Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
rigorous numerics; fundamental matrix solution; Floquet theory; analytical category
rigorous numerics; fundamental matrix solution; Floquet theory; analytical category
Summary:
This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form is validated in the space of analytic functions. The technique combines analytical estimates and rigorous numerical computations and no rigorous integration is needed. An application to the theory of dynamical system is presented, together with a comparison with the results obtained by computing the enclosure in the $C^s$ category.
References:
[1] Cabré, X., Fontich, E., Llave, R. de la: The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J. 52 (2003), 283-328. DOI 10.1512/iumj.2003.52.2245 | MR 1976079 | Zbl 1034.37016
[2] Cabré, X., Fontich, E., Llave, R. de la: The parameterization method for invariant manifolds II: Regularity with respect to parameters. Indiana Univ. Math. J. 52 (2003), 329-360. MR 1976080 | Zbl 1034.37017
[3] Cabré, X., Fontich, E., Llave, R. de la: The parameterization method for invariant manifolds III: Overview and applications. J. Differ. Equations 218 (2005), 444-515. DOI 10.1016/j.jde.2004.12.003 | MR 2177465 | Zbl 1101.37019
[4] Castelli, R., Lessard, J.-P.: A method to rigorously enclose eigenpairs of complex interval matrices. Internat. Conf. Appl. Math. In Honor of the 70th Birthday of K. Segeth Academy of Sciences of the Czech Republic, Institute of Mathematics, Prague (2013), 21-31. MR 3204427
[5] Castelli, R., Lessard, J.-P.: Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits. SIAM J. Appl. Dyn. Syst. (electronic only) 12 (2013), 204-245. DOI 10.1137/120873960 | MR 3032858 | Zbl 1293.37033
[6] Castelli, R., Lessard, J.-P., James, J. D. Mireles: Parameterization of invariant manifolds for periodic orbits I: Efficient numerics via the Floquet normal form. SIAM J. Appl. Dyn. Syst. (electronic only) 14 (2015), 132-167. DOI 10.1137/140960207 | MR 3304254
[7] Day, S., Lessard, J.-P., Mischaikow, K.: Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal. 45 (2007), 1398-1424. DOI 10.1137/050645968 | MR 2338393 | Zbl 1151.65074
[8] Gameiro, M., Lessard, J.-P.: Analytic estimates and rigorous continuation for equilibria of higher-dimensional PDEs. J. Differ. Equations 249 (2010), 2237-2268. DOI 10.1016/j.jde.2010.07.002 | MR 2718657 | Zbl 1256.35196
[9] Hungria, A., Lessard, J.-P., James, J. D. Mireles: Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach. (to appear) in Math. Comput. (2015).
[10] Lessard, J.-P., James, J. D. M., Reinhardt, C.: Computer assisted proof of transverse saddle-to-saddle connecting orbits for first order vector fields. J. Dyn. Differ. Equations 26 (2014), 267-313. DOI 10.1007/s10884-014-9367-0 | MR 3207723
[11] James, J. D. Mireles, Mischaikow, K.: Rigorous a posteriori computation of (un)stable manifolds and connecting orbits for analytic maps. SIAM J. Appl. Dyn. Syst. (electronic only) 12 (2013), 957-1006. DOI 10.1137/12088224X | MR 3068557
[12] Rump, S. M.: INTLAB---INTerval LABoratory. T. Csendes Developments in Reliable Computing SCAN-98 conference, Budapest. Kluwer Academic Publishers Dordrecht (1999), 77-104, http://www.ti3.tu-harburg.de/rump/ Zbl 0949.65046
[13] Yakubovich, V. A., Starzhinskij, V. M.: Linear Differential Equations with Periodic Coefficients, Vol. 1, 2. Wiley, New York Halsted, Jerusalem (1975). MR 0364740 | Zbl 0308.34001
Search
Partner of
