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On embedded bifurcation structure in some discretized vector fields.

Hunseok Kang1, Ichiro Tsuda

  • 1Research Institute for Electronic Science, Hokkaido University, Kita-ku, Sapporo, Japan. kang@math.sci.hokudai.ac.jp

Chaos (Woodbury, N.Y.)
|October 2, 2009
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Summary
This summary is machine-generated.

Researchers discovered an embedded bifurcation structure within the discretized Brusselator model. This structure, revealed by dynamical orbits, connects to the logistic map and the original ordinary differential equations (ODEs).

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Area of Science:

  • Dynamical Systems
  • Mathematical Chemistry
  • Computational Mathematics

Background:

  • The Brusselator model is a well-known system of two-dimensional ordinary differential equations (ODEs) used to study chemical reactions.
  • Discretization of continuous systems like the Brusselator can lead to complex dynamic behaviors.
  • Understanding the underlying structures within discretized models is crucial for predicting system behavior.

Purpose of the Study:

  • To investigate the dynamic structure of discretized vector fields derived from the Brusselator model.
  • To identify and analyze any embedded mathematical structures within these discretized fields.
  • To establish a connection between the behavior of the discretized system and known mathematical maps.

Main Methods:

  • Discretization of the two-dimensional Brusselator ordinary differential equations (ODEs).
  • Analysis of the resulting discrete dynamical system's vector fields.
  • Examination of dynamical orbits and their convergence properties.
  • Mathematical analysis to elucidate the observed structures.

Main Results:

  • A bifurcation structure, characteristic of the logistic map, was found to be embedded within the discretized Brusselator vector fields.
  • Dynamical orbits within the discretized system were observed to converge to a fixed point, revealing the embedded structure.
  • A mathematical framework was developed to explain the emergence of this structure.

Conclusions:

  • The study reveals a hidden connection between the Brusselator model and the logistic map through discretization.
  • The findings demonstrate that complex mathematical structures can emerge from the discretization of chemical reaction models.
  • This research provides a deeper understanding of the dynamics of discretized ODEs and their relationship to fundamental mathematical concepts.