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Consecutive Reactions

Consecutive reactions involve a sequence where the product of a preceding reaction becomes the reactant for the subsequent one. In a simple scheme, A transforms into B, which further reacts to form C, with rate constants k1 and k2, respectively. This concept is evident in the radioactive decay series. Assuming an initial state with only A present, the conservation of matter leads to three coupled differential equations, determining the concentrations of A, B, and C over time.The rate of change...
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Updated: May 27, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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Amplitude equations for reaction-diffusion systems with cross diffusion.

Evgeny P Zemskov1, Vladimir K Vanag, Irving R Epstein

  • 1Department of Chemistry, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 9, 2011
PubMed
Summary
This summary is machine-generated.

This study develops amplitude equations for reaction-diffusion systems, revealing how cross-diffusion impacts instabilities like Hopf and Turing. The findings highlight opposite effects in the Oregonator and Brusselator models due to their distinct structures.

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Area of Science:

  • Chemical kinetics
  • Nonlinear dynamics
  • Mathematical modeling

Background:

  • Reaction-diffusion systems are fundamental to pattern formation.
  • Cross-diffusion introduces complex dynamics not captured by standard models.
  • Understanding instabilities like Hopf and Turing is crucial for predicting system behavior.

Purpose of the Study:

  • To develop general amplitude equations for two-variable reaction-diffusion systems with cross-diffusion.
  • To analyze the impact of cross-diffusion on Hopf and Turing instabilities.
  • To investigate the contrasting effects of cross-diffusion in the Oregonator and Brusselator models.

Main Methods:

  • Taylor series expansion
  • Multiscaling analysis
  • Expansion in powers of a small parameter
  • Application to Oregonator and Brusselator models

Main Results:

  • General amplitude equations derived for systems with cross-diffusion.
  • Inhibitor and activator cross-diffusion exhibit opposing effects in Oregonator and Brusselator models.
  • Identified regions for supercritical/subcritical bifurcations and wave phenomena (including turbulent waves) under Hopf instability.

Conclusions:

  • Cross-diffusion significantly alters the dynamics of reaction-diffusion systems.
  • Model-specific differences in community matrices lead to contrasting cross-diffusion effects.
  • The developed framework aids in predicting complex spatio-temporal patterns and bifurcations.