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Diffusion-induced instability in two coupled reactors: Chaos

Diffusion-induced instability in two coupled reactors: Chaos. case of Degn -Harrison Model. Diffusion-induced instability. Normal idea: Diffusion is normally thought as acting to equalize concentration differences in space. “The Chemical Basis of Morphogenesis”

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Diffusion-induced instability in two coupled reactors: Chaos

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  1. Diffusion-induced instability in two coupled reactors: Chaos case of Degn-Harrison Model

  2. Diffusion-induced instability • Normal idea: Diffusion is normally thought as acting to equalize concentration differences in space. • “The Chemical Basis of Morphogenesis” • - By A. M. Turing, in Aug.1952 • In this paper, Turing showed diffusion can have the opposite effect: When diffusion is coupled to suitable chemical reactions, it can destabilize a stable homogeneous steady state.

  3. Homogeneous reaction and steady state • A two-variable homogeneous reaction • The system has a single stable steady state

  4. Coupled reactor system • Two reactors have the same unique stable steady state • The same chemical reactions occur with the same parameters in the two reactors • The reactors are coupled through a permeable wall (membrane)

  5. Coupled reactor system • Couple the system with an identical system ci- coupling strength Dx ,Dy – diffusion coefficients

  6. The Degn-Harrison Model • “Theory of Oscillations of Respiration rate in Continuous Culture of Klebsiella aerogenes” -by H. Degn & D. E. F. Harrison, in Sep. 1969 • It describes the respiratory behavior of a Klebsiella Aerogenes bacterial culture • The model consists of three steps:

  7. The Degn-Harrison Model • equations for the temporal evolution of the system • The steady-state solution of the equation

  8. Degn-Harrison model of coupled reaction Where we choose a = 8.951; b = 11.0; q = 0.5; Dx = 10-5; Dy = 10-3

  9. Dynamics of the coupled system • Varying the coupling parameter c, we observe the stable state of x1:

  10. Fractional order extension for Degn-Harrison Model • the fractional case of the system can be described as follow Where

  11. Questions • Does chaos exist in this fractional-order system? • If it does, does chaotic behavior exist for all 0<α<1, or only for some α in a certain range?

  12. α=0.98

  13. α=0.95 Is it Chaos?

  14. Conclusion • Chaos does exit in the fractional order case of Degn-Harrison Model. • It only exists in a very small range of α where it is close to 1. Future work • Confirm the result and conclusion. • Understand the different dynamics of integer order and fractional order Degn-Harrison Model.

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