Computer assisted proof to symmetry-breaking bifurcation phenomena in nonlinear vibration
- Tadashi Kawanago
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Using the numerical verification method, we analyze a nonlinear vibration system with friction described by a wave equation related to the Duffing equation. We present a complete proof of the existence of a symmetry-breaking bifurcation point, which was first found in the numerical simulation. Our method is based on some general theorems established by the author in another paper  and is applicable to various systems described by semilinear partial differential equations including elliptic and parabolic ones. All of the numerical results in the proof can be accurately reproduced, though some of them are comparatively large scale. This reproducibility is realized by introducing a method for completely controlling the numerical calculations.
- V. Barbu and N.H. Pavel, Periodic solutions to nonlinear one dimensional wave equation with X-dependent coefficients. Trans. Amer. Math. Soc.,349 (1997), 2035–2048. CrossRef
- J.G. Heywood, W. Nagata and W. Xie, A numerically based existence theorem for the Navier-Stokes equations. J. Math. Fluid. Mech.,1 (1999), 5–23. CrossRef
- T. Kawanago, A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Japan J. Indust. Appl. Math.,21 (2004), 57–74. CrossRef
- T. Kawanago, Analysis for bifurcation phenomena of nonlinear vibrations. Numerical Solution of Partial Differential Equations and Related Topics II, RIMS Kokyuroku1198 (2001), 13–20.
- T. Kawanago, Applications of computer algebra to some bifurcation problems in nonlinear vibrations. Computer Algebra — Algorithms, Implementations and Applications, RIMS Kokyuroku1295 (2002), 137–143.
- T. Kawanago, Approximation of linear differential operators by almost diagonal operators and its applications. Preprint.
- Y. Komatsu, A bifurcation phenomenon for the periodic solutions of a semilinear dissipative wave equation. J. Math. Kyoto Univ.,41 (2001), 669–692.
- O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, 1968.
- A. Matsumura, Bifurcation phenomena for the Duffing equation. Advances in Nonlinear Partial Differential Equations and Stochastics (eds. Kawashima and Yanagisawa), World Scientific, 1998.
- S. Oishi, Introduction to Nonlinear Analysis (in Japanese). Corona-sha, 1997.
- T. Nishida, Y. Teramoto and H. Yoshihara, Bifurcation problems for equations of fluid dynamics and computer aided proof. Lecture Notes in Numer. Appl. Anal.14, Springer, 1995, 145–157.
- M. Plum, Enclosures for two-point boundary value problems near bifurcation points. Scientific Computing and Validated Numerics (Wuppertal, 1995), Math. Res.90, Akademie Verlag, Berlin, 1996, 265–279.
- T. Tsuchiya, Numerical verification of simple bifurcation points. RIMS Kokyuroku831 (1993), 129–140.
- T. Tsuchiya and M.T. Nakao, Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points. Japan J. Indust. Appl. Math.,14 (1997), 357–372. CrossRef
- Computer assisted proof to symmetry-breaking bifurcation phenomena in nonlinear vibration
Japan Journal of Industrial and Applied Mathematics
Volume 21, Issue 1 , pp 75-108
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- nonlinear vibration
- computer assisted proof
- almost diagonal operator
- complete reproducibility
- Tadashi Kawanago (1)
- Author Affiliations
- 1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Oh-Okayama, Meguro-ku, 152-8551, Tokyo, Japan