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Advances in Numerical Analysis
Volume 2013 (2013), Article ID 420897, 9 pages
Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Received 20 December 2012; Revised 13 April 2013; Accepted 18 April 2013
Academic Editor: William J. Layton
Copyright © 2013 Tadashi Kawanago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- T. Kawanago, “A symmetry-breaking bifurcation theorem and some related theoremsapplicable to maps having unbounded derivatives,” Japan Journal of Industrial and Applied Mathematics, vol. 21, no. 1, pp. 57–74, 2004, Corrigendum to this paper: Japan Journal of Industrial and Applied Mathematics, vol. 22, pp. 147, 2005.
- T. Kawanago, “Computer assisted proof to symmetry-breaking bifurcation phenomena in nonlinear vibration,” Japan Journal of Industrial and Applied Mathematics, vol. 21, no. 1, pp. 75–108, 2004.
- Y. Watanabe and M. T. Nakao, “Numerical verification method of solutions for elliptic equations and its application to the Rayleigh-Bénard problem,” Japan Journal of Industrial and Applied Mathematics, vol. 26, no. 2-3, pp. 443–463, 2009.
- M. T. Nakao, Y. Watanabe, N. Yamamoto, T. Nishida, and M.-N. Kim, “Computer assisted proofs of bifurcating solutions for nonlinear heat convection problems,” Journal of Scientific Computing, vol. 43, no. 3, pp. 388–401, 2010.
- M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” Functional Analysis, vol. 8, pp. 321–340, 1971.
- M. Reed and B. Simon, Functional analysis I, Academic Press, New York, NY, USA, 2nd edition, 1980.
- M. Kubíček and M. Holodniok, “Algorithms for determination of period-doubling bifurcation points in ordinary differential equations,” Journal of Computational Physics, vol. 70, no. 1, pp. 203–217, 1987.
- T. Kawanago, “Error analysis of Galerkin's method for semilinear equations,” Journal of Applied Mathematics, vol. 2012, Article ID 298640, 15 pages, 2012.
- T. Kawanago, “Improved convergence theorems of Newton's method designed for the numerical verification for solutions of differential equations,” Journal of Computational and Applied Mathematics, vol. 199, no. 2, pp. 365–371, 2007.
- J. K. Hale, “Bifurcation from simple eigenvalues for several parameter families,” Nonlinear Analysis, vol. 2, no. 4, pp. 491–497, 1978.
- J. López-Gómez, “Multiparameter local bifurcation based on the linear part,” Journal of Mathematical Analysis and Applications, vol. 138, no. 2, pp. 358–370, 1989.
- M. G. Crandall and P. H. Rabinowitz, “The Hopf bifurcation theorem in infinite dimensions,” Archive for Rational Mechanics and Analysis, vol. 67, no. 1, pp. 53–72, 1977.
- T. Iohara, T. Nishida, Y. Teramoto, and H. Yoshihara, “Benard-marangoni convection with a deformable surface,” Sūrikaisekikenkyūsho Kōkyūroku, no. 974, pp. 30–42, 1996.
- T. Nishida, Y. Teramoto, and H. Yoshihara, “Bifurcation problems for equations of fluid dynamics and computer aided proof,” in Proceedings of the 2nd Japan-China Seminar on Numerical Mathematics, vol. 14 of Lecture Notes in Numerical and Applied Analysis, pp. 145–157, Kinokuniya, Tokyo, Japan, 1995, Advances in numerical mathematics.