolds of periodic orbits in delay differential equations, J. Comput. Math., 186 (2003), pp. 230–249, . · Zbl 1017.65102
[48] B. Krauskopf and J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Nin͂o southern oscillation, Proc. R. Soc. A, 470 (2014), 2169, . · Zbl 1320.86004
[49] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, Discrete Contin. Dyn. Syst. Ser. A, 9 (2003), pp. 993–1028, . · Zbl 1048.34123
[50] T. Krisztin, \(C^1\)-smoothness of center manifolds for differential equations with state-dependent delay, in Nonlinear Dynamics and Evolution Equations, X.-Q. Z. Hermann Brunner and X. Zou, eds., Fields Inst. Commun. 48, Amer. Math. Soc., Providence, RI, 2006, pp. 213–226. · Zbl 1109.34057
[51] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci. 112, Springer-Verlag, New York, 2004. · Zbl 1082.37002
[52] Y. N. Kyrychko, K. B. Blyuss, and E. Schöll, Amplitude and phase dynamics in oscillators with distributed-delay coupling, Phil. Trans. R. Soc. A, 371 (2013), . · Zbl 1353.34045
[53] V. G. LeBlanc, Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators, J. Differential Equations, 254 (2013), pp. 637–647, . · Zbl 1263.34099
[54] K. Lüdge, ed., Nonlinear Laser Dynamics. From Quantum Dots to Cryptography, Wiley-VCH, New York, 2012.
[55] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), pp. 287–289, . · Zbl 1383.92036
[56] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags, I. Arch. Ration. Mech. Anal., 120 (1992), pp. 99–146, . · Zbl 0763.34056
[57] 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), pp. 129–197. · Zbl 0854.34072
[58] J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time lags: III, J. Differential Equations, 189 (2003), pp. 640–692, . · Zbl 1035.34089
[59] J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differential Equations, 250 (2011), pp. 4085–4103, . · Zbl 1250.34057
[60] J. Mallet-Paret and R. D. Nussbaum, Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations, J. Differential Equations, 250 (2011), pp. 4037–4084, . · Zbl 1227.34072
[61] J. Mallet-Paret and R. D. Nussbaum, Periodic Solutions of Differential Equations with Two State-Dependent Delays, (2017), in preparation. · Zbl 1250.34057
[62] J. Mallet-Paret, R. D. Nussbaum, and P. Paraskevopoulos, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), pp. 101–162. · Zbl 0808.34080
[63] Mathworks, MATLAB 2015b, Mathworks, Natick, MA, 2015.
[64] J. Milton, J. L. Townsend, M. A. King, and T. Ohira, Balancing with positive feedback: The case for discontinuous control, Phil. Trans. R. Soc. A, 367 (2009), pp. 1181–1193, . · Zbl 1185.34119
[65] C. M. Postlethwaite, Stabilization of long-period periodic orbits using time-delayed feedback control, SIAM J. Appl. Dyn. Syst., 8 (2009), pp. 21–39, . · Zbl 1164.37011
[66] A. S. Purewal, C. M. Postlethwaite, and B. Krauskopf, A global bifurcation analysis of the subcritical Hopf normal form subject to Pyragas time-delayed feedback control, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 1879–1915, . · Zbl 1323.37034
[67] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), pp. 421–428, .
[68] V. Pyragas and K. Pyragas, Adaptive modification of the delayed feedback control algorithm with a continuously varying time delay, Phys. Lett. A, 375 (2011), pp. 3866–3871, . · Zbl 1254.34083
[69] R. Qesmi and H.-O. Walther, Center-stable manifolds for differential equations with state-dependent delays, Discrete Contin. Dyn. Syst. Ser. A, 23 (2009), pp. 1009–1033, . · Zbl 1163.34047
[70] E. Schöll, G. Hiller, P. Hövel, and M. A. Dahlem, Time-delayed feedback in neurosystems, Phil. Trans. R. Soc. A, 367 (2009), pp. 1079–1096, . · Zbl 1185.34108
[71] J. Sieber, Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), pp. 2607–2651, . · Zbl 1259.34059
[72] J. Sieber, K. Engelborghs, T. Luzyanina, G. Samaey, and D. Roose, DDE-BIFTOOL Manual—Bifurcation analysis of delay differential equations, 2015, .
[73] R. Sipahi, F. M. Atay, and S.-I. Niculescu, Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J. Appl. Math., 68 (2007), pp. 738–759, . · Zbl 1146.34058
[74] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Appl. Math., Springer, New York, 2011. · Zbl 1227.34001
[75] G. Stépán, Retarded dynamical systems: Stability and characteristic functions, Longman Scientific & Technical, Harlow, UK, 1989.
[76] E. Stumpf, On a differential equation with state-dependent delay: A center-unstable manifold connecting an equilibrium and a periodic orbit, J. Dynam. Differential Equations, 24 (2012), pp. 197–248, . · Zbl 1267.34125
[77] R. Szalai and D. Roose, Continuation and bifurcation analysis of delay differential equations, in Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems, B. Krauskopf, H. M. Osinga, and J. Galán-Vioque, eds., Springer-Verlag, New York, Berlin, 2007, pp. 359–399. · Zbl 1132.34001
[78] B. I. Wage, Normal Form Computations for Delay Differential Equations in DDE-BIFTOOL, Master’s thesis, Faculty of Science, Universiteit Utrecht, The Netherlands, 2014, .
[79] E. Wall, F. Guichard, and A. R. Humphries, Synchronization in ecological systems by weak dispersal coupling with time delay, Theor. Ecol., 6 (2013), pp. 405–418, .
[80] H.-O. Walther, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004), pp. 5193–5207, . · Zbl 1069.37015
[81] H.-O. Walther, Complicated histories close to a homoclinic loop generated by variable delay, Adv. Differential Equations, 19 (2014), pp. 911–946, . · Zbl 1300.34162
[82] Y. Yuan and J. Bélair, Threshold dynamics in an SEIRS model with latency and temporary immunity, J. Math. Biol., 69 (2014), pp. 875–904, . · Zbl 1345.92160
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