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Article type: Cover
2005 Volume 15 Issue 2 Pages
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Article type: Index
2005 Volume 15 Issue 2 Pages
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Hisashi Okamoto
Article type: Article
2005 Volume 15 Issue 2 Pages
89-
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Toshiyuki Ogawa, Kenichi Nakamura
Article type: Article
2005 Volume 15 Issue 2 Pages
90-92
Published: June 24, 2005
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Konstantin Mischaikow, Ken-ichi Nakamura (translation)
Article type: Article
2005 Volume 15 Issue 2 Pages
93-107
Published: June 24, 2005
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We argue that computational homology can play an important role in the analysis of complicated temporal and/or spatial behavior associated with nonlinear systems. A brief description concerning the homology of spaces and maps is provided. This is followed by examples that indicate how the homology of spaces can be used to study the evolution of complicated patterns arising from numerical simulations and physical experiments. We also indicate how in conjunction with numerical computations, homology can be used to obtain computer assisted proofs in dynamics.
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Zin Arai
Article type: Article
2005 Volume 15 Issue 2 Pages
108-119
Published: June 24, 2005
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This paper is a survey article on computer assisted analysis of discrete dynamical systems. The focus is on the application of the Conley index theory. We discuss the definition and some properties of the Conley index and how to perform rigorous computations of it on a computer. We sketch the steps of our algorithm for computing the Conley index, which is a combination of computational homology theory and interval arithmetic. Finally, we propose a rigorous computational method for detecting homoclinic tangencies in a discrete dynamical system as an example of our argument. The problem of finding homoclinic tangencies is translated to that of finding connecting orbits in an associated dynamics on the projectivized tangent bundle and then the Conley index theory is used to find the connecting orbit. Applying this method, we prove the existence of homoclinic tangencies in the Henon family.
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Yasuaki Hiraoka, Toshiyuki Ogawa
Article type: Article
2005 Volume 15 Issue 2 Pages
120-131
Published: June 24, 2005
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A rigorous numerical method to verify bifurcation branches in infinite dimensional dynamical systems is studied in this paper. Key tools of this method are the Conley index and interval arithmetic. At first, we briefly explain the Conley index theory and consider a numerical technique to rigorously verify the existence of equilibrium point in R^m. The essential points are to construct a box in R^m which is expected to contain an equilibrium point and to check the vector field on the boundary. If the Conley index determined by the vector field is the same as that for a hyperbolic equilibrium point, then the Conley index theory assures the existence of equilibrium point in the box. Next, this method is extended to infinite dimensional problems to prove the existence of stationary solutions for evolution equations. The difficulty caused by infinite dimensional problems is that it is impossible to directly check all the vector fields by using computers. This problem is overcome in such a way that we impose the power decay property on higer modes. Then the infinite dimensional problem can be essentially reduced to finite dimensional one.
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Masaru Shibata
Article type: Article
2005 Volume 15 Issue 2 Pages
132-147
Published: June 24, 2005
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A theoretical study of general relativistic and dynamical phenomena in the universe requires to solve Einstein's equations which are highly nonlinear and partial differential equations. For a solution of a problem with no spacetime symmetry such as merger of binary neutron stars and black holes, numerical simulation is the unique approach. "Numerical relativity" is the research field in which the methods for numerical solutions of Einstein's equations are studied. The purpose of this article is to introduce the numerical relativity. First, we describe the main procedures in numerical relativity particularly emphasizing the latest progress for the formulation of the Einstein's evolution equations. Then, the current status and the issues to be resolved are summarized. Finally, we present our latest numerical simulations for merger of binary neutron stars to a black hole and a neutron star.
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Nariaki Horinouchi, Masahide Inagaki, Yoshihiro Kato
Article type: Article
2005 Volume 15 Issue 2 Pages
148-152
Published: June 24, 2005
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Susumu Yamada, Masahiko Machida, Toshiyuki Imamura
Article type: Article
2005 Volume 15 Issue 2 Pages
153-158
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Masahisa Tabata
Article type: Article
2005 Volume 15 Issue 2 Pages
159-164
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Toru Maruyama
Article type: Article
2005 Volume 15 Issue 2 Pages
165-168
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Hiroshi Kashiwagi
Article type: Article
2005 Volume 15 Issue 2 Pages
169-172
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Shannon Jacobs
Article type: Article
2005 Volume 15 Issue 2 Pages
173-175
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Hiroshi Akiba
Article type: Article
2005 Volume 15 Issue 2 Pages
176-178
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Yusaku Yamamoto
Article type: Article
2005 Volume 15 Issue 2 Pages
179-180
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Wuyi Yue
Article type: Article
2005 Volume 15 Issue 2 Pages
180-181
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Sin Hitotumatu
Article type: Article
2005 Volume 15 Issue 2 Pages
181-182
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Yoshimasa Nakamura
Article type: Article
2005 Volume 15 Issue 2 Pages
182-183
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Takashi Sakajo
Article type: Article
2005 Volume 15 Issue 2 Pages
184-185
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Yoichiro Takahashi
Article type: Article
2005 Volume 15 Issue 2 Pages
185-186
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Article type: Appendix
2005 Volume 15 Issue 2 Pages
187-190
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Article type: Appendix
2005 Volume 15 Issue 2 Pages
190-191
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Article type: Appendix
2005 Volume 15 Issue 2 Pages
191-
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Article type: Index
2005 Volume 15 Issue 2 Pages
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Article type: Appendix
2005 Volume 15 Issue 2 Pages
App1-
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Article type: Appendix
2005 Volume 15 Issue 2 Pages
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Article type: Cover
2005 Volume 15 Issue 2 Pages
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Published: June 24, 2005
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