July  2021, 8(3): 353-401. doi: 10.3934/jcd.2021015

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

Department of Mathematics, VU Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The Netherlands

* Corresponding author: Jan Bouwe van den Berg

Received  March 2020 Revised  March 2021 Published  July 2021 Early access  July 2021

Fund Project: The first author was partially supported by NWO-VICI grant 639033109

In this paper we present a rigorous numerical method for validating analytic solutions of nonlinear ODEs by using Chebyshev-series and domain decomposition. The idea is to define a Newton-like operator, whose fixed points correspond to solutions of the ODE, on the space of geometrically decaying Chebyshev coefficients, and to use the so-called radii-polynomial approach to prove that the operator has an isolated fixed point in a small neighborhood of a numerical approximation. The novelty of the proposed method is the use of Chebyshev series in combination with domain decomposition. In particular, a heuristic procedure based on the theory of Chebyshev approximations for analytic functions is presented to construct efficient grids for validating solutions of boundary value problems. The effectiveness of the proposed method is demonstrated by validating long periodic and connecting orbits in the Lorenz system for which validation without domain decomposition is not feasible.

Citation: Jan Bouwe van den Berg, Ray Sheombarsing. Rigorous numerics for ODEs using Chebyshev series and domain decomposition. Journal of Computational Dynamics, 2021, 8 (3) : 353-401. doi: 10.3934/jcd.2021015
References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  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, Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133.  doi: 10.1137/10078298X.  Google Scholar

[4]

A. W. BakerM. Dellnitz and O. Junge, A topological method for rigorously computing periodic orbits using Fourier modes, Discrete Contin. Dyn. Syst., 13 (2005), 901-920.  doi: 10.3934/dcds.2005.13.901.  Google Scholar

[5]

R. BarrioA. Dena and W. Tucker, A database of rigorous and high-precision periodic orbits of the Lorenz model, Comput. Phys. Comm., 194 (2015), 76-83.  doi: 10.1016/j.cpc.2015.04.007.  Google Scholar

[6]

M. Berz and K. Makino, Rigorous reachability analysis and domain decomposition of Taylor models, in International Workshop on Numerical Software Verification, Lecture Notes in Computer Science, 10381, Springer, Cham, 2017, 90-97. doi: 10.1007/978-3-319-63501-9_7.  Google Scholar

[7]

M. Berz and K. Makino, Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models, Reliab. Comput., 4 (1998), 361-369.  doi: 10.1023/A:1024467732637.  Google Scholar

[8]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[9]

M. Breden and J.-P. Lessard, Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2825-2858.  doi: 10.3934/dcdsb.2018164.  Google Scholar

[10]

M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[11]

B. BreuerJ. HorákP. J. McKenna and M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Differential Equations, 224 (2006), 60-97.  doi: 10.1016/j.jde.2005.07.016.  Google Scholar

[12]

F. Bünger, A Taylor model toolbox for solving ODEs implemented in MATLAB/INTLAB, J. Comput. Appl. Math., 368 (2020), 20pp. doi: 10.1016/j.cam.2019.112511.  Google Scholar

[13]

J. Burgos-García, J.-P. Lessard and J. D. Mireles James, Spatial periodic orbits in the equilateral circular restricted four-body problem: Computer-assisted proofs of existence, Celestial Mech. Dynam. Astronom., 131 (2019), 36pp. doi: 10.1007/s10569-018-9879-8.  Google Scholar

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X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. â…¢. Overview and applications, J. Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.  Google Scholar

[15]

CAPD, Computer assisted proofs in dynamics., Available from: http://capd.ii.uj.edu.pl. Google Scholar

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B. A. CoomesH. Koçak and K. J. Palmer, Transversal connecting orbits from shadowing, Numer. Math., 106 (2007), 427-469.  doi: 10.1007/s00211-007-0065-2.  Google Scholar

[17]

COSY, Center for Beam Theory and Dynamical Systems., Available from: http://bt.pa.msu.edu/index.htm. Google Scholar

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S. DayY. HiraokaK. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31.  doi: 10.1137/040604479.  Google Scholar

[19]

S. DayJ.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[20]

T. A. Driscoll, N. Hale and L. N. Trefethen, Chebfun Guide, Pafnuty Publications, Oxford, 2014. Available from: http://www.chebfun.org/. Google Scholar

[21]

O. FogelklouW. TuckerG. Kreiss and M. Siklosi, A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1227-1243.  doi: 10.1016/j.cnsns.2010.07.008.  Google Scholar

[22]

M. GameiroT. GedeonW. KaliesH. KokubuK. Mischaikow and H. Oka, Topological horseshoes of traveling waves for a fast-slow predator-prey system, J. Dynam. Differential Equations, 19 (2007), 623-654.  doi: 10.1007/s10884-006-9013-6.  Google Scholar

[23]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations, 249 (2010), 2237-2268.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[24]

P. GonnetR. Pachón and L. N. Trefethen, Robust rational interpolation and least-squares, Electron. Trans. Numer. Anal., 38 (2011), 146-167.   Google Scholar

[25]

A. HungriaJ.-P. Lessard and J. D. Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[26]

M. Kashiwagi, KV - A C++ library for verified numerical computation., Available from: http://verifiedby.me/kv/index-e.html. Google Scholar

[27]

J.-P. LessardJ. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.  Google Scholar

[28]

J.-P. LessardJ. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.  doi: 10.1007/s10884-014-9367-0.  Google Scholar

[29]

J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal, 52 (2014), 1-22.  doi: 10.1137/13090883X.  Google Scholar

[30]

R. J. Lohner, Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, in Computational Ordinary Differential Equations, Inst. Math. Appl. Conf. Ser. New Ser., 39, Oxford Univ. Press, New York, 1992,425-435.  Google Scholar

[31]

R. J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, in Computerarithmetic, Teubner, Stuttgart, 1987,255-286.  Google Scholar

[32]

K. Makino and M. Berz, Verified computations using Taylor models and their applications, in International Workshop on Numerical Software Verification, Lecture Notes in Computer Science, 10381, Springer, Cham, 2017, 3-13. doi: 10.1007/978-3-319-63501-9_1.  Google Scholar

[33]

J. C. Mason and D. C. Hanscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[34]

R. PachónP. Gonnet and J. van Deun, Fast and stable rational interpolation in roots of unity and Chebyshev points, SIAM J. Numer. Anal, 50 (2012), 1713-1734.  doi: 10.1137/100797291.  Google Scholar

[35]

L. N. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. Available from: http://www.ti3.tuhh.de/rump/. Google Scholar

[36]

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

[37]

J. B. van den BergC. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem, SIAM J. Appl. Dyn. Syst, 14 (2015), 423-447.  doi: 10.1137/140987973.  Google Scholar

[38]

J. B. van den BergJ.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[39]

J. B. van den BergJ. D. Mireles JamesJ.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal., 43 (2011), 1557-1594.  doi: 10.1137/100812008.  Google Scholar

[40]

J. B. van den Berg and R. Sheombarsing, MATLABcode for rigorous numerics for ODEs using Chebyshev-series and domain decomposition, 2021., Available from: http://www.few.vu.nl/janbouwe/code/domaindecomposition. Google Scholar

[41]

J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373.  doi: 10.1016/j.indag.2020.01.007.  Google Scholar

[42]

M. Webb, Computing complex singularities of differential equations with Chebfun, SIAM Undergraduate Research Online, 2011. Available from: http://evoq-eval.siam.org/Portals/0/Publications/SIURO/Vol6/COMPUTING_COMPLEX_SINGULARITIES_Differential.pdf?ver=2018-04-06-151849-873. Google Scholar

[43]

D. Wilczak and P. Zgliczynski, $C^n$-Lohner algorithm, Scheade Informaticae, 20 (2011), 9-46.   Google Scholar

[44]

D. Wilczak and P. Zgliczynski, Heteroclinic connections between periodic orbits in planar restricted circular three-body problem - A computer assisted proof, Comm. Math. Phys., 234 (2003), 37-75.  doi: 10.1007/s00220-002-0709-0.  Google Scholar

[45]

N. Yamamoto, A numerical verification method for solutions of of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal, 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[46]

P. Zgliczynski, $C^1$-Lohner algorithm, Found. Comput. Math., 2 (2002), 429-465.  doi: 10.1007/s102080010025.  Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Ration. Mech. Anal., 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  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, Integration of dissipative partial differential equations: A case study, SIAM J. Appl. Dyn. Syst., 9 (2010), 1119-1133.  doi: 10.1137/10078298X.  Google Scholar

[4]

A. W. BakerM. Dellnitz and O. Junge, A topological method for rigorously computing periodic orbits using Fourier modes, Discrete Contin. Dyn. Syst., 13 (2005), 901-920.  doi: 10.3934/dcds.2005.13.901.  Google Scholar

[5]

R. BarrioA. Dena and W. Tucker, A database of rigorous and high-precision periodic orbits of the Lorenz model, Comput. Phys. Comm., 194 (2015), 76-83.  doi: 10.1016/j.cpc.2015.04.007.  Google Scholar

[6]

M. Berz and K. Makino, Rigorous reachability analysis and domain decomposition of Taylor models, in International Workshop on Numerical Software Verification, Lecture Notes in Computer Science, 10381, Springer, Cham, 2017, 90-97. doi: 10.1007/978-3-319-63501-9_7.  Google Scholar

[7]

M. Berz and K. Makino, Verified integration of ODEs and flows using differential algebraic methods on high-order Taylor models, Reliab. Comput., 4 (1998), 361-369.  doi: 10.1023/A:1024467732637.  Google Scholar

[8]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[9]

M. Breden and J.-P. Lessard, Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2825-2858.  doi: 10.3934/dcdsb.2018164.  Google Scholar

[10]

M. BredenJ.-P. Lessard and M. Vanicat, Global bifurcation diagrams of steady states of systems of PDEs via rigorous numerics: A 3-component reaction-diffusion system, Acta Appl. Math., 128 (2013), 113-152.  doi: 10.1007/s10440-013-9823-6.  Google Scholar

[11]

B. BreuerJ. HorákP. J. McKenna and M. Plum, A computer-assisted existence and multiplicity proof for travelling waves in a nonlinearly supported beam, J. Differential Equations, 224 (2006), 60-97.  doi: 10.1016/j.jde.2005.07.016.  Google Scholar

[12]

F. Bünger, A Taylor model toolbox for solving ODEs implemented in MATLAB/INTLAB, J. Comput. Appl. Math., 368 (2020), 20pp. doi: 10.1016/j.cam.2019.112511.  Google Scholar

[13]

J. Burgos-García, J.-P. Lessard and J. D. Mireles James, Spatial periodic orbits in the equilateral circular restricted four-body problem: Computer-assisted proofs of existence, Celestial Mech. Dynam. Astronom., 131 (2019), 36pp. doi: 10.1007/s10569-018-9879-8.  Google Scholar

[14]

X. CabréE. Fontich and R. de la Llave, The parameterization method for invariant manifolds. â…¢. Overview and applications, J. Differential Equations, 218 (2005), 444-515.  doi: 10.1016/j.jde.2004.12.003.  Google Scholar

[15]

CAPD, Computer assisted proofs in dynamics., Available from: http://capd.ii.uj.edu.pl. Google Scholar

[16]

B. A. CoomesH. Koçak and K. J. Palmer, Transversal connecting orbits from shadowing, Numer. Math., 106 (2007), 427-469.  doi: 10.1007/s00211-007-0065-2.  Google Scholar

[17]

COSY, Center for Beam Theory and Dynamical Systems., Available from: http://bt.pa.msu.edu/index.htm. Google Scholar

[18]

S. DayY. HiraokaK. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 4 (2005), 1-31.  doi: 10.1137/040604479.  Google Scholar

[19]

S. DayJ.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM J. Numer. Anal, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[20]

T. A. Driscoll, N. Hale and L. N. Trefethen, Chebfun Guide, Pafnuty Publications, Oxford, 2014. Available from: http://www.chebfun.org/. Google Scholar

[21]

O. FogelklouW. TuckerG. Kreiss and M. Siklosi, A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1227-1243.  doi: 10.1016/j.cnsns.2010.07.008.  Google Scholar

[22]

M. GameiroT. GedeonW. KaliesH. KokubuK. Mischaikow and H. Oka, Topological horseshoes of traveling waves for a fast-slow predator-prey system, J. Dynam. Differential Equations, 19 (2007), 623-654.  doi: 10.1007/s10884-006-9013-6.  Google Scholar

[23]

M. Gameiro and J.-P. Lessard, Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs, J. Differential Equations, 249 (2010), 2237-2268.  doi: 10.1016/j.jde.2010.07.002.  Google Scholar

[24]

P. GonnetR. Pachón and L. N. Trefethen, Robust rational interpolation and least-squares, Electron. Trans. Numer. Anal., 38 (2011), 146-167.   Google Scholar

[25]

A. HungriaJ.-P. Lessard and J. D. Mireles James, Rigorous numerics for analytic solutions of differential equations: The radii polynomial approach, Math. Comp., 85 (2016), 1427-1459.  doi: 10.1090/mcom/3046.  Google Scholar

[26]

M. Kashiwagi, KV - A C++ library for verified numerical computation., Available from: http://verifiedby.me/kv/index-e.html. Google Scholar

[27]

J.-P. LessardJ. D. Mireles James and J. Ransford, Automatic differentiation for Fourier series and the radii polynomial approach, Phys. D, 334 (2016), 174-186.  doi: 10.1016/j.physd.2016.02.007.  Google Scholar

[28]

J.-P. LessardJ. D. Mireles James and C. Reinhardt, Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields, J. Dynam. Differential Equations, 26 (2014), 267-313.  doi: 10.1007/s10884-014-9367-0.  Google Scholar

[29]

J.-P. Lessard and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal, 52 (2014), 1-22.  doi: 10.1137/13090883X.  Google Scholar

[30]

R. J. Lohner, Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems, in Computational Ordinary Differential Equations, Inst. Math. Appl. Conf. Ser. New Ser., 39, Oxford Univ. Press, New York, 1992,425-435.  Google Scholar

[31]

R. J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, in Computerarithmetic, Teubner, Stuttgart, 1987,255-286.  Google Scholar

[32]

K. Makino and M. Berz, Verified computations using Taylor models and their applications, in International Workshop on Numerical Software Verification, Lecture Notes in Computer Science, 10381, Springer, Cham, 2017, 3-13. doi: 10.1007/978-3-319-63501-9_1.  Google Scholar

[33]

J. C. Mason and D. C. Hanscomb, Chebyshev Polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[34]

R. PachónP. Gonnet and J. van Deun, Fast and stable rational interpolation in roots of unity and Chebyshev points, SIAM J. Numer. Anal, 50 (2012), 1713-1734.  doi: 10.1137/100797291.  Google Scholar

[35]

L. N. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 1999, 77-104. Available from: http://www.ti3.tuhh.de/rump/. Google Scholar

[36]

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

[37]

J. B. van den BergC. M. Groothedde and J. F. Williams, Rigorous computation of a radially symmetric localized solution in a Ginzburg-Landau problem, SIAM J. Appl. Dyn. Syst, 14 (2015), 423-447.  doi: 10.1137/140987973.  Google Scholar

[38]

J. B. van den BergJ.-P. Lessard and K. Mischaikow, Global smooth solution curves using rigorous branch following, Math. Comp., 79 (2010), 1565-1584.  doi: 10.1090/S0025-5718-10-02325-2.  Google Scholar

[39]

J. B. van den BergJ. D. Mireles JamesJ.-P. Lessard and K. Mischaikow, Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation, SIAM J. Math. Anal., 43 (2011), 1557-1594.  doi: 10.1137/100812008.  Google Scholar

[40]

J. B. van den Berg and R. Sheombarsing, MATLABcode for rigorous numerics for ODEs using Chebyshev-series and domain decomposition, 2021., Available from: http://www.few.vu.nl/janbouwe/code/domaindecomposition. Google Scholar

[41]

J. B. van den Berg and R. Sheombarsing, Validated computations for connecting orbits in polynomial vector fields, Indag. Math. (N.S.), 31 (2020), 310-373.  doi: 10.1016/j.indag.2020.01.007.  Google Scholar

[42]

M. Webb, Computing complex singularities of differential equations with Chebfun, SIAM Undergraduate Research Online, 2011. Available from: http://evoq-eval.siam.org/Portals/0/Publications/SIURO/Vol6/COMPUTING_COMPLEX_SINGULARITIES_Differential.pdf?ver=2018-04-06-151849-873. Google Scholar

[43]

D. Wilczak and P. Zgliczynski, $C^n$-Lohner algorithm, Scheade Informaticae, 20 (2011), 9-46.   Google Scholar

[44]

D. Wilczak and P. Zgliczynski, Heteroclinic connections between periodic orbits in planar restricted circular three-body problem - A computer assisted proof, Comm. Math. Phys., 234 (2003), 37-75.  doi: 10.1007/s00220-002-0709-0.  Google Scholar

[45]

N. Yamamoto, A numerical verification method for solutions of of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal, 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[46]

P. Zgliczynski, $C^1$-Lohner algorithm, Found. Comput. Math., 2 (2002), 429-465.  doi: 10.1007/s102080010025.  Google Scholar

Figure 1.  Approximations of the solution of $ u' = u(1-u) $ with $ u(0) = \frac{1}{2} $. In blue (partly obscured by red) based on a truncated Chebyshev series for $ t\in[0,50] $ with $ N = 75 $ terms and geometric decay factor $ \nu = 1.05 $ (see Section 2) with error at most $ 6.3 \cdot 10^{-10} $ (uniformly). In red based on a truncated Taylor series for $ t\in [0,3] $ with $ N = 225 $ terms and $ \nu = 1 $ with uniform error control $ 2.7 \cdot 10^{-4} $. The computation times (including the proof of the error bound) for both cases are comparable. The rigorous error bounds are based on the truncated series only, not on the availability of an explicit expression for the solution
Figure 2.  The connecting orbit from the positive eye to the origin in the Lorenz equation with classical parameters. The time of flight between the local (un)stable manifolds is $ L = 30 $. The geometric objects colored in red and green are representations of $ W_{ \rm{loc}}^{s} \left( q^{+} \right) $ and $ W_{ \rm{loc}}^{u} \left( {\bf{0}} \right) $, respectively
Figure 3.  (a) A periodic orbit on the Lorenz attractor of period $ L \approx 25.03 $ validated with $ m = 35 $ subdomains. (b) A semi-logarithmic plot of the coefficients in the Chebyshev series on all subdomains for the three components of the solution
Figure 4.  (a) A typical periodic orbit near the homoclinic connection. The period of the orbit is $ L \approx 100.25 $. Notice the sharp turn of the orbit near the origin. (b) The $ x $, $ y $ and $ z $ components of the orbit. The three components are fairly flat for a relatively long time. These flat parts correspond to the part of the orbit near the equilibrium where the dynamics are slow
Figure 5.  The fifteen components of the validated periodic orbit in the coupled Lorenz system (4). The period of the periodic orbit is $ L \approx 1.70 $. We have grouped the components, which in the synchronized setting would correspond to the "same component", together. In each case the components with higher indices are colored with a lighter shade of blue
Figure 6.  (a) A validated solution of the initial value problem with initial condition $ p_{0} = \begin{bmatrix} 6 & 10 & 8 \end{bmatrix}^{T} $ and integration time $ L = 11.2 $. (b) The three components of the validated orbit. The dashed lines in grey indicate the six chosen integration times $ L \in \{2.3, 4.8, 6.1, 7.6, 9.5, 11.2\} $. At the $ k $-th integration time, ordered from small to large, the orbit transitioned $ k $ times between the eyes
Figure 7.  The approximate complex singularities of the validated periodic orbit. Note that the time variable has been rescaled to $ [0,1] $. The complex singularities were computed with the procedure described in Section 4.2
Figure 8.  (a) A plot of the two grids $ \left(i, t_{i} \right)_{i = 0}^{33} $ and $ \left(i, \tau_{i} \right)_{i = 0}^{34} $ corresponding to $ m = 33 $ and $ m = 34 $, respectively. (b) A plot of the grid-points $ t_{32} $, $ \tau_{33} $ and the complex singularities (colored in red) which determine the size of the Bernstein ellipses associated to $ \left[ t_{32},1 \right] $ and $ \left[ \tau_{33},1 \right] $
Figure 9.  The dependence of the period $ L $ as a function of $ \rho $ obtained via non-rigorous pseudo-arclength continuation
Figure 10.  (a) The complex singularities of the connecting orbit. (b) The grid, determined by the algorithm in Section 4, on which the connecting orbit was validated
Figure 11.  (a) The complex singularities of the validated periodic orbit in the coupled Lorenz system for $ K = 5 $. (b) A semi-logarithmic plot of the coefficients in the Chebyshev series of the periodic orbit for $ K = 5 $ for all subdomains and components. The decay rates are approximately the same for each subdomain and component. The black line corresponds to the theoretically predicted decay rate determined by the size of the smallest Bernstein-ellipse. The theoretically predicted decay rate coincides (approximately) with the observed decay rate, which indicates that the approximation of the complex singularities (near the real-axis) was accurate
Figure 12.  The thirty components of the validated periodic orbit in the coupled Lorenz system (4). The period of the periodic orbit is $ L \approx 1.68 $. We have grouped the components, which in the synchronized setting would correspond to the "same component", together. In each case the components with higher indices are colored with a lighter shade of blue
Figure 13.  (a) Step sizes used by awa for $ L = 11.2 $. (b) Step sizes used by verifyode for $ L = 6.1 $. Approximately $ 45 \% $, $ 40 \% $ and $ 10 \% $ of the time steps are located around small neighborhoods of $ t = 4 $, $ t = 5 $ and (just after) $ t = 6 $, respectively. Moreover, these time steps are of minimal size
Table 1.  Numerical results for a validated periodic orbit of period $ L \approx 25.0271 $ in the classical Lorenz system. In each case the number of modes $ N_{i} $ per domain was approximately the same. The number $ \bar N $ denotes the (rounded) average number of modes per domain
$ m $ $ \bar N $ $ \dim \mathcal{X}^{N}_{\nu} $ $ \nu $ $ \nu_{e} $ $ I_{m,\nu} $
$ 32 $ $ 79 $ $ 7618 $ $ 1.1148 $ $ 1.6413 $ $ \left[ 4.5581 \cdot 10^{-10}, \ 1.0217 \cdot 10^{-7} \right] $
$ 33 $ $ 76 $ $ 7498 $ $ 1.1278 $ $ 1.6837 $ $ \left[ 3.1192 \cdot 10^{-10}, \ 1.5371 \cdot 10^{-7} \right] $
$ 34 $ $ 63 $ $ 6433 $ $ 1.1432 $ $ 1.8749 $ $ \left[ 3.8158 \cdot 10^{-10}, \ 1.1950 \cdot 10^{-7} \right] $
$ 35 $ $ 61 $ $ 6457 $ $ 1.1529 $ $ 1.9056 $ $ \left[ 3.3529 \cdot 10^{-10}, \ 1.0320 \cdot 10^{-7} \right] $
$ 36 $ $ 61 $ $ 6610 $ $ 1.1564 $ $ 1.9115 $ $ \left[ 3.1282 \cdot 10^{-10}, \ 1.1097 \cdot 10^{-7} \right] $
$ m $ $ \bar N $ $ \dim \mathcal{X}^{N}_{\nu} $ $ \nu $ $ \nu_{e} $ $ I_{m,\nu} $
$ 32 $ $ 79 $ $ 7618 $ $ 1.1148 $ $ 1.6413 $ $ \left[ 4.5581 \cdot 10^{-10}, \ 1.0217 \cdot 10^{-7} \right] $
$ 33 $ $ 76 $ $ 7498 $ $ 1.1278 $ $ 1.6837 $ $ \left[ 3.1192 \cdot 10^{-10}, \ 1.5371 \cdot 10^{-7} \right] $
$ 34 $ $ 63 $ $ 6433 $ $ 1.1432 $ $ 1.8749 $ $ \left[ 3.8158 \cdot 10^{-10}, \ 1.1950 \cdot 10^{-7} \right] $
$ 35 $ $ 61 $ $ 6457 $ $ 1.1529 $ $ 1.9056 $ $ \left[ 3.3529 \cdot 10^{-10}, \ 1.0320 \cdot 10^{-7} \right] $
$ 36 $ $ 61 $ $ 6610 $ $ 1.1564 $ $ 1.9115 $ $ \left[ 3.1282 \cdot 10^{-10}, \ 1.1097 \cdot 10^{-7} \right] $
Table 2.  Numerical results for two periodic orbits near the homoclinic connection. The periodic orbit of period $ L \approx 100.2554 $ was in both cases validated on a grid for which (the same) six subdomains were used to approximate the non-flat part of the orbit
$ L $ $ m $ $ \dim \mathcal{X}^{N}_{\nu} $ $ I_{m,\nu} $
$ 4.5473 $ $ 1 $ $ 566 $ $ \left[ 4.4568 \cdot 10^{-11}, \ 8.4403 \cdot 10^{-6} \right] $
$ 100.2554 $ $ 7 $ $ 5894 $ $ \left[ 1.5186 \cdot 10^{-11}, \ 7.4915 \cdot 10^{-8} \right] $
$ 100.2554 $ $ 506 $ $ 8441 $ $ \left[ 1.5174 \cdot 10^{-11}, \ 4.2914 \cdot 10^{-6} \right] $
$ L $ $ m $ $ \dim \mathcal{X}^{N}_{\nu} $ $ I_{m,\nu} $
$ 4.5473 $ $ 1 $ $ 566 $ $ \left[ 4.4568 \cdot 10^{-11}, \ 8.4403 \cdot 10^{-6} \right] $
$ 100.2554 $ $ 7 $ $ 5894 $ $ \left[ 1.5186 \cdot 10^{-11}, \ 7.4915 \cdot 10^{-8} \right] $
$ 100.2554 $ $ 506 $ $ 8441 $ $ \left[ 1.5174 \cdot 10^{-11}, \ 4.2914 \cdot 10^{-6} \right] $
Table 3.  Numerical results for the connecting orbit from $ q^{+} $ to the origin. The interval $ I_{m,\nu} $ is the set of admissible radii on which the radii-polynomials were proven to be strictly negative
$ m $ $ \bar N $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 55 $ $ 42 $ $ 6864 $ $ 1.3710 $ $ \left[ 6.4412 \cdot 10^{-9}, \ r^{\ast} \right] $
$ m $ $ \bar N $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 55 $ $ 42 $ $ 6864 $ $ 1.3710 $ $ \left[ 6.4412 \cdot 10^{-9}, \ r^{\ast} \right] $
Table 4.  Numerical results for a validated periodic orbit in systems of five and 10 coupled Lorenz vector fields. We took $ m = 4 $ for both proofs. The interval $ I_{m,\nu} $ is the set of admissible radii on which the radii-polynomials were proven to be strictly negative
$ 3K $ $ L $ $ N $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 15 $ $ 1.7046 $ $ [57 \; \; \; 54 \; \; \; 59 \; \; \; 55] $ $ 3376 $ $ 1.1807 $ $ \left[ 4.39 \cdot 10^{-10}, \ 1.66 \cdot 10^{-6} \right] $
$ 30 $ $ 1.6839 $ $ [54 \; \; \; 53 \; \; \; 56 \; \; \; 54 ] $ $ 6511 $ $ 1.1849 $ $ \left[ 3.33 \cdot 10^{-10}, \ 2.21 \cdot 10^{-6} \right] $
$ 3K $ $ L $ $ N $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 15 $ $ 1.7046 $ $ [57 \; \; \; 54 \; \; \; 59 \; \; \; 55] $ $ 3376 $ $ 1.1807 $ $ \left[ 4.39 \cdot 10^{-10}, \ 1.66 \cdot 10^{-6} \right] $
$ 30 $ $ 1.6839 $ $ [54 \; \; \; 53 \; \; \; 56 \; \; \; 54 ] $ $ 6511 $ $ 1.1849 $ $ \left[ 3.33 \cdot 10^{-10}, \ 2.21 \cdot 10^{-6} \right] $
Table 5.  Numerical results for the validated integration of an IVP in the Lorenz system with initial condition $ p_{0} = [10 \; \; 6 \; \; 8]^{T} $ and different integration times $ L $. The orbits were validated using the verified integrators awa and verifyode in INTLAB using the default settings. In particular, $ 10 $-th order Taylor approximations were used and the minimal step size was $ 10^{-4} $. The integer $ n_{q^{\pm}} $ denotes the number of transitions between the eyes. The third and fourth column correspond to the number of time steps used for integration. The integrator verifyode failed, due to too large enclosures, for $ n_{q^{\pm}} \geq 4 $
Steps Steps $ C^{0} $-error $ C^{0} $-error
$ L $ $ n_{q^{\pm}} $ awa verifyode awa verifyode
$ 2.3 $ $ 1 $ $ 719 $ $ 150 $ $ 5.3774 \cdot 10^{-11} $ $ 1.5439 \cdot 10^{-3} $
$ 4.8 $ $ 2 $ $ 1471 $ $ 2643 $ $ 9.9833 \cdot 10^{-10} $ $ 9.8674 \cdot 10^{-2} $
$ 6.1 $ $ 3 $ $ 1873 $ $ 5213 $ $ 1.186 \cdot 10^{-8} $ $ 4.0664 $
$ 7.6 $ $ 4 $ $ 2342 $ $ - $ $ 1.186 \cdot 10^{-8} $ $ - $
$ 9.5 $ $ 5 $ $ 2901 $ $ - $ $ 7.1604 \cdot 10^{-8} $ $ - $
$ 11.2 $ $ 6 $ $ 3411 $ $ - $ $ 5.1561 \cdot 10^{-7} $ $ - $
Steps Steps $ C^{0} $-error $ C^{0} $-error
$ L $ $ n_{q^{\pm}} $ awa verifyode awa verifyode
$ 2.3 $ $ 1 $ $ 719 $ $ 150 $ $ 5.3774 \cdot 10^{-11} $ $ 1.5439 \cdot 10^{-3} $
$ 4.8 $ $ 2 $ $ 1471 $ $ 2643 $ $ 9.9833 \cdot 10^{-10} $ $ 9.8674 \cdot 10^{-2} $
$ 6.1 $ $ 3 $ $ 1873 $ $ 5213 $ $ 1.186 \cdot 10^{-8} $ $ 4.0664 $
$ 7.6 $ $ 4 $ $ 2342 $ $ - $ $ 1.186 \cdot 10^{-8} $ $ - $
$ 9.5 $ $ 5 $ $ 2901 $ $ - $ $ 7.1604 \cdot 10^{-8} $ $ - $
$ 11.2 $ $ 6 $ $ 3411 $ $ - $ $ 5.1561 \cdot 10^{-7} $ $ - $
Table 6.  Numerical results for the validated integration of an IVP in the Lorenz system with initial condition $ p_{0} = [10 \; \; 6 \; \; 8]^{T} $ and different integration times $ L $ using the proposed method. The integer $ n_{q^{\pm}} $ denotes the number of transitions between the eyes. The number $ \bar N $ denotes the (rounded) average number of modes per subdomain. The interval $ I_{m,\nu} $ is the set of admissible radii on which the radii-polynomials were proven to be strictly negative and $ \nu $ is the associated weight with which the validation was performed. Note that the left-endpoints of $ I_{m,\nu} $ are rigorous upper bounds for the $ C^{0} $-error
$ L $ $ n_{q^{\pm}} $ $ m $ $ \bar{N} $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 2.3 $ $ 1 $ $ 2 $ $ 118 $ $ 705 $ $ 1.057 $ $ \left[2.0218 \cdot 10^{-10}, 9.0595 \cdot 10^{-4}\right] $
$ 4.8 $ $ 2 $ $ 4 $ $ 122 $ $ 1458 $ $ 1.087 $ $ \left[6.6631 \cdot 10^{-9}, 2.7927 \cdot 10^{-5}\right] $
$ 6.1 $ $ 3 $ $ 5 $ $ 125 $ $ 1881 $ $ 1.1 $ $ \left[2.1101 \cdot 10^{-8}, 9.4164 \cdot 10^{-6} \right] $
$ 7.6 $ $ 4 $ $ 8 $ $ 122 $ $ 2937 $ $ 1.094 $ $ \left[2.1476 \cdot 10^{-8}, 5.1128 \cdot 10^{-6} \right] $
$ 9.5 $ $ 5 $ $ 11 $ $ 124 $ $ 4101 $ $ 1.113 $ $ \left[7.7316 \cdot 10^{-8}, 1.1828 \cdot 10^{-6} \right] $
$ 11.2 $ $ 6 $ $ 15 $ $ 123 $ $ 5547 $ $ 1.136 $ $ \left[1.6234 \cdot 10^{-7}, 5.1904 \cdot 10^{-7} \right] $
$ L $ $ n_{q^{\pm}} $ $ m $ $ \bar{N} $ $ \dim \Pi_{N} \left( \mathcal{X}_{\nu} \right) $ $ \nu $ $ I_{m,\nu} $
$ 2.3 $ $ 1 $ $ 2 $ $ 118 $ $ 705 $ $ 1.057 $ $ \left[2.0218 \cdot 10^{-10}, 9.0595 \cdot 10^{-4}\right] $
$ 4.8 $ $ 2 $ $ 4 $ $ 122 $ $ 1458 $ $ 1.087 $ $ \left[6.6631 \cdot 10^{-9}, 2.7927 \cdot 10^{-5}\right] $
$ 6.1 $ $ 3 $ $ 5 $ $ 125 $ $ 1881 $ $ 1.1 $ $ \left[2.1101 \cdot 10^{-8}, 9.4164 \cdot 10^{-6} \right] $
$ 7.6 $ $ 4 $ $ 8 $ $ 122 $ $ 2937 $ $ 1.094 $ $ \left[2.1476 \cdot 10^{-8}, 5.1128 \cdot 10^{-6} \right] $
$ 9.5 $ $ 5 $ $ 11 $ $ 124 $ $ 4101 $ $ 1.113 $ $ \left[7.7316 \cdot 10^{-8}, 1.1828 \cdot 10^{-6} \right] $
$ 11.2 $ $ 6 $ $ 15 $ $ 123 $ $ 5547 $ $ 1.136 $ $ \left[1.6234 \cdot 10^{-7}, 5.1904 \cdot 10^{-7} \right] $
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