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July  2020, 40(7): 4533-4564. doi: 10.3934/dcds.2020190

Large spectral gap induced by small delay and its application to reduction

1. 

Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China

2. 

School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Jun Shen, junshen85@163.com

Received  October 2019 Published  April 2020

Fund Project: This work was partly supported by NSFC grant (No.11831012), the Fundamental Research Funds for the Central Universities (No.YJ201646), International Visiting Program for Excellent Young Scholars of SCU

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

Citation: Shuang Chen, Jun Shen. Large spectral gap induced by small delay and its application to reduction. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190
References:
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Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

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J. K. Hale and S. M. Verduyn Lunel, Effects of small delays on stability and control, in Operator Theory and Analysis (Amsterdam, 1997), Birkhäuser, Basel, (2001), 275–301. doi: 10.1007/978-3-0348-8283-5_10.  Google Scholar

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D. Henry, Topics in analysis, Publ. Sec. Mat. Univ. Autònoma Barcelona, 31 (1987), 29-84.   Google Scholar

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M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

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M. Kaashoek and S. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations, 112 (1994), 374-406.  doi: 10.1006/jdeq.1994.1109.  Google Scholar

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A. Kelly, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  doi: 10.1016/0022-0396(67)90016-2.  Google Scholar

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M. Krupa and J. Touboul, Canard explosion in delay differential equations, J. Dynam. Differential Equations, 28 (2016), 471-491.  doi: 10.1007/s10884-015-9478-2.  Google Scholar

[32]

X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations, 63 (1986), 227-254.  doi: 10.1016/0022-0396(86)90048-3.  Google Scholar

[33]

X.-B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations, 108 (1994), 36-63.  doi: 10.1006/jdeq.1994.1024.  Google Scholar

[34] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton University Press, Princeton, N.J., 1947.   Google Scholar
[35] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London, 1966.   Google Scholar
[36]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[39]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.  Google Scholar

[40]

W. Rudin, Real and Complex Analysis, 3$rd$ edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[41]

J. A. Rjabov, Certain asymptotic properties of linear systems with small time-lag, Trudy Sem. Teor. Differencial. Uravneniĭ s Otklon. Argumentom Univ. Družby Narodov Patrisa Lumumby, 3 (1965), 153-164.   Google Scholar

[42]

G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  doi: 10.2307/2374346.  Google Scholar

[43]

S. M. Verduyn Lunel, Series expansions for functional-differential equations, Integral Equations Operator Theory, 22 (1995), 93-122.  doi: 10.1007/BF01195491.  Google Scholar

[44]

W. N. Zhang, Generalized exponential dichotomies and invariant manifolds for differential equations, Adv. in Math. (China), 22 (1993), 1-45.   Google Scholar

show all references

References:
[1]

O. Arino and M. Pituk, More on linear differential systems with small delays, J. Differential Equations, 170 (2001), 381-407.  doi: 10.1006/jdeq.2000.3824.  Google Scholar

[2]

L. Barreira, D. Dragičević and C. Valls, Admissibility and Hyperbolicity, Springer, Cham, 2018. doi: 10.1007/978-3-319-90110-7.  Google Scholar

[3]

P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in Dynamics Reported, Wiley, Chinchester, 1989, 1–38. doi: 10.1007/978-3-322-96657-5_1.  Google Scholar

[4]

S. CampbellE. Stone and T. Erneux, Delay induced canards in a model of high speed machining, Dyn. Syst., 24 (2009), 373-392.  doi: 10.1080/14689360902852547.  Google Scholar

[5]

J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[6]

M. Carter and B. van Brunt, The Lebesgue-Stieltjes Integral: A Practical Introduction, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1174-7.  Google Scholar

[7]

C. Chicone, Inertial and slow manifolds for delay equations with small delays, J. Differential Equations, 190 (2003), 364-406.  doi: 10.1016/S0022-0396(02)00148-1.  Google Scholar

[8]

C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/surv/070.  Google Scholar

[9]

S.-N. Chow and K. Lu, Invariant manifolds for flows in Banach spaces, J. Differential Equations, 74 (1988), 285-317.  doi: 10.1016/0022-0396(88)90007-1.  Google Scholar

[10]

W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629, Springer, New York, 1978.  Google Scholar

[11]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[12]

J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-1, Academic Press, New York-London, 1969.  Google Scholar

[13]

R. D. Driver, On Ryabov's asymptotic characterizaiton of the solutions of quasi-linear differential equations with small delays, SIAM Rev., 10 (1968), 329-341.  doi: 10.1137/1010058.  Google Scholar

[14]

R. D. Driver, Linear differential systems with small delays, J. Differential Equations, 21 (1976), 149-166.  doi: 10.1016/0022-0396(76)90022-X.  Google Scholar

[15]

Z. DuJ. Li and X. Li, The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach, J. Funct. Anal., 275 (2018), 988-1007.  doi: 10.1016/j.jfa.2018.05.005.  Google Scholar

[16]

T. Faria and W. Huang, Special solutions for linear functional differential equations and asymptotic behaviour, Differential Integral Equations, 18 (2005), 337-360.   Google Scholar

[17]

C. FoiasG. R. Sell and R. Teman, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.  doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[18]

G. Greiner and M. Schwarz, Weak spectral mapping theorems for functional-differential equations, J. Differential Equations, 94 (1991), 205-216.  doi: 10.1016/0022-0396(91)90089-R.  Google Scholar

[19]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, Vol. 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.  Google Scholar

[20]

I. Györi and M. Pituk, Special solutions for neutral functional differential equations, J. Inequal. Appl., 6 (2001), 99-117.   Google Scholar

[21]

J. Hadamard, Surl'iteration et les solutions asymptotiquesd es equations differentielles, Bull. Soc. Math. France, 29 (1901), 224-228.   Google Scholar

[22]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[23]

J. K. Hale and S. M. Verduyn Lunel, Effects of small delays on stability and control, in Operator Theory and Analysis (Amsterdam, 1997), Birkhäuser, Basel, (2001), 275–301. doi: 10.1007/978-3-0348-8283-5_10.  Google Scholar

[24]

D. Henry, The adjoint of a linear functional differential equation and boundary value problems, J. Differential Equations, 9 (1971), 55-66.  doi: 10.1016/0022-0396(70)90153-1.  Google Scholar

[25]

D. Henry, Linear autonomous neutral functional differential equations, J. Differential Equations, 15 (1974), 106-128.  doi: 10.1016/0022-0396(74)90089-8.  Google Scholar

[26]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[27]

D. Henry, Topics in analysis, Publ. Sec. Mat. Univ. Autònoma Barcelona, 31 (1987), 29-84.   Google Scholar

[28]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[29]

M. Kaashoek and S. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differential Equations, 112 (1994), 374-406.  doi: 10.1006/jdeq.1994.1109.  Google Scholar

[30]

A. Kelly, The stable, center-stable, center, center-unstable, unstable manifolds, J. Differential Equations, 3 (1967), 546-570.  doi: 10.1016/0022-0396(67)90016-2.  Google Scholar

[31]

M. Krupa and J. Touboul, Canard explosion in delay differential equations, J. Dynam. Differential Equations, 28 (2016), 471-491.  doi: 10.1007/s10884-015-9478-2.  Google Scholar

[32]

X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations, 63 (1986), 227-254.  doi: 10.1016/0022-0396(86)90048-3.  Google Scholar

[33]

X.-B. Lin, Exponential dichotomies in intermediate spaces with applications to a diffusively perturbed predator-prey model, J. Differential Equations, 108 (1994), 36-63.  doi: 10.1006/jdeq.1994.1024.  Google Scholar

[34] A. M. Lyapunov, Problème Géneral de la Stabilité du Mouvement, Princeton University Press, Princeton, N.J., 1947.   Google Scholar
[35] J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York-London, 1966.   Google Scholar
[36]

K. J. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.  doi: 10.1090/S0002-9939-1988-0958058-1.  Google Scholar

[37]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[38]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssysteme, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.  Google Scholar

[39]

S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991-1011.  doi: 10.1017/S0308210500003590.  Google Scholar

[40]

W. Rudin, Real and Complex Analysis, 3$rd$ edition, McGraw-Hill Book Co., New York, 1987.  Google Scholar

[41]

J. A. Rjabov, Certain asymptotic properties of linear systems with small time-lag, Trudy Sem. Teor. Differencial. Uravneniĭ s Otklon. Argumentom Univ. Družby Narodov Patrisa Lumumby, 3 (1965), 153-164.   Google Scholar

[42]

G. R. Sell, Smooth linearization near a fixed point, Amer. J. Math., 107 (1985), 1035-1091.  doi: 10.2307/2374346.  Google Scholar

[43]

S. M. Verduyn Lunel, Series expansions for functional-differential equations, Integral Equations Operator Theory, 22 (1995), 93-122.  doi: 10.1007/BF01195491.  Google Scholar

[44]

W. N. Zhang, Generalized exponential dichotomies and invariant manifolds for differential equations, Adv. in Math. (China), 22 (1993), 1-45.   Google Scholar

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