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January  2015, 35(1): 537-553. doi: 10.3934/dcds.2015.35.537

## Spectrum and amplitude equations for scalar delay-differential equations with large delay

 1 Humboldt-University of Berlin, Institute of Mathematics, Unter den Linden 6, 10099, Berlin, Germany, Germany 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany, Germany

Received  August 2013 Revised  June 2014 Published  August 2014

The subject of the paper is scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.
Citation: Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537
##### References:
 [1] S.-N. Chow, J. K. Hale and W. Huang, From sine waves to square waves in delay equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 223-229. doi: 10.1017/S0308210500032108.  Google Scholar [2] S.-N. Chow, X.-B. Lin and J. Mallet-Paret, Transition layers for singularly perturbed delay differential equations with monotone nonlinearities, J. Dynam. Diff. Equations, 1 (1989), 3-43. doi: 10.1007/BF01048789.  Google Scholar [3] P. Collet and J. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153. doi: 10.1007/BF02278004.  Google Scholar [4] B. Fiedler, C. Rocha and M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138. doi: 10.1016/j.jde.2003.10.027.  Google Scholar [5] G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627. doi: 10.1088/0951-7715/12/6/311.  Google Scholar [6] G. Giacomelli and A. Politi, Multiple scale analysis of delayed dynamical systems, Phys. D, 117 (1998), 26-42. doi: 10.1016/S0167-2789(97)00318-7.  Google Scholar [7] G. Giacomelli and A. Politi, Relationship between delayed and spatially extended dynamical systems, Phys. Rev. Lett., 76 (1996), 2686-2689. doi: 10.1103/PhysRevLett.76.2686.  Google Scholar [8] J. Giannoulis and A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 493-523. doi: 10.3934/dcdsb.2006.6.493.  Google Scholar [9] J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities, Nonlinearity, 17 (2004), 551-565. doi: 10.1088/0951-7715/17/2/011.  Google Scholar [10] J. Hale and W. Huang, Periodic solutions of singularly perturbed delay equations, Z. Angew. Math. Phys., 47 (1996), 57-88. doi: 10.1007/BF00917574.  Google Scholar [11] S. Heiligenthal, Th. Dahms, S. Yanchuk, Th. Jüngling, V. Flunkert, I. Kanter, E. Schöll and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett., 107 (2011), 234102. doi: 10.1103/PhysRevLett.107.234102.  Google Scholar [12] W. Huang, Stability of square wave periodic solution for singularly perturbed delay differential equations, J. Differential Equations, 168 (2000), 239-269. doi: 10.1006/jdeq.2000.3886.  Google Scholar [13] O. D'Huys, S. Zeeb, Th. Jüngling, S. Heiligenthal, S. Yanchuk and W. Kinzel, Synchronisation and scaling properties of chaotic networks with multiple delays, EPL (Europhysics Letters), 103 (2013), 10013. doi: 10.1209/0295-5075/103/10013.  Google Scholar [14] A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynamics Reported. Expositions in Dynamical Systems (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Springer-Verlag, 1992, 164-224.  Google Scholar [15] S. A. Kashchenko, The Ginzburg-Landau equation as a normal form for a second-order difference-differential equation with a large delay, Comput. Meth. Math. Phys., 38 (1998), 443-451.  Google Scholar [16] P. Kirrmann, G. Schneider and A. Mielke, The validity of mudulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91. doi: 10.1017/S0308210500020989.  Google Scholar [17] B. Krauskopf and D. Lenstra, editors, Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, New York, 2000. Google Scholar [18] M. Lichtner, M. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802. doi: 10.1137/090766796.  Google Scholar [19] J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.  Google Scholar [20] J. Mallet-Paret and R. D. Nussbaum, A differential delay equations arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292. doi: 10.1137/0520019.  Google Scholar [21] A. Mielke, editor, Analysis, Modeling and Simulation of Multiscale Problems, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-35657-6.  Google Scholar [22] A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, WIAS-preprint 1914 (2014), submitted to Discrete Contin. Dyn. Syst. Ser. B. Google Scholar [23] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 759-834. doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar [24] A. Mielke, G. Schneider and A. Ziegra, Comparison of inertial manifolds and application to modulated systems, Math. Nachr., 214 (2000), 53-69. doi: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4.  Google Scholar [25] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. doi: 10.1017/S0022112069000176.  Google Scholar [26] G. Schneider, A new estimate for the Ginzburg-Landau approximation on the real axis, J. Nonlinear Sci., 4 (1994), 23-34. doi: 10.1007/BF02430625.  Google Scholar [27] J. Sieber, M. Wolfrum, M. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst. A, 33 (2013), 3109-3134. doi: 10.3934/dcds.2013.33.3109.  Google Scholar [28] M. C. Soriano, J. García-Ojalvo, C. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470. doi: 10.1103/RevModPhys.85.421.  Google Scholar [29] A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422. doi: 10.1007/BF02429847.  Google Scholar [30] M. Wolfrum and S. Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett, 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar [31] M. Wolfrum, S. Yanchuk, P. Hövel and E. Schöll, Complex dynamics in delay-differential equations with large delay, Eur. Phys. J. Special Topics, 191 (2010), 91-103. doi: 10.1140/epjst/e2010-01343-7.  Google Scholar

show all references

##### References:
 [1] S.-N. Chow, J. K. Hale and W. Huang, From sine waves to square waves in delay equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 223-229. doi: 10.1017/S0308210500032108.  Google Scholar [2] S.-N. Chow, X.-B. Lin and J. Mallet-Paret, Transition layers for singularly perturbed delay differential equations with monotone nonlinearities, J. Dynam. Diff. Equations, 1 (1989), 3-43. doi: 10.1007/BF01048789.  Google Scholar [3] P. Collet and J. Eckmann, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys., 132 (1990), 139-153. doi: 10.1007/BF02278004.  Google Scholar [4] B. Fiedler, C. Rocha and M. Wolfrum, Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle, J. Differential Equations, 201 (2004), 99-138. doi: 10.1016/j.jde.2003.10.027.  Google Scholar [5] G. Friesecke and R. L. Pego, Solitary waves on FPU lattices. I. Qualitative properties, renormalization and continuum limit, Nonlinearity, 12 (1999), 1601-1627. doi: 10.1088/0951-7715/12/6/311.  Google Scholar [6] G. Giacomelli and A. Politi, Multiple scale analysis of delayed dynamical systems, Phys. D, 117 (1998), 26-42. doi: 10.1016/S0167-2789(97)00318-7.  Google Scholar [7] G. Giacomelli and A. Politi, Relationship between delayed and spatially extended dynamical systems, Phys. Rev. Lett., 76 (1996), 2686-2689. doi: 10.1103/PhysRevLett.76.2686.  Google Scholar [8] J. Giannoulis and A. Mielke, Dispersive evolution of pulses in oscillator chains with general interaction potentials, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 493-523. doi: 10.3934/dcdsb.2006.6.493.  Google Scholar [9] J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities, Nonlinearity, 17 (2004), 551-565. doi: 10.1088/0951-7715/17/2/011.  Google Scholar [10] J. Hale and W. Huang, Periodic solutions of singularly perturbed delay equations, Z. Angew. Math. Phys., 47 (1996), 57-88. doi: 10.1007/BF00917574.  Google Scholar [11] S. Heiligenthal, Th. Dahms, S. Yanchuk, Th. Jüngling, V. Flunkert, I. Kanter, E. Schöll and W. Kinzel, Strong and weak chaos in nonlinear networks with time-delayed couplings, Phys. Rev. Lett., 107 (2011), 234102. doi: 10.1103/PhysRevLett.107.234102.  Google Scholar [12] W. Huang, Stability of square wave periodic solution for singularly perturbed delay differential equations, J. Differential Equations, 168 (2000), 239-269. doi: 10.1006/jdeq.2000.3886.  Google Scholar [13] O. D'Huys, S. Zeeb, Th. Jüngling, S. Heiligenthal, S. Yanchuk and W. Kinzel, Synchronisation and scaling properties of chaotic networks with multiple delays, EPL (Europhysics Letters), 103 (2013), 10013. doi: 10.1209/0295-5075/103/10013.  Google Scholar [14] A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in Dynamics Reported. Expositions in Dynamical Systems (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Springer-Verlag, 1992, 164-224.  Google Scholar [15] S. A. Kashchenko, The Ginzburg-Landau equation as a normal form for a second-order difference-differential equation with a large delay, Comput. Meth. Math. Phys., 38 (1998), 443-451.  Google Scholar [16] P. Kirrmann, G. Schneider and A. Mielke, The validity of mudulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 85-91. doi: 10.1017/S0308210500020989.  Google Scholar [17] B. Krauskopf and D. Lenstra, editors, Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, New York, 2000. Google Scholar [18] M. Lichtner, M. Wolfrum and S. Yanchuk, The spectrum of delay differential equations with large delay, SIAM J. Math. Anal., 43 (2011), 788-802. doi: 10.1137/090766796.  Google Scholar [19] J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl., 145 (1986), 33-128. doi: 10.1007/BF01790539.  Google Scholar [20] J. Mallet-Paret and R. D. Nussbaum, A differential delay equations arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292. doi: 10.1137/0520019.  Google Scholar [21] A. Mielke, editor, Analysis, Modeling and Simulation of Multiscale Problems, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-35657-6.  Google Scholar [22] A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, WIAS-preprint 1914 (2014), submitted to Discrete Contin. Dyn. Syst. Ser. B. Google Scholar [23] A. Mielke, The Ginzburg-Landau equation in its role as a modulation equation, in Handbook of Dynamical Systems, Vol. 2, North-Holland, Amsterdam, 2002, 759-834. doi: 10.1016/S1874-575X(02)80036-4.  Google Scholar [24] A. Mielke, G. Schneider and A. Ziegra, Comparison of inertial manifolds and application to modulated systems, Math. Nachr., 214 (2000), 53-69. doi: 10.1002/1522-2616(200006)214:1<53::AID-MANA53>3.0.CO;2-4.  Google Scholar [25] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. doi: 10.1017/S0022112069000176.  Google Scholar [26] G. Schneider, A new estimate for the Ginzburg-Landau approximation on the real axis, J. Nonlinear Sci., 4 (1994), 23-34. doi: 10.1007/BF02430625.  Google Scholar [27] J. Sieber, M. Wolfrum, M. Lichtner and S. Yanchuk, On the stability of periodic orbits in delay equations with large delay, Discrete Contin. Dyn. Syst. A, 33 (2013), 3109-3134. doi: 10.3934/dcds.2013.33.3109.  Google Scholar [28] M. C. Soriano, J. García-Ojalvo, C. R. Mirasso and I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers, Rev. Mod. Phys., 85 (2013), 421-470. doi: 10.1103/RevModPhys.85.421.  Google Scholar [29] A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422. doi: 10.1007/BF02429847.  Google Scholar [30] M. Wolfrum and S. Yanchuk, Eckhaus instability in systems with large delay, Phys. Rev. Lett, 96 (2006), 220201. doi: 10.1103/PhysRevLett.96.220201.  Google Scholar [31] M. Wolfrum, S. Yanchuk, P. Hövel and E. Schöll, Complex dynamics in delay-differential equations with large delay, Eur. Phys. J. Special Topics, 191 (2010), 91-103. doi: 10.1140/epjst/e2010-01343-7.  Google Scholar
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