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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 and Continuous Dynamical Systems, 2020, 40 (7) : 4533-4564. doi: 10.3934/dcds.2020190
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.

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[38]

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

[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.

[40]

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

[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. 

[42]

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

[43]

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

[44]

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

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.

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[25]

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

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[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.

[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.

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

[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.

[38]

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

[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.

[40]

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

[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. 

[42]

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

[43]

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

[44]

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

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