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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

B. Krauskopf and D. Lenstra, editors, Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, New York, 2000.

[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.

[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.

[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.

[21]

A. Mielke, editor, Analysis, Modeling and Simulation of Multiscale Problems, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-35657-6.

[22]

A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, WIAS-preprint 1914 (2014), submitted to Discrete Contin. Dyn. Syst. Ser. B.

[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.

[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.

[25]

A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. doi: 10.1017/S0022112069000176.

[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.

[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.

[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.

[29]

A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422. doi: 10.1007/BF02429847.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

B. Krauskopf and D. Lenstra, editors, Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, 548, AIP, New York, 2000.

[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.

[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.

[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.

[21]

A. Mielke, editor, Analysis, Modeling and Simulation of Multiscale Problems, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-35657-6.

[22]

A. Mielke, Deriving amplitude equations via evolutionary $\Gamma$-convergence, WIAS-preprint 1914 (2014), submitted to Discrete Contin. Dyn. Syst. Ser. B.

[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.

[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.

[25]

A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303. doi: 10.1017/S0022112069000176.

[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.

[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.

[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.

[29]

A. van Harten, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci., 1 (1991), 397-422. doi: 10.1007/BF02429847.

[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.

[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.

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