Connecting Orbit Structure of Monotone Solutions in the Shadow System

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概要

• 論文の詳細を見る

• The shadow system [snip] is a scalar reaction diffusion equation coupled with an ODE. The extra freedom coming from the ODE drastically influences the solution structure and dynamics as compared to that of a single scalar reaction diffusion system. In fact, it causes secondary bifurcations and coexistence of multiple stable equilibria. Our long term goal is a complete description of the global dynamics on its global attractor A as a function of ε, f, and g. Since this is still far beyond our capabilities, we focus on describing the dynamics of solutions to the shadow system which are monotone in x, and classify the global connecting orbit structures in the monotone solution space up to the semi-conjugacy. The maximum principle and hence the lap number arguments, which have played a central role in the analysis of one dimensional scalar reaction diffusion equations, cannot be directly applied to the shadow system, although there is a Lyapunov function in an appropriate parameter regime. In order to overcome this difficulty, we resort to the Conley index theory. This method is topological in nature, and allows us to reduce the connection problem to a series of algebraic computations. The semi-conjugacy property can be obtained once the connection problem is solved. The shadow system turns out to exhibit minimal dynamics which displays the mechanism of basic pattern formation, namely it explains the dynamic relation among the trivial rest states (constant solutions) and the event patterns (large amplitude inhomogeneous solutions).

• Academic Pressの論文
• 1997-11-01

著者

• Kokubu Hiroshi

  Department Of Mathematics Kyoto University・jst-crest

• Mischaikow Konstantin

  Department of Mathematics and BioMaPS, Hill Center-Busch Campus

• Mischaikow Konstantin

  Department Of Mathematics And Biomaps Hill Center-busch Campus

• Oka Hiroe

  Department Of Applied Mathematics And Informatics Faculty Of Science And Technology Ryukoku University

関連論文

• A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems
• Connecting Orbit Structure of Monotone Solutions in the Shadow System
• Graph-based topological approximation of saddle-node bifurcation in maps : Dedicated to Professor Yasumasa Nishiura for his sixtieth birthday (Far-From-Equilibrium Dynamics)

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