June  2020, 7(1): 123-158. doi: 10.3934/jcd.2020005

A functional analytic approach to validated numerics for eigenvalues of delay equations

1. 

Department of Math. and Stat., McGill University, Burnside Hall, Room 1119,805 Sherbrooke West, Montreal, QC, H3A 0B9, CANADA

2. 

Department of Mathematical Sciences, Florid Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA

* Corresponding author: J. D. Mireles James

Received  June 2019 Revised  January 2020 Published  March 2020

Fund Project: The first author is supported by the NSERC. The second author was partially supported by NSF grant DMS - 1813501

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.

Citation: J. P. Lessard, J. D. Mireles James. A functional analytic approach to validated numerics for eigenvalues of delay equations. Journal of Computational Dynamics, 2020, 7 (1) : 123-158. doi: 10.3934/jcd.2020005
References:
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G. Arioli and H. Koch, Non-symmetric low-index solutions for a symmetric boundary value problem, J. Differential Equations, 252 (2012), 448-458.  doi: 10.1016/j.jde.2011.08.014.  Google Scholar

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G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113 (2015), 51-70.  doi: 10.1016/j.na.2014.09.023.  Google Scholar

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G. Arioli and H. Koch, Spectral stability for the wave equation with periodic forcing, J. Differential Equations, 265 (2018), 2470-2501.  doi: 10.1016/j.jde.2018.04.040.  Google Scholar

[4]

I. BalázsJ. B. van den BergJ. CourtoisJ. DudásJ.-P. LessardA. Vörös-KissJ. F. Williams and X. Y. Yin, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003.  Google Scholar

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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.  Google Scholar

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B. Barker, Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, J. Differential Equations, 257 (2014), 2950-2983.  doi: 10.1016/j.jde.2014.06.005.  Google Scholar

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B. Barker and K. Zumbrun, Numerical proof of stability of viscous shock profiles, Math. Models Methods Appl. Sci., 26 (2016), 2451-2469.  doi: 10.1142/S0218202516500585.  Google Scholar

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F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

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D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: Experimental comparison, Sci. Math. Jpn., 58 (2003), 377-388.   Google Scholar

[11]

D. BredaO. DiekmannM. GyllenbergF. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), 1-23.  doi: 10.1137/15M1040931.  Google Scholar

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D. BredaS. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal., 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.  Google Scholar

[13]

D. BredaS. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.  Google Scholar

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P. BrunovskýA. Erdélyi and H.-O. Walther, On a model of a currency exchange rate—local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432.  doi: 10.1007/s10884-004-4285-1.  Google Scholar

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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.  Google Scholar

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R. Castelli and J.-P. Lessard, Rigorous numerics in Floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245.  doi: 10.1137/120873960.  Google Scholar

[17]

R. Castelli and H. Teismann, Rigorous numerics for NLS: Bound states, spectra, and controllability, Phys. D, 334 (2016), 158-173.  doi: 10.1016/j.physd.2016.01.005.  Google Scholar

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J. Cyranka and T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki Model, SIAM J. Appl. Dyn. Syst., 17 (2018), 694-731.  doi: 10.1137/17M111938X.  Google Scholar

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T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, 3. Springer, New York, 2009.  Google Scholar

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T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.  Google Scholar

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Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261-4272.  doi: 10.1142/S0218127407019937.  Google Scholar

[22]

M. Gameiro and 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 (2017), 687-728.  doi: 10.1137/16M1073789.  Google Scholar

[23]

G. H. Golub and H. A. van der Vorst, Eigenvalue computation in the 20th century. Numerical analysis 2000, Vol. Ⅲ. Linear algebra, J. Comput. Appl. Math., 123 (2000), 35-65.  doi: 10.1016/S0377-0427(00)00413-1.  Google Scholar

[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.  Google Scholar

[25]

M. HladíkD. Daney and E. Tsigaridas, Bounds on real eigenvalues and singular values of interval matrices, SIAM J. Matrix Anal. Appl., 31 (2009/10), 2116-2129.  doi: 10.1137/090753991.  Google Scholar

[26]

K. IkedaH. Daido and O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Phys. Rev. Lett., 45 (1980), 709-712.   Google Scholar

[27]

K. Ikeda and K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D: Nonlinear Phenomena, 29 (1987), 223-235.  doi: 10.1016/0167-2789(87)90058-3.  Google Scholar

[28]

J. JaquetteJ.-P. Lessard and K. Mischaikow, Stability and uniqueness of slowly oscillating periodic solutions to Wright's equation, J. Differential Equations, 263 (2017), 7263-7286.  doi: 10.1016/j.jde.2017.08.018.  Google Scholar

[29]

H. Koch, On hyperbolicity in the renormalization of near-critical area-preserving maps, Discrete Contin. Dyn. Syst., 36 (2016), 7029-7056.  doi: 10.3934/dcds.2016106.  Google Scholar

[30]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287–289. doi: 10.1126/science.267326.  Google Scholar

[34]

J. M. MahaffyJ. B'elair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146.  doi: 10.1006/jtbi.1997.0537.  Google Scholar

[35]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge Studies in Modern Optics, 21. Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[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 (2017), 1467-1523.  doi: 10.1007/s10208-016-9325-9.  Google Scholar

[38]

R. E. Moore, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14 (1977), 611-615.  doi: 10.1137/0714040.  Google Scholar

[39]

R. E. Moore, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1966.  Google Scholar

[40]

K. NagatouM. Plum and M. T. Nakao, Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 545-562.  doi: 10.1098/rspa.2011.0159.  Google Scholar

[41]

M. T. NakaoN. Yamamoto and K. Nagatou, Numerical verifications for eigenvalues of second-order elliptic operators, Japan J. Indust. Appl. Math., 16 (1999), 307-320.  doi: 10.1007/BF03167360.  Google Scholar

[42]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4), 101 (1974), 263-306.  doi: 10.1007/BF02417109.  Google Scholar

[43]

J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660.  doi: 10.2307/2313800.  Google Scholar

[44]

C. Reinhardt and 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 (2019), 39-80.  doi: 10.1016/j.indag.2018.08.003.  Google Scholar

[45]

S. M. Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl., 324 (2001), 209-226.  doi: 10.1016/S0024-3795(00)00279-2.  Google Scholar

[46]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numer., 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[47]

S. M. Rump and J.-P. M. Zemke, On eigenvector bounds, BIT, 43 (2003), 823-837.  doi: 10.1023/B:BITN.0000009941.51707.26.  Google Scholar

[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/. Google Scholar

[49]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[50]

J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007), 397-402.  doi: 10.1016/j.physleta.2007.01.083.  Google Scholar

[51]

L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.  Google Scholar

[52] W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.   Google Scholar
[53]

A. Uçar, A prototype model for chaos studies, Internat. J. Engrg. Sci., 40 (2002), 251-258.  doi: 10.1016/S0020-7225(01)00060-X.  Google Scholar

[54]

Y. WatanabeK. NagatouM. Plum and M. T. Nakao, Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM J. Numer. Anal., 52 (2014), 975-992.  doi: 10.1137/120894683.  Google Scholar

[55]

Y. WatanabeM. Plum and M. T. Nakao, A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow, ZAMM Z. Angew. Math. Mech., 89 (2009), 5-18.  doi: 10.1002/zamm.200700158.  Google Scholar

[56]

T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math., 34 (1980), 189-199.  doi: 10.1007/BF01396059.  Google Scholar

[57]

T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors. Ⅱ, Numer. Math., 40 (1982), 201-206.  doi: 10.1007/BF01400539.  Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, Non-symmetric low-index solutions for a symmetric boundary value problem, J. Differential Equations, 252 (2012), 448-458.  doi: 10.1016/j.jde.2011.08.014.  Google Scholar

[2]

G. Arioli and H. Koch, Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation, Nonlinear Anal., 113 (2015), 51-70.  doi: 10.1016/j.na.2014.09.023.  Google Scholar

[3]

G. Arioli and H. Koch, Spectral stability for the wave equation with periodic forcing, J. Differential Equations, 265 (2018), 2470-2501.  doi: 10.1016/j.jde.2018.04.040.  Google Scholar

[4]

I. BalázsJ. B. van den BergJ. CourtoisJ. DudásJ.-P. LessardA. Vörös-KissJ. F. Williams and X. Y. Yin, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003.  Google Scholar

[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.  Google Scholar

[6]

B. Barker, Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flow, J. Differential Equations, 257 (2014), 2950-2983.  doi: 10.1016/j.jde.2014.06.005.  Google Scholar

[7]

B. Barker and K. Zumbrun, Numerical proof of stability of viscous shock profiles, Math. Models Methods Appl. Sci., 26 (2016), 2451-2469.  doi: 10.1142/S0218202516500585.  Google Scholar

[8]

M. Brackstone and M. McDonald, Car-following: A historical review, Transport. Res. Part F, 2 (1999), 181-196.  doi: 10.1016/S1369-8478(00)00005-X.  Google Scholar

[9]

F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[10]

D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: Experimental comparison, Sci. Math. Jpn., 58 (2003), 377-388.   Google Scholar

[11]

D. BredaO. DiekmannM. GyllenbergF. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), 1-23.  doi: 10.1137/15M1040931.  Google Scholar

[12]

D. BredaS. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J. Numer. Anal., 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.  Google Scholar

[13]

D. BredaS. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.  Google Scholar

[14]

P. BrunovskýA. Erdélyi and H.-O. Walther, On a model of a currency exchange rate—local stability and periodic solutions, J. Dynam. Differential Equations, 16 (2004), 393-432.  doi: 10.1007/s10884-004-4285-1.  Google Scholar

[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.  Google Scholar

[16]

R. Castelli and J.-P. Lessard, Rigorous numerics in Floquet theory: Computing stable and unstable bundles of periodic orbits, SIAM J. Appl. Dyn. Syst., 12 (2013), 204-245.  doi: 10.1137/120873960.  Google Scholar

[17]

R. Castelli and H. Teismann, Rigorous numerics for NLS: Bound states, spectra, and controllability, Phys. D, 334 (2016), 158-173.  doi: 10.1016/j.physd.2016.01.005.  Google Scholar

[18]

J. Cyranka and T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki Model, SIAM J. Appl. Dyn. Syst., 17 (2018), 694-731.  doi: 10.1137/17M111938X.  Google Scholar

[19]

T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, 3. Springer, New York, 2009.  Google Scholar

[20]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.  Google Scholar

[21]

Z. Galias and P. Zgliczyński, Infinite-dimensional Krawczyk operator for finding periodic orbits of discrete dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4261-4272.  doi: 10.1142/S0218127407019937.  Google Scholar

[22]

M. Gameiro and 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 (2017), 687-728.  doi: 10.1137/16M1073789.  Google Scholar

[23]

G. H. Golub and H. A. van der Vorst, Eigenvalue computation in the 20th century. Numerical analysis 2000, Vol. Ⅲ. Linear algebra, J. Comput. Appl. Math., 123 (2000), 35-65.  doi: 10.1016/S0377-0427(00)00413-1.  Google Scholar

[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.  Google Scholar

[25]

M. HladíkD. Daney and E. Tsigaridas, Bounds on real eigenvalues and singular values of interval matrices, SIAM J. Matrix Anal. Appl., 31 (2009/10), 2116-2129.  doi: 10.1137/090753991.  Google Scholar

[26]

K. IkedaH. Daido and O. Akimoto, Optical turbulence: Chaotic behavior of transmitted light from a ring cavity, Phys. Rev. Lett., 45 (1980), 709-712.   Google Scholar

[27]

K. Ikeda and K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D: Nonlinear Phenomena, 29 (1987), 223-235.  doi: 10.1016/0167-2789(87)90058-3.  Google Scholar

[28]

J. JaquetteJ.-P. Lessard and K. Mischaikow, Stability and uniqueness of slowly oscillating periodic solutions to Wright's equation, J. Differential Equations, 263 (2017), 7263-7286.  doi: 10.1016/j.jde.2017.08.018.  Google Scholar

[29]

H. Koch, On hyperbolicity in the renormalization of near-critical area-preserving maps, Discrete Contin. Dyn. Syst., 36 (2016), 7029-7056.  doi: 10.3934/dcds.2016106.  Google Scholar

[30]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287–289. doi: 10.1126/science.267326.  Google Scholar

[34]

J. M. MahaffyJ. B'elair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146.  doi: 10.1006/jtbi.1997.0537.  Google Scholar

[35]

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics, Cambridge Studies in Modern Optics, 21. Cambridge University Press, 1997. Google Scholar

[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. Google Scholar

[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 (2017), 1467-1523.  doi: 10.1007/s10208-016-9325-9.  Google Scholar

[38]

R. E. Moore, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14 (1977), 611-615.  doi: 10.1137/0714040.  Google Scholar

[39]

R. E. Moore, Interval Analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1966.  Google Scholar

[40]

K. NagatouM. Plum and M. T. Nakao, Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468 (2012), 545-562.  doi: 10.1098/rspa.2011.0159.  Google Scholar

[41]

M. T. NakaoN. Yamamoto and K. Nagatou, Numerical verifications for eigenvalues of second-order elliptic operators, Japan J. Indust. Appl. Math., 16 (1999), 307-320.  doi: 10.1007/BF03167360.  Google Scholar

[42]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4), 101 (1974), 263-306.  doi: 10.1007/BF02417109.  Google Scholar

[43]

J. M. Ortega, The Newton-Kantorovich theorem, Amer. Math. Monthly, 75 (1968), 658-660.  doi: 10.2307/2313800.  Google Scholar

[44]

C. Reinhardt and 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 (2019), 39-80.  doi: 10.1016/j.indag.2018.08.003.  Google Scholar

[45]

S. M. Rump, Computational error bounds for multiple or nearly multiple eigenvalues, Linear Algebra Appl., 324 (2001), 209-226.  doi: 10.1016/S0024-3795(00)00279-2.  Google Scholar

[46]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numer., 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[47]

S. M. Rump and J.-P. M. Zemke, On eigenvector bounds, BIT, 43 (2003), 823-837.  doi: 10.1023/B:BITN.0000009941.51707.26.  Google Scholar

[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/. Google Scholar

[49]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[50]

J. C. Sprott, A simple chaotic delay differential equation, Phys. Lett. A, 366 (2007), 397-402.  doi: 10.1016/j.physleta.2007.01.083.  Google Scholar

[51]

L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013.  Google Scholar

[52] W. Tucker, Validated Numerics. A Short Introduction to Rigorous Computations, Princeton University Press, Princeton, NJ, 2011.   Google Scholar
[53]

A. Uçar, A prototype model for chaos studies, Internat. J. Engrg. Sci., 40 (2002), 251-258.  doi: 10.1016/S0020-7225(01)00060-X.  Google Scholar

[54]

Y. WatanabeK. NagatouM. Plum and M. T. Nakao, Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces, SIAM J. Numer. Anal., 52 (2014), 975-992.  doi: 10.1137/120894683.  Google Scholar

[55]

Y. WatanabeM. Plum and M. T. Nakao, A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow, ZAMM Z. Angew. Math. Mech., 89 (2009), 5-18.  doi: 10.1002/zamm.200700158.  Google Scholar

[56]

T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors, Numer. Math., 34 (1980), 189-199.  doi: 10.1007/BF01396059.  Google Scholar

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Figure 1.  Schematic illustration of the meaning of Theorem 2.2: The idea behind why a Newton-Kantorovich type theorem works is that we find a point $ \bar z $ where $ g(\bar z) $ is small and evaluate the derivative $ g'(\bar z) $. If $ g'(z) \neq 0 $ then the tangent line (best linear approximation of $ g $ at $ \bar z $) must have a zero. Does it follow that $ g $ has a zero? The answer depends on the size of the second derivative, which lets us construct a confining "envelope" around the linear approximation. In the left frame we have a situation where the envelope forces the function to have a zero, and in the right frame the envelope is not tight enough and the function could "escape". Of course for a given bound on the second derivative (size of the envelope) changing the value of $ g(\bar z) $ (quality of the approximate zero) or the value of $ g'(\bar z) $ (slope of the tangent line). Changing any of these values effects the location of the envelope, and hence the outcome of the proof. Given the data $ g(\bar z) $, $ g'(\bar z) $, and the second derivative bound the role of $ p(r) $ is to determine if the data is good enough to imply the existence of a zero for explicit values of $ r $. The actual proof of the theorem involves changing coordinates to flatten out the function over its tangent line and applying the contraction mapping theorem, and this introduces factors of $ a = 1/g(\bar z) $ throughout the hypotheses. In the proof the "envelope" is the size of the neighborhood on which the resulting map is a contraction
Figure 2.  Validated Morse index by contour integration: imagine that $ \lambda_1 $ and $ \lambda_2 $ zeros inside the unit circle for the function $ \tilde{g} $ defined in Equation (25), so that $ \lambda_1^{-1}, \lambda_2^{-1} $ are unstable eigenvalues for the step map $ F $. To validate the eigenvalue count we choose a branch of the complex logarithm defined on $ \mathbb{C} \backslash(-\infty, 0] $, and consider the "key hole" contour $ \Gamma = \alpha + \beta + \gamma + \delta $ illustrated in the figure. If $ \tilde g $ is analytic inside the region enclosed by $ \Gamma $ then the argument principle counts the number of zeros inside. Supposing that there are no zeros or poles of $ \tilde g $ on $ [-1, 0) $, taking the limit as $ R \to 0 $ and $ d \to 0 $ gives the unstable eigenvalue count, i.e. the Morse index. By choosing other circles we could count the number of stable eigenvalue with modulus larger than some desired bound
Figure 3.  Representation of real analytic functions: Illustration of the complex extension of a real analytic function $ y(t) $ defined on $ [-\tau, 0] $ whose nearest complex pole is at $ z_0 \in \mathbb{C} $ with $ \mbox{dist}([-\tau, 0], z_0) < \tau $. There is a mesh $ \tau_0 = -\tau, \ldots, \tau_M = 0 $, and power series expansions $ y_0(t), \ldots, y_M(t) $ so that $ y(t) = y_j(t) $ for any $ t $ where the power series converges. Each power series $ y_j(t) $ is centered at the point $ t_j $ and converges on a disk of radius $ R_j = |t_j - z_0| $. We refer to these disks as $ C_j $, $ 0 \leq j \leq M $. The same function $ y(t) $ can be represented by a Chebyshev series converging absolutely and uniformly on the Bernstein ellipse $ E $ whose semi-minor axis is at least $ \rho = |\mbox{imag}(z_0)| $. The fact that one Chebyshev series is always sufficient to represent a real analytic function on $ [-\tau, 0] $ is a significant advantage for our discretization scheme
Figure 4.  On the left, the spectrum computations for $ DF(u_0) $ and on the right, the spectrum computations for $ DF(u_1) $. The circles of radii $ r $ used in the computation of the generalized Morse indices in the Mackey-Glass equation (14) at the parameter values $ \tau = 2 $, $ \gamma = 1 $, $ \beta = 2 $ and $ \rho = 10 $ are plotted. On each plot, the unit circle is the largest one and is portrayed in red
Table 1.  Parameters used in the proof of Theorem 4.4 to obtain the generalized Morse indices in the Mackey-Glass equation (14) at the parameter values $ \tau = 2 $, $ \gamma = 1 $, $ \beta = 2 $ and $ \rho = 10 $
$ u_0 $ $ r $ $ N $ $ \nu $ $ m $ $ \mu_r(u_0) $
0 1 32 1.3 10 1
0 0.6 70 1.2 10 3
0 0.29 200 1.15 40 5
0 0.2 600 1.05 60 7
1 1 310 1.1 100 2
1 0.85 130 1.2 30 4
1 0.46 250 1.1 30 6
1 0.341 500 1.05 90 8
$ u_0 $ $ r $ $ N $ $ \nu $ $ m $ $ \mu_r(u_0) $
0 1 32 1.3 10 1
0 0.6 70 1.2 10 3
0 0.29 200 1.15 40 5
0 0.2 600 1.05 60 7
1 1 310 1.1 100 2
1 0.85 130 1.2 30 4
1 0.46 250 1.1 30 6
1 0.341 500 1.05 90 8
Table 2.  Parameters used in the proofs of the higher dimensional examples
$ d $ $ \tau $ $ N $ $ \nu $ $ m $ $ C_1 $ $ C_2 $ $ C_3 $ $ C $ $ \mu_1(u_0) $ Elapsed time (in ${\tt secs}$)
3 1 5 2.4 15 4.1174 4.0101 0.013124 0.21669 2 1.27
6 1 5 2.4 15 4.3696 4.2823 0.014524 0.27176 2 1.41
12 1 5 2.4 15 4.8091 4.9555 4.9555 0.4729 2 1.63
24 1 5 2.4 15 5.3788 5.9556 0.029061 0.9309 2 2.67
$ d $ $ \tau $ $ N $ $ \nu $ $ m $ $ C_1 $ $ C_2 $ $ C_3 $ $ C $ $ \mu_1(u_0) $ Elapsed time (in ${\tt secs}$)
3 1 5 2.4 15 4.1174 4.0101 0.013124 0.21669 2 1.27
6 1 5 2.4 15 4.3696 4.2823 0.014524 0.27176 2 1.41
12 1 5 2.4 15 4.8091 4.9555 4.9555 0.4729 2 1.63
24 1 5 2.4 15 5.3788 5.9556 0.029061 0.9309 2 2.67
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