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2003 Fiscal Year Final Research Report Summary

Global Bifurcational Approach to Complex Spatio^temporal Patterns in Dissipative Systems

Research Project

Project/Area Number 13440027
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionHokkaido University

Principal Investigator

NISHIURA Yasumasa  Hokkaido Univ., Research Institute for Electronic Science, Prof., 電子科学研究所, 教授 (00131277)

Co-Investigator(Kenkyū-buntansha) YANAGITA Tatsuo  Hokkaido Univ., Research Institute for Electronic Science, Res.Asso., 電子科学研究所, 助手 (80242262)
KOBAYASHI Ryo  Hokkaido Univ., Research Institute for Electronic Science, Asso.Prof., 電子科学研究所, 助教授 (60153657)
TSUDA Ichiro  Hokkaido Univ., Faculty of Science, Prof., 大学院・理学研究科, 教授 (10207384)
UEYAMA Daishin  Hiroshima Univ., Graduate School of Science, Res.Asso., 大学院・理学研究科, 助手 (20304389)
KOKUBU Hiroshi  Kyoto Univ., Graduate School of Science, Asso.Prof., 大学院・理学研究科, 助教授 (50202057)
Project Period (FY) 2001 – 2003
Keywordsreaction diffusion systems / bifurcation / self-replicating pattern / chaos / scattering / pattern formation / 散乱
Research Abstract

In a regime of far-from equilibrium there appears a diversity of complex patterns such as self-replication, spatio-temporal patterns, and collisions among particle-like patterns. One of the powerful tools to understand these things is dynamical system theory, however its naive application in general does not work partly due to the high dimensionality of phase space and large deformation of solutions. What should be the clue for us to start with in understanding such behaviors? We need to alter our way of thinking, namely "Let us think about the geometric structures that guide solution orbits creating such a chaotic dynamism, rather than keeping track of the deformations of solutions in detail". In other words, we should try to characterize geometric structures of the infinite dimensional phase space in which behaviors of solution orbits become easily detectable. Taking this viewpoint, we accomplished the following two main things. Please refer to the published papers for other aspect o … More f achievements.
1.Unfolding of generalized heteroclinic cycle implies spatio-temporal chaos.
Chimerical methods, such as AUTO, give us a great amount of information on an unstable solution, as well as on the behavior of its unstable manifold. Heteroclinic cycle connecting several stationary patterns was identified as a key to understand the complex behaviors like spatio-temporal chaos for the Gray-Scott model. The mechanism itself has much wider applicability to other model systems.
2.Role of "Scattors" for collision process among particle-like patterns.
Scattering of particle-like patterns in dissipative systems has much attention from various fields. We focused on the issue how the input-output relation is controlled at a head-on collision where traveling pulses or spots interact strongly.
It had remained an open problem due to the large deformation of patterns at a colliding point. We found that special type of unstable steady or time-periodic solutions called scattors and their stable and unstable manifolds direct the traffic flow of orbits.
Such scattors are in general highly unstable even in ID case which causes a variety of input-output relations through the scattering process. We illustrate the ubiquity of scattors by using the complex Ginzburg-Landau equation, the Gray-Scott model and a three-component reaction diffusion model arising in gas-discharge phenomena. Less

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Y.Nishiura, D.Ueyama: "Spatio-temporal chaos for the Gray-Scott model"Physica D. 150. 137-162 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Ei, Y.Nishiura, K.I.Ueda: "2^n-splitting or Edge-splitting-A manner of splitting in dissipative systems"Japan J.of Ind.Appl.Math. 18(2). 181-205 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Nishiura, T.Teramoto, K.I.Ueda: "Scattering and separators in dissipative systems"PHYSICAL REVIEW E. Vol.67. 056210-1-056210-7 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Nishiura, T.Teramoto, K.I.Ueda: "Dynamic transitions through scattors in dissipative systems"CHAOS. Vol.13,No.3. 962-972 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.Nonomura, R.Kobayashi, Y.Nishiura, M.Shimomura: "Periodic Precipitation during Droplet Evaporation on a Substrate"Journal of the Physical Society of Japan. Vol.72 No.10,October. 2468-2471 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] I.Tsuda, T.Umemura: "Chaotic itinerancy generated by coupling of Milnor attractors"Chaos. 13(3). 937-946 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Nishiura: "Far-from-equilibrium Dynamics"AMS. 311 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Nishiura, D.Ueyama: "Spatio-temporal chaos for the Gray-Scott model"Physica D. 150. 137-162 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S.Ei, Y.Nishiura, K.I.Ueda: "2^n-splitting or Edge-splitting-A manner of splitting in dissipative systems"Japan J.of Ind.Appl.Math.. 18(2). 181-205 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Nishiura, T.Teramoto, K.I.Ueda: "Scattering and separators in dissipative systems"PHYSICAL REVIEW E. Vol.67. 056210-1-056210-7 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Nishiura, T.Teramoto, K.I.Ueda: "Dynamic transitions through scattors in dissipative systems"CHAOS. Vol.13, No.3. 962-972 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.Nonomura, R.Kobayashi, Y.Nishiura, M.Shimomura: "Periodic Precipitation during Droplet Evaporation on a Substrate"Journal of the Physical Society of Japan. Vol.72, No.10. 2468-2471 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] I.Tsuda, T.Umemura: "Chaotic itinerancy generated by coupling of Milnor attractors"Chaos. 13(3). 937-946 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Nishiura: "Far-from-equilibrium Dynamics"AMS. 311 (2002)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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