今後の展望

精度保証法を力学系解析の道具とすることには、大きな発展が見込まれる。本研究と関連するも のに限っても、Lyapunov関数の軌道積分への応用や、非双曲型平衡点の解析（特にHamilton^形 式）への拡張などが考えらる。また、本論文で述べた諸々の手法を計算ライブラリの形で整備し、

強力な実用解析ツールとして提供することも視野に入れたい。

謝辞

本論文をまとめるにあたり、電気通信大学大学院 情報理工学研究科 山本 野人教授にご懇切なる ご指導および助言いただきましたこと、感謝いたします。

また、学位論文を審査していただいた緒方 秀教教授、仲谷 栄伸教授、山本 有作教授、小山 大介 助教、および九州大学 情報基盤研究開発センター 渡部 善隆准教授に感謝申し上げます。

共同研究者である松江 要助教には力学系の専門的立場からのご指導と助言いただきましたこと、

厚くお礼申し上げます。

参考文献

[1] Valentin S Afraimovich, V.V Bykov, and Leonid P Shilnikov. On the origin and structure of the lorenz attractor. InAkademiia Nauk SSSR Doklady, Vol. 234, pp. 336–339, 1977.

[2] G Alefeld. Inclusion methods for systems of nonlinear equations - the interval newton method and modifications. In J. Herzberger, editor, Topics in Validated Computations, pp. 7–26. Elsevier Amsterdam 1994, 1993.

[3] Fabio Camilli, Lars Gr¨une, and Fabian Wirth. A generalization of zubov’s method to perturbed systems. SIAM Journal on Control and Optimization, Vol. 40, No. 2, pp.

496–515, 2001.

[4] Z Galias and P Zgliczy´nski. Computer assisted proof of chaos in the lorenz equations.

Physica D: Nonlinear Phenomena, Vol. 115, No. 3, pp. 165–188, 1998.

[5] Marcio Gameiro and Jean-Philippe Lessard. Rigorous computation of smooth branches of equilibria for the three dimensional cahn–hilliard equation. Numerische Mathematik, Vol. 117, No. 4, pp. 753–778, 2011.

[6] Peter A Giesl. Construction of global Lyapunov functions using radial basis functions.

Springer, 2007.

[7] Peter A Giesl and Sigurdur Hafstein. Computation and verification of lyapunov functions.

SIAM Journal on Applied Dynamical Systems, Vol. 14, No. 4, pp. 1663–1698, 2015.

[8] Peter A Giesl and Sigurdur F Hafstein. Revised cpa method to compute lyapunov func-tions for nonlinear systems.Journal of Mathematical Analysis and Applications, Vol. 410, No. 1, pp. 292–306, 2014.

[9] Peter A Giesl and Sigurdur F Hafstein. Review on computational methods for lyapunov functions. Discrete and Continuous Dynamical Systems - Series B, Vol. 20, No. 8, pp.

2291–2331, 2015.

[10] Peter Giesl and Holger Wendland. Meshless collocation: Error estimates with application to dynamical systems. SIAM Journal on Numerical Analysis, Vol. 45, No. 4, pp. 1723–

1741, 2007.

[11] Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, W Kalies, and Dinesh Kasti.

Efficient computation of lyapunov functions for morse decompositions. DCDS-S, 2015.

[12] Lars Gr¨une. An adaptive grid scheme for the discrete hamilton-jacobi-bellman equation.

Numerische Mathematik, Vol. 75, No. 3, pp. 319–337, 1997.

[13] Lars Gr¨une.Asymptotic behavior of dynamical and control systems under pertubation and discretization. No. 1783 in Lecture Notes in Mathematics. Springer Science & Business Media, 2002.

[14] Sigurdur Freyr Hafstein. A constructive converse lyapunov theorem on asymptotic

stabil-ity for nonlinear autonomous ordinary differential equations.Dynamical Systems, Vol. 20, No. 3, pp. 281–299, 2005.

[15] Sigurdur Freyr Hafstein. An algorithm for constructing lyapunov functions, 2007.

[16] Morris Hirsch, Stephen Smale, Robert L Devaney, ^訳:^桐木紳, ^三波篤郎, ^谷川清隆,^辻井正 人. ^{力学系入門} : 微分方程式からカオスまで. Differential equations, dynamical systems &

an introduction to chaos.共立出版,東京, Japan, 2007.8 2007.

[17] Peter Al Hokayem and Eduardo Gallestey. Lyapunov stability theory.

http://people.ee.ethz.ch/ apnoco/Lectures2015/03 Lyapunov.pdf, 2015. ^{参 照} :2016-11-07.

[18] Chieh Su Hsu. Cell-to-cell mapping: a method of global analysis for nonlinear systems, Vol. 64. Springer Science & Business Media, 2013.

[19] William D Kalies, Konstantin Mischaikow, and Robert CAM Vandervorst. An algorithmic approach to chain recurrence. Foundations of Computational Mathematics, Vol. 5, No. 4, pp. 409–449, 2005.

[20] Hiroshi Kokubu, Daniel Wilczak, and Piotr Zgliczy´nski. Rigorous verification of cocoon bifurcations in the michelson system. Nonlinearity, Vol. 20, No. 9, p. 2147, 2007.

[21] Bernd Krauskopf and Hinke M Osinga. Computing invariant manifolds via the continua-tion of orbit segments. InNumerical Continuation Methods for Dynamical Systems, pp.

117–154. Springer, 2007.

[22] Bernd Krauskopf, Hinke M Osinga, Eusebius J Doedel, Michael E Henderson, John Guckenheimer, Alexander Vladimirsky, Michael Dellnitz, and Oliver Junge. A survey of methods for computing (un) stable manifolds of vector fields. International Journal of Bifurcation and Chaos, Vol. 15, No. 03, pp. 763–791, 2005.

[23] A. Liapounoff. Probl`eme g´en´eral de la stabilit´e du mouvement. Annales de la Facult´e des sciences de Toulouse : Math´ematiques, Vol. 9, pp. 203–474, 1907.

[24] R. J. Lohner. Enclosing the solutions of ordinary initial and boundary value problems.

E.W. Kaucher, U.W. Kulisch, C. Ullrich (Eds.), Computer Arithmetic: Scientific Com-putation and Programming Languages, Wiley-Teubner Series in Computer Science, pp.

255–286, 1987.

[25] Kaname Matsue, Tomohiro Hiwaki, and Nobito Yamamoto. On the construction of lya-punov functions with computer assistance. Preprint, 2016.

[26] A. I. Mees. Dynamics of Feedback Systems. John Wiley & Sons, Inc., New York, NY, USA, 1981.

[27] Antonis Papachristodoulou, James Anderson, Giorgio Valmorbida, Stephen Prajna, Pete Seiler, and Pablo Parrilo. Sostools version 3.00 sum of squares optimization toolbox for matlab. arXiv preprint arXiv:1310.4716, 2013.

[28] Pablo A Parrilo. Structured semidefinite programs and semialgebraic geometry methods

in robustness and optimization. PhD thesis, Citeseer, 2000.

[29] Matthew M Peet. Exponentially stable nonlinear systems have polynomial lyapunov functions on bounded regions. IEEE Transactions on Automatic Control, Vol. 54, No. 5, pp. 979–987, 2009.

[30] Matthew M Peet and Antonis Papachristodoulou. A converse sum of squares lyapunov result with a degree bound. IEEE Transactions on Automatic Control, Vol. 57, No. 9, pp. 2281–2293, 2012.

[31] Michael Plum. Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing, Vol. 49, No. 1, pp. 25–44, 1992.

[32] Clark Robinson, ^訳:^国府寛司,^柴山健伸,^岡宏枝. ^{力学系 上}. シュプリンガーフェアラーク東 京, 2001.

[33] Siegfried M Rump. Verification methods: Rigorous results using floating-point arithmetic.

Acta Numerica, Vol. 19, pp. 287–449, 2010.

[34] Siegfried M Rump. Verified bounds for singular values, in particular for the spectral norm of a matrix and its inverse. BIT Numerical Mathematics, Vol. 51, No. 2, pp. 367–384, 2011.

[35] Ja G Sinai and EB Vul. Discovery of closed orbits of dynamical systems with the use of computers. Journal of Statistical Physics, Vol. 23, No. 1, pp. 27–47, 1980.

[36] Akitoshi Takayasu, Kaname Matsue, Takiko Sasaki, Kazuaki Tanaka, Makoto Mizuguchi, Shin ichiOishi. Numerical validation of blow-up solutions of ordinary differential equa-tions. Journal of Computational and Applied Mathematics, Vol. 314, pp. 10–29, 2017.

[37] Daniel Wilczak. The existence of shilnikov homoclinic orbits in the michelson system: a computer assisted proof. Foundations of Computational Mathematics, Vol. 6, No. 4, pp.

495–535, 2006.

[38] Daniel Wilczak and Piotr Zgliczynski. Topological method for symmetric periodic orbits for maps with a reversing symmetry. arXiv preprint math/0401145, 2004.

[39] Nobito Yamamoto. A numerical verification method for solutions of boundary value prob-lems with local uniqueness by banach’s fixed-point theorem. SIAM Journal on Numerical Analysis, Vol. 35, No. 5, pp. 2004–2013, 1998.

[40] Piotr Zgliczynski. C 1 lohner algorithm. Foundations of Computational Mathematics, Vol. 2, No. 4, pp. 429–465, 2002.

[41] Piotr Zgliczy´nski. Covering relations, cone conditions and the stable manifold theorem.

Journal of Differential Equations, Vol. 246, No. 5, pp. 1774–1819, 2009.

[42] Piotr Zgliczynski and Konstantin Mischaikow. Rigorous numerics for partial differential equations: The kuramoto^―sivashinsky equation. Foundations of Computational Mathe-matics, Vol. 1, No. 3, pp. 255–288, 2001.

[43] ^貝ヶ石亘,^松尾宇泰. 力学系の閉軌道を求めるニュートン・ラフソン・ミーズ法と樋脇・山本

の方法について. 応用数理学会研究部会連合発表会, 2013.

[44] ^宮路智行. Poincar´e 写像による極限周期軌道の解析とその周辺. 第１回山梨精度保証研究会,

2014.

[45] ^山口昌哉,^吉原英昭,^西田孝明. 精度保証付き数値計算とその応用：微分方程式に対する保征 付き数値計算. ^情報処理, Vol. 31, No. 9, sep 1990.

[46] 山本野人. 常微分方程式の解の精度保証法. シミュレーション, Vol. 31, No. 3, 2012.

[47] 山野駿. 連続力学系におけるホモクリニック軌道の精度保証による検証について. Master’s thesis,^{電気通信大学}, 2016.

[48] ^松田望. 中心値・半径方式による精度保証付き多倍長区間演算ライブラリの開発. PhD thesis, 電気通信大学, 2016.

[49] 新田光輝,中山大輔, 三宅智大,山本野人. Hybrid 力学系の不動点およびlyapunov関数につ いての精度保証. 日本応用数理学会2016 年度年会, 2016.

[50] ^大石進一. ^{精度保証付き数値計算}. ^{精度保証付き数値計算}.^コロナ社,^東京, Japan, 2000.

[51] ^中尾充宏,^山本野人. ^{精度保証付き数値計算}: コンピュータによる無限への挑戦. ^{精度保証付} き数値計算.^{日本評論社},^東京, Japan, 1998.

[52] ^中尾充宏,^渡部善隆. 実例で学ぶ精度保証付き数値計算: ^{理論と実装}. For senior & graduate courses.サイエンス社,東京, Japan, 2011.

関連論文

[1] 樋脇知広、山本野人、「力学系における閉軌道の存在領域の精度保証法による同定」、日本応用 数理学会論文誌，vol 22, No.4, 269-276, 2012.

[2] T. Hiwaki and N. Yamamoto. Some remarks on numerical verification of closed orbits in dynamical systems., Nonlinear Theory and Its Applications, IEICE, 6(3):397-403, 2015.

関連論文の印刷公表の方法及び時期

• ^全著者名: 樋脇 知広、山本 野人

論文題目: 力学系における閉軌道の存在領域の精度保証法による同定 印刷公表の方法及び時期 : 日本応用数理学会論文誌22^（4^）,2012^年12^月

（4章の内容と関連）

• ^全著者名: Hiwaki Tomohiro^、Yamamoto Nobito

論文題目: Some remarks on numerical verification of closed orbits in dynamical systems 印刷公表の方法及び時期 : Nonlinear Theory and Its Applications, IEICE, Vol.6, No.3, 2015年6月

（4章の内容と関連）