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Global smooth solution curves using rigorous branch following


Authors: Jan Bouwe van den Berg, Jean-Philippe Lessard and Konstantin Mischaikow
Journal: Math. Comp. 79 (2010), 1565-1584
MSC (2010): Primary 37M99; Secondary 65G20, 65N30
DOI: https://doi.org/10.1090/S0025-5718-10-02325-2
Published electronically: March 11, 2010
MathSciNet review: 2630003
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Abstract: In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators $ f:\mathbb{R}^{l_1} \times B_1 \rightarrow \re^{l_2} \times B_2$, where $ B_1$ and $ B_2$ are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.


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  • 1. J. B. van den Berg and J.-P. Lessard. Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 7(3):988-1031 (electronic), 2008. MR 2443030 (2009g:37030)
  • 2. B. Breuer, J. Horák, P.J. McKenna and M. Plum. A computer-assisted existence and multiplicity proof for traveling waves in a nonlinearly supported beam. J. Differential Equations, 224(1):60-97, 2006. MR 2220064 (2007a:34072)
  • 3. S. N. Chow and J. K. Hale. Methods of bifurcation theory, volume 251 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York, 1982. MR 660633 (84e:58019)
  • 4. S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa. Rigorous numerics for global dynamics: a study of the Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst., 4(1):1-31 (electronic), 2005. MR 2136516 (2006i:37160)
  • 5. S. Day, J.-P. Lessard and K. Mischaikow. Validated continuation for equilibria of PDEs. SIAM J. Numer. Anal., 45(4):1398-1424 (electronic), 2007. MR 2338393 (2008k:37169)
  • 6. M. Gameiro and J.-P. Lessard. A Priori estimates and rigorous continuation for equilibria of higher-dimensional PDEs. To appear in Journal of Differential Equations, 2010.
  • 7. M. Gameiro, J.-P. Lessard and K. Mischaikow. Validated continuation over large parameter ranges for equilibria of PDEs. Mathematics and Computers in Simulation, 79(4): 1368-1382, 2008. MR 2487806
  • 8. H. B. Keller. Lectures on numerical methods in bifurcation problems, volume 79 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Bombay, 1987. MR 910499 (89f:58031)
  • 9. J.-P. Lessard. Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation. To appear in Journal of Differential Equations, 2010.
  • 10. U. Miller. Rigorous numerics using Conley index theory, volume 9 of Augsburger Schriften zur Mathematik, Physik und Informatik [Augsburger Publications of Mathematics, Physics and Information Sciences]. Logos Verlag Berlin, Berlin, 2005. MR 2275297 (2008h:37086)
  • 11. T. Minamoto and M.T. Nakao. Numerical method for verifying the existence and local uniqueness of a double turning point for a radially symmetric solution of the perturbed Gelfand equation. J. Comput. Appl. Math. 202(2):177-185, 2007. MR 2319947 (2008a:35108)
  • 12. M. T. Nakao and N. Yamamoto. Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite elements. J. Comput. Appl. Math. 60:271-279, 1995. MR 1354660 (96k:65075)
  • 13. M.T. Nakao and Y. Watanabe. An efficient approach to the numerical verification for solutions of elliptic differential equations. Numer. Algorithms, 37(1-4):311-323, 2004. MR 2109916 (2005i:65215)
  • 14. M. Plum. Computer-assisted enclosure methods for elliptic differential equations. Special issue on linear algebra in self-validating methods. Linear Algebra Appl., 324(1-3):147-187, 2009. MR 1810529 (2002a:65080)
  • 15. M. Plum. Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems. J. Comput. Appl. Math. 60(1-2):187-200, 1995. MR 1354655 (96i:65097)
  • 16. E. M. Wright. A non-linear difference-differential equation. J. Reine Angew. Math., 194:66-87, 1955. MR 0072363 (17:272b)
  • 17. N. Yamamoto. A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem. SIAM J. Numer. Anal., 35(5):2004-2013 (electronic), 1998. MR 1639986 (99f:65180)
  • 18. P. Zgliczyński and K. Mischaikow. Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation. Found. Comput. Math., 1(3):255-288, 2001. MR 1838755 (2002e:65143)

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Additional Information

Jan Bouwe van den Berg
Affiliation: VU University Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Email: janbouwe@few.vu.nl

Jean-Philippe Lessard
Affiliation: Rutgers University, Department of Mathematics, Hill Center-Busch Campus, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019 and VU University Amsterdam, Department of Mathematics, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Email: lessard@math.rutgers.edu

Konstantin Mischaikow
Affiliation: Rutgers Univeristy, Department of Mathematics, Hill Center-Busch Campus, 110 Frelinghuysen Rd, Piscataway, New Jersey 08854-8019
Email: mischaik@math.rutgers.edu

DOI: https://doi.org/10.1090/S0025-5718-10-02325-2
Received by editor(s): September 29, 2009
Received by editor(s) in revised form: May 18, 2009
Published electronically: March 11, 2010
Additional Notes: The second author was supported in part by NSF Grant DMS-0511115, by DARPA, and by DOE Grant DE-FG02-05ER25711.
The third author was supported by NSF Grant DMS-0638131, DMS-0835621, DMS-0915019, DARPA, DOE Grant DE-FG02-05ER25711, and by AFOSR
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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