Construction of stability theory of ordinary differential systems by fractal analysis

• Project/Area Number: 17K14226
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Okayama University of Science
• Principal Investigator: Onitsuka Masakazu 岡山理科大学, 理学部, 准教授 (20548367)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Project Status: Discontinued (Fiscal Year 2020)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: フラクタル解析 / 2次元系 / 安定性理論 / ウラム安定性 / ウラム定数 / 摂動問題 / ハイヤーズ－ウラム安定性(HUS) / HUS定数 / ボックス次元 / 線形差分方程式 / 刻み幅 / 関数方程式論 / 解析学

Outline of Final Research Achievements

The main results of this research project are the construction of stability theory for two-dimensional linear systems and nonlinear systems by using fractal analysis, and the derivation of Ulam stability and best Ulam constants in linear differential equations and difference equations. Sufficient conditions and a necessary and sufficient condition for the rectifiability of spiral orbits for two-dimensional nonlinear systems (including quasi-linear systems) are obtained. We also succeeded in giving precise results on the perturbation problem of a two-dimensional quasi-linear system by polar coordinate transformation using generalized trigonometric functions.

Academic Significance and Societal Importance of the Research Achievements

2次元系の平衡点に巻きつく解軌道の複雑さの度合いは、解軌道の長さが有限か無限かの2つに分けられ、さらに無限の長さのとき、フラクタル次元により明確な数値として表せる。ここで、複雑さの度合いを安定性の度合いと言い換えれば、本研究は、新たな安定性解析の確立を実現したと言える。また、摂動問題の一つであるウラム安定性は、実方程式と摂動方程式（近似方程式）との解の誤差を精密に研究することで、現象を記述する数理モデルへ応用できる。実際に、本研究では、カラテオドリ型微分方程式のウラム安定性とそのウラム定数を導出し、得た結果を物体の表面温度に関する数理モデルへ応用し、厳密解と近似解の精確な誤差を与えた。

Research Products

• [Int'l Joint Research] Concordia College/University of Arkansas at Little Rock(米国)
• [Int'l Joint Research] Universidade de Brasilia(ブラジル)
• [Journal Article] A necessary and sufficient condition for Hyers-Ulam stability of first-order periodic linear differential equations (2020)
  • Author(s): Fukutaka Ryuma、Onitsuka Masakazu
  • Journal Title: Applied Mathematics Letters Volume: 100 Pages: 1-7
  • DOI: 10.1016/j.aml.2019.106040
  • Peer Reviewed / Open Access
• [Journal Article] Hyers-Ulam Stability and Best Constant for Cayley h-Difference Equations (2020)
  • Author(s): Anderson Douglas R.、Onitsuka Masakazu
  • Journal Title: Bulletin of the Malaysian Mathematical Sciences Society Volume: 2020 Issue: 6 Pages: 1-16
  • DOI: 10.1007/s40840-020-00920-z
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hyers-Ulam stability of second-order nonhomogeneous linear difference equations with a constant stepsize (2020)
  • Author(s): Onitsuka Masakazu
  • Journal Title: Journal of Computational Analysis and Applications Volume: 28 Pages: 152-165
  • Peer Reviewed / Open Access
• [Journal Article] Best Constant for Hyers-Ulam Stability of Second-Order h-Difference Equations with Constant Coefficients (2019)
  • Author(s): Anderson Douglas R.、Onitsuka Masakazu
  • Journal Title: Results in Mathematics Volume: 74 Issue: 4 Pages: 1-16
  • DOI: 10.1007/s00025-019-1077-9
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hyers-Ulam Stability of a Discrete Diamond-Alpha Derivative Equation (2019)
  • Author(s): Anderson Douglas R.、Onitsuka Masakazu
  • Journal Title: Frontiers in Functional Equations and Analytic Inequalities, Springer New york Volume: 2019 Pages: 237-254
  • DOI: 10.1007/978-3-030-28950-8_14
  • ISBN: 9783030289492, 9783030289508
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Ulam Stability for a Class of Hill’s Equations (2019)
  • Author(s): Fukutaka Ryuma、Onitsuka Masakazu
  • Journal Title: Symmetry Volume: 11 Issue: 12 Pages: 1-15
  • DOI: 10.3390/sym11121483
  • Peer Reviewed / Open Access
• [Journal Article] Hyers-Ulam stability for a discrete time scale with two step sizes (2019)
  • Author(s): Anderson Douglas R.、Onitsuka Masakazu
  • Journal Title: Applied Mathematics and Computation Volume: 344-345 Pages: 128-140
  • DOI: 10.1016/j.amc.2018.10.014
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hyers-Ulam stability of first order linear differential equations of Caratheodory type and its application (2019)
  • Author(s): Onitsuka Masakazu
  • Journal Title: Applied Mathematics Letters Volume: 90 Pages: 61-68
  • DOI: 10.1016/j.aml.2018.10.013
  • Peer Reviewed / Open Access
• [Journal Article] Best constant in Hyers?Ulam stability of first-order homogeneous linear differential equations with a periodic coefficient (2019)
  • Author(s): Fukutaka Ryuma、Onitsuka Masakazu
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 473 Issue: 2 Pages: 1432-1446
  • DOI: 10.1016/j.jmaa.2019.01.030
  • Peer Reviewed / Open Access
• [Journal Article] Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales (2018)
  • Author(s): Anderson Douglas R.、Onitsuka Masakazu
  • Journal Title: Demonstratio Mathematica Volume: 51 Issue: 1 Pages: 198-210
  • DOI: 10.1515/dema-2018-0018
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Hyers-Ulam stability of first-order nonhomogeneous linear difference equations with a constant stepsize (2018)
  • Author(s): M. Onitsuka
  • Journal Title: Applied Mathematics and Computation Volume: 330 Pages: 143-151
  • DOI: 10.1016/j.amc.2018.02.036
  • Peer Reviewed / Open Access
• [Presentation] Hill方程式のHyers-Ulam安定性とHUS定数 (2020)
  • Author(s): 福髙 龍馬, 鬼塚 政一
  • Organizer: 日本数学会 中国・四国支部例会
• [Presentation] Hyers-Ulam stability for first-order linear differential equations with periodic coefficient (2019)
  • Author(s): M. Onitsuka
  • Organizer: The 11th Colloquium on the Qualitative Theory of Differential Equations
  • Int'l Joint Research
• [Presentation] Hyers-Ulam stability and best constant for second-order linear difference equations (2019)
  • Author(s): M. Onitsuka
  • Organizer: The 25th International Conference on Difference Equations and Applications
  • Int'l Joint Research
• [Presentation] ダイヤモンドアルファ差分方程式のウラム安定性 (2019)
  • Author(s): 鬼塚 政一
  • Organizer: 日本数学会2019年度秋季総合分科会
  • Invited
• [Presentation] Hill方程式のHyers-Ulam安定性 (2019)
  • Author(s): 福髙 龍馬, 鬼塚 政一
  • Organizer: 第六回ODE若手セミナー
• [Presentation] 1階周期線形微分方程式に対する Hyers－Ulam 安定性と最小のHUS定数 (2019)
  • Author(s): 福髙 龍馬, 鬼塚 政一
  • Organizer: 日本数学会中国・四国支部例会
• [Presentation] ２次元半分線形系の摂動について (2019)
  • Author(s): 板倉 健太, 鬼塚 政一, 田中 敏
  • Organizer: 日本数学会中国・四国支部例会
• [Presentation] ダイヤモンドアルファ差分方程式におけるウラム安定性 (2019)
  • Author(s): 鬼塚 政一
  • Organizer: RIMS 共同研究（グループ型）「常微分方程式の手法による非線形問題の探求」
• [Presentation] Hyers-Ulam stability of second-order linear difference equations with constant coefficients (2019)
  • Author(s): 鬼塚 政一
  • Organizer: 日本数学会2019年度年会
• [Presentation] Best constant in Hyers-Ulam stability of first-order nonhomogeneous linear difference equations with a constant stepsize (2018)
  • Author(s): M. Onitsuka
  • Organizer: The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Int'l Joint Research / Invited
• [Presentation] Hyers-Ulam stability for periodic linear differential equations of first order (2018)
  • Author(s): R. Fukutaka, M. Onitsuka
  • Organizer: Japan-China Joint Workshop on Differential and Difference Equations and Related Topics in Osaka 2018
  • Int'l Joint Research
• [Presentation] 周期係数をもつ1階同次線形微分方程式の Hyers－Ulam 安定性と最良定数 (2018)
  • Author(s): 福髙 龍馬, 鬼塚 政一
  • Organizer: 第五回ODE若手セミナー
• [Presentation] Characteristic equation for autonomous planar half-linear differential systems (2018)
  • Author(s): 田中 敏, 鬼塚 政一
  • Organizer: 日本数学会2018年度年会
• [Presentation] Box dimension of solution curves for a class of two-dimensional linear differential systems (2018)
  • Author(s): 鬼塚 政一, 田中 敏
  • Organizer: 日本数学会2018年度年会
• [Presentation] 1階非同次線形差分方程式の Hyers－Ulam 安定性と刻み幅の関係 (2018)
  • Author(s): 鬼塚 政一
  • Organizer: RIMS 共同研究（グループ型）「非線形問題への常微分方程式の手法によるアプローチ」
• [Presentation] 微小な刻み幅をもつ 1 階同次線形差分方程式の Aoki－Rassias 安定性 (2018)
  • Author(s): 眞鍋 佳菜子, 鬼塚 政一
  • Organizer: 日本数学会中国・四国支部例会
• [Presentation] 変数係数をもつ 1 階非同次線形微分方程式の Hyers－Ulam 安定性 (2018)
  • Author(s): 福高 龍馬, 鬼塚 政一
  • Organizer: 日本数学会中国・四国支部例会
• [Presentation] 1階同次線形差分方程式の刻み幅が Hyers－Ulam 安定性に与える影響 (2017)
  • Author(s): 鬼塚 政一, 眞鍋 佳菜子
  • Organizer: 第四回ODE若手セミナー
• [Presentation] 時変係数をもつ1階非同次線形微分方程式の Hyers－Ulam 安定性 (2017)
  • Author(s): 福高 龍馬, 鬼塚 政一
  • Organizer: 第四回ODE若手セミナー
• [Presentation] 2次元定数係数線形差分方程式系の原点の幾何学的分類 (2017)
  • Author(s): 谷本 陽輝, 鬼塚 政一
  • Organizer: 第四回ODE若手セミナー
• [Presentation] Best constant in Hyers-Ulam stability of first-order nonhomogeneous linear difference equations (2017)
  • Author(s): M. Onitsuka
  • Organizer: Japan-China Joint Workshop on Ordinary Differential Equations and Related Topics in Osaka 2017
  • Int'l Joint Research
• [Presentation] On the Hyers-Ulam stability of first-order nonhomogeneous linear difference equations (2017)
  • Author(s): M. Onitsuka
  • Organizer: The 17th International Conference on Functional Equations and Inequalities
  • Int'l Joint Research
• [Presentation] Integral Average Conditions for Oscillation of Damped Half-linear Differential Equations (2017)
  • Author(s): M. Onitsuka
  • Organizer: International Conference on Differential & Difference Equations and Applications 2017
  • Int'l Joint Research
• [Remarks] 常微分方程式における最近の動向とその発展
  • URL: http://www.xmath.ous.ac.jp/~onitsuka/workshop2019_1.html
• [Remarks] 鬼塚政一 (Onitsuka Masakazu) 研究室ホームページ
  • URL: https://www.xmath.ous.ac.jp/~onitsuka/
• [Remarks] 岡山理科大学 教員データベース
  • URL: https://portal.pub.ous.ac.jp/KyoinDB/KyoinDB_Search.asp
• [Remarks] 鬼塚政一 (Onitsuka Masakazu) 研究室ホームページ
  • URL: http://www.xmath.ous.ac.jp/~onitsuka/
• [Remarks] 岡山理科大学 教員データベース
  • URL: https://edb.pub.ous.ac.jp/OUSEducatorDB
• [Funded Workshop] RIMS Workshop Recent Trends in Ordinary Differential Equations and Their Developments (2019)