Analysis of non-hyperbolic systems through theories and numerical analysis

• Project/Area Number: 16K17609
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Hitotsubashi University
• Principal Investigator: Shinohara Katsutoshi 一橋大学, 大学院商学研究科, 准教授 (50740429)
• Project Period (FY): 2016-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 非双曲型力学系 / 部分双曲型力学系 / 野生的力学系 / 周期点の増大度 / ブレンダー / 微分力学系 / 異次元ヘテロクリニックサイクル

Outline of Final Research Achievements

1. In general, if non-hyperbolicity exists, it is expected that the increase of the number of periodic points of the system (with respect to the period) will be large. I proved this fact for a class of systems called partial hyperbolic dynamical system with some condition.2. A blender is an important mechanism for occurrence of non hyperbolic robustness against robustness (robustness). I investigated the visualization of the limiting set of systems having blenders which are given as quadratic polynomial maps.3. I studied the class of non-hyperbolic dynamical system called wild. I proved the existence of wild dynamical system among the class called volume hyperbolic systems.

Research Products

• [Int'l Joint Research] Inperial College London(英国)
• [Int'l Joint Research] Universite de Bourgogne(フランス)
• [Int'l Joint Research] University of Auckland(ニュージーランド)
• [Int'l Joint Research] Univerisity of Auckland(ニュージーランド)
• [Int'l Joint Research] Imperial College London(英国)
• [Int'l Joint Research] Universite de Bourgogne(フランス)
• [Journal Article] Estimation of mean squared errors of non-binary A/D-encoders through Fredholm determinants of piecewise-linear transformations (2018)
  • Author(s): Katsutoshi SHINOHARA and Kenta Kobayashi
  • Journal Title: Nonlinear Theory and Its Applications, IEICE Volume: 9 Issue: 2 Pages: 243-258
  • DOI: 10.1587/nolta.9.243
  • NAID: 130006602776
  • ISSN: 2185-4106
  • Peer Reviewed / Open Access
• [Journal Article] Volume hyperbolicity and wildness (2016)
  • Author(s): C. Bonatti and K. Shinohara
  • Journal Title: Ergodic Theory and Dynamical Systems Volume: - Issue: 3 Pages: 886-920
  • DOI: 10.1017/etds.2016.51
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics (2016)
  • Author(s): M. Asaoka, K. Shinohara, D.Turaev
  • Journal Title: Mathematische Annalen Volume: - Issue: 3-4 Pages: 1277-1309
  • DOI: 10.1007/s00208-016-1468-0
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Presentation] Super exponential growth of number of periodic points for skew product maps (2017)
  • Author(s): 篠原克寿
  • Organizer: The Workshop on Dynamical Systems 2017
  • Int'l Joint Research
• [Presentation] The geometry of blenders in a three-dimensional Henon-like family (2017)
  • Author(s): Stefanie Hittmeyer
  • Organizer: Dynamics Beyond Uniform Hyperbolicity 2017
  • Int'l Joint Research / Invited
• [Presentation] Volume hyperbolicity and wildness (2016)
  • Author(s): K Shinohara
  • Organizer: Dynamics, Bifurcations, and Strange Attractors
  • Place of Presentation: Lobachevsky State University of Nizhny Novgorod，ニジニノブゴロド（ロシア）
  • Year and Date: 2016-07-18
  • Int'l Joint Research
• [Remarks] The geometry of blenders
  • URL: http://www.math.auckland.ac.nz/~berndk/transfer/hkos_blender_prep.pdf