×

A functional analytic approach to validated numerics for eigenvalues of delay equations. (English) Zbl 07235863

Summary: This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by \(N \times N\) matrices for large enough \(N\). We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.

MSC:

65G20 Algorithms with automatic result verification
37B30 Index theory for dynamical systems, Morse-Conley indices
34K08 Spectral theory of functional-differential operators
34K20 Stability theory of functional-differential equations
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Software:

INTLAB
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Arioli; H. Koch, Non-symmetric low-index solutions for a symmetric boundary value problem, J. Differential Equations, 252, 448-458 (2012) · Zbl 1232.35047 · doi:10.1016/j.jde.2011.08.014
[2] G. Arioli; H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113, 51-70 (2015) · Zbl 1304.35181 · doi:10.1016/j.na.2014.09.023
[3] G. Arioli; H. Koch, Spectral stability for the wave equation with periodic forcing, J. Differential Equations, 265, 2470-2501 (2018) · Zbl 1402.35039 · doi:10.1016/j.jde.2018.04.040
[4] I. Balázs; J. B. van den Berg; J. Courtois; J. Dudás; J.-P. Lessard; A. Vörös-Kiss; J. F. Williams; X. Y. Yin, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5, 61-80 (2018) · Zbl 1409.35017 · doi:10.3934/jcd.2018003
[5] B. Barker, Numerical Proof of Stability of Roll Waves in the Small-Amplitude Limit for Inclined Thin Film Flow, Thesis (Ph.D.)-Indiana University. 2014. 482 pp. · Zbl 1300.35121
[6] B. Barker, Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, J. Differential Equations, 257, 2950-2983 (2014) · Zbl 1300.35121 · doi:10.1016/j.jde.2014.06.005
[7] B. Barker; K. Zumbrun, Numerical proof of stability of viscous shock profiles, Math. Models Methods Appl. Sci., 26, 2451-2469 (2016) · Zbl 1356.35170 · doi:10.1142/S0218202516500585
[8] M. Brackstone; M. McDonald, Car-following: A historical review, Transport. Res. Part F, 2, 181-196 (1999) · doi:10.1016/S1369-8478(00)00005-X
[9] F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40. Springer-Verlag, New York, 2001. · Zbl 0967.92015
[10] D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: Experimental comparison, Sci. Math. Jpn., 58, 377-388 (2003) · Zbl 1065.65099
[11] D. Breda; O. Diekmann; M. Gyllenberg; F. Scarabel; R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15, 1-23 (2016) · Zbl 1352.34101 · doi:10.1137/15M1040931
[12] D. Breda; S. Maset; R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal., 24, 1-19 (2004) · Zbl 1054.65079 · doi:10.1093/imanum/24.1.1
[13] D. Breda; S. Maset; R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27, 482-495 (2005) · Zbl 1092.65054 · doi:10.1137/030601600
[14] P. Brunovský; A. Erdélyi; H.-O. Walther, On a model of a currency exchange rate—local stability and periodic solutions, J. Dynam. Differential Equations, 16, 393-432 (2004) · Zbl 1078.34061 · doi:10.1007/s10884-004-4285-1
[15] R. Castelli and J.-P. Lessard, A method to rigorously enclose eigenpairs of complex interval matrices, Applications of Mathematics 2013, Acad. Sci. Czech Repub. Inst. Math., Prague, (2013), 21-31. · Zbl 1340.65057
[16] R. Castelli; J.-P. Lessard, Rigorous numerics in Floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12, 204-245 (2013) · Zbl 1293.37033 · doi:10.1137/120873960
[17] R. Castelli; H. Teismann, Rigorous numerics for NLS: Bound states, spectra, and controllability, Phys. D, 334, 158-173 (2016) · Zbl 1417.65193 · doi:10.1016/j.physd.2016.01.005
[18] J. Cyranka; T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki Model, SIAM J. Appl. Dyn. Syst., 17, 694-731 (2018) · Zbl 1415.35057 · doi:10.1137/17M111938X
[19] T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, 3. Springer, New York, 2009. · Zbl 1201.34002
[20] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254, 433-463 (2001) · Zbl 0973.35034 · doi:10.1006/jmaa.2000.7182
[21] Z. Galias; P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17, 4261-4272 (2007) · Zbl 1148.37041 · doi:10.1142/S0218127407019937
[22] M. Gameiro; J.-P. Lessard, A posteriori verification of invariant objects of evolution equations: Periodic orbits in the Kuramoto-Sivashinsky PDE, SIAM J. Appl. Dyn. Syst., 16, 687-728 (2017) · Zbl 1365.35222 · doi:10.1137/16M1073789
[23] G. H. Golub; H. A. van der Vorst, Eigenvalue computation in the 20th century. Numerical analysis 2000, Vol. Ⅲ. Linear algebra, J. Comput. Appl. Math., 123, 35-65 (2000) · Zbl 0965.65057 · doi:10.1016/S0377-0427(00)00413-1
[24] R. Haberman, Mathematical Models. Mechanical Vibrations, Population Dynamics, and Traffic Flow. An Introduction to Applied Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. · Zbl 0354.93002
[25] M. Hladík; D. Daney; E. Tsigaridas, Bounds on real eigenvalues and singular values of interval matrices, SIAM J. Matrix Anal. Appl., 31, 2116-2129 (2009/10) · Zbl 1203.65076 · doi:10.1137/090753991
[26] K. Ikeda; H. Daido; O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Phys. Rev. Lett., 45, 709-712 (1980)
[27] K. Ikeda; K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D: Nonlinear Phenomena, 29, 223-235 (1987) · Zbl 0626.58014 · doi:10.1016/0167-2789(87)90058-3
[28] J. Jaquette; J.-P. Lessard; K. Mischaikow, Stability and uniqueness of slowly oscillating periodic solutions to Wright’s equation, J. Differential Equations, 263, 7263-7286 (2017) · Zbl 1382.34069 · doi:10.1016/j.jde.2017.08.018
[29] H. Koch, On hyperbolicity in the renormalization of near-critical area-preserving maps, Discrete Contin. Dyn. Syst., 36, 7029-7056 (2016) · Zbl 1369.37053 · doi:10.3934/dcds.2016106
[30] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. · Zbl 0777.34002
[31] J.-P. Lessard and J. D. Mireles James, An implicit \({C}^1\) time-stepping scheme for delay differential equations, (Submitted), (2019), http://cosweb1.fau.edu/ jmirelesjames/methodOfSteps_CAP_DDE.html.
[32] J.-P. Lessard and J. D. Mireles James, http://www.math.mcgill.ca/jplessard/ResearchProjects/spectrumDDE/home.html, MATLAB codes to perform the proofs, 2019.
[33] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. · Zbl 1383.92036
[34] J. M. Mahaffy; J. B’elair; M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190, 135-146 (1998) · doi:10.1006/jtbi.1997.0537
[35] P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge Studies in Modern Optics, 21. Cambridge University Press, 1997.
[36] G. Mayer, Result verification for eigenvectors and eigenvalues, Topics in validated computations (Oldenburg, 1993), Stud. Comput. Math., North-Holland, Amsterdam, 5 (1994), 209-276. · Zbl 0813.65077
[37] J. D. Mireles James, Fourier-Taylor approximation of unstable manifolds for compact maps: Numerical implementation and computer-assisted error bounds, Found. Comput. Math., 17, 1467-1523 (2017) · Zbl 1383.37066 · doi:10.1007/s10208-016-9325-9
[38] R. E. Moore, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14, 611-615 (1977) · Zbl 0365.65034 · doi:10.1137/0714040
[39] R. E. Moore, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1966. · Zbl 0176.13301
[40] K. Nagatou; M. Plum; M. T. Nakao, Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468, 545-562 (2012) · Zbl 1364.34124 · doi:10.1098/rspa.2011.0159
[41] M. T. Nakao; N. Yamamoto; K. Nagatou, Numerical verifications for eigenvalues of second-order elliptic operators, Japan J. Indust. Appl. Math., 16, 307-320 (1999) · Zbl 1306.65278 · doi:10.1007/BF03167360
[42] R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4), 101, 263-306 (1974) · Zbl 0323.34061 · doi:10.1007/BF02417109
[43] J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75, 658-660 (1968) · Zbl 0183.43004 · doi:10.2307/2313800
[44] C. Reinhardt; J. D. Mireles James, Fourier-Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation, Indag. Math. (N.S.), 30, 39-80 (2019) · Zbl 1419.35131 · doi:10.1016/j.indag.2018.08.003
[45] S. M. Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl., 324, 209-226 (2001) · Zbl 0986.65031 · doi:10.1016/S0024-3795(00)00279-2
[46] S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numer., 19, 287-449 (2010) · Zbl 1323.65046 · doi:10.1017/S096249291000005X
[47] S. M. Rump; J.-P. M. Zemke, On eigenvector bounds, BIT, 43, 823-837 (2003) · Zbl 1042.65029 · doi:10.1023/B:BITN.0000009941.51707.26
[48] S. M. Rump, INTLAB - INTerval LABoratory, Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, (1999), 77-104, http://www.ti3.tu-harburg.de/rump/. · Zbl 0949.65046
[49] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. · Zbl 1227.34001
[50] J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366, 397-402 (2007) · Zbl 1203.37064 · doi:10.1016/j.physleta.2007.01.083
[51] L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. · Zbl 1264.41001
[52] A. Uçar, A prototype model for chaos studies, Internat. J. Engrg. Sci., 40, 251-258 (2002) · Zbl 1211.37041 · doi:10.1016/S0020-7225(01)00060-X
[53] A. Uçar, A prototype model for chaos studies, Internat. J. Engrg. Sci., 40, 251-258 (2002) · Zbl 1294.35047 · doi:10.1016/S0020-7225(01)00060-X
[54] Y. Watanabe; K. Nagatou; M. Plum; M. T. Nakao, Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM J. Numer. Anal., 52, 975-992 (2014) · Zbl 1156.76025 · doi:10.1137/120894683
[55] Y. Watanabe; M. Plum; M. T. Nakao, A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow, ZAMM Z. Angew. Math. Mech., 89, 5-18 (2009) · Zbl 0411.65022 · doi:10.1002/zamm.200700158
[56] T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math., 34, 189-199 (1980) · Zbl 0491.65021 · doi:10.1007/BF01396059
[57] T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors. Ⅱ, Numer. Math., 40, 201-206 (1982) · Zbl 0491.65021 · doi:10.1007/BF01400539
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.