Invariant grids: method of complexity reduction in reaction networksAndrei ZinovyevInstitut Curie, ParisInstitut des Hautes tudes Scientifiques

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Stoichiometric equationsas1A1 + + asnAn bs1A1 + + bsnAnn number of species, s number of reactionsc1c2cn

European conference on complex systems, Paris, November 2005

What is Model Reduction?1 Shorten list of species eliminate some create integrated components2 Shorten list of reactions eliminate some freeze fast reactions3 Decompose motion into fast and slow

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Approaching steady state

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Positively Invariant ManifoldSteady statefast motionslow motionWhy Invariant? once the point on the manifold, the trajectory will stay on it until the equilibriumW

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Why for do we need invariant manifold?Model reduction:Macroscopic system descriptionxRN detailed description yRm macroscopic description. m

Why for do we need invariant manifold?Dynamics visualization

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Other useful non-invariant manifoldsQuasy steady-stateFast variables are steady

Quasi-equilibriumManifolds maximizing entropy

Intrinsic low-dimensional manifoldDecomposition of Jacobian fields

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Projector Pc on (some) manifoldinduces new (reduced) dynamicsJinduced dynamicsPc Jtangent space = (1-Pc)J - invariance defectWTxW

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Quasi-equilibrium manifoldis not necessarily invariantentropy Smaxmacroscopic (reduced) variables

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Class of dissipative systemscGceqLyapunov function

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Thermodynamic projectorJPc JThe induced dynamics is dissipative only if

European conference on complex systems, Paris, November 2005

Correction of invariance defect0.20.40.60.81.00.050.100.150.20C1C3equilibriuminvariant manifoldinitial approximationcorrections(1-Pc)J = 0invariance equationNewton iterations

European conference on complex systems, Paris, November 2005

Invariant gridEQUILIBRIUMtangent spacetangent spaceJinvariance defect is corrected for every node independently

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Invariant grid

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Growing Invariant FlagPhase space

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Entropic scalar product012-1-2equilibriumnatural parameterentropy

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Hydrogen burning model reaction1: H2 2H 2: O2 2O 3: H2O H + OH 4: H2 + O H + OH5: O2 + H O + OH6: H2 + O H2O Conservation laws: 2cH2 +2cH2O+cH+cOH = bH2cO2+cH2O+cO+cOH = bO

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One-dimensional dynamicsequilibrium

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Separation of timesl is the eigen valueof symmetrised

matrix

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Two-dimensional dynamics

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Visualizing functions:concentration of HFast coordinate

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Visualizing functions:concentration of H2Slow coordinate

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Visualizing functions:concentration of OH

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Visualizing functions:Entropy and entropy productionEntropyEntropy production

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Visualizing functions:Separation of relaxation times2/13/2

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Open system as closed system in a flowflow Entropy does not increase everywhere

Non-uniqueness of stationary states, auto-oscillations, etc.

inertial manifold often exists

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Zero-order approximationConstruct the invariant manifold W for W

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First-order approximationNew invariance equation:Fast and slow flow:W

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ConclusionsInvariant grids: constructive method for chemical kineticsclass of dissipative systemsextension to open systems

Use of thermodynamics: metrics in the phase spaceunique thermodynamic projector

Possibility to visualize and explore system dynamics globally

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PapersGorban A, Karlin I, Zinovyev A.Constructive Methods of Invariant Manifolds for Kinetic Problems 2004. Physics Reports 396, pp.197-403.

Gorban A, Karlin I, Zinovyev A.Invariant Grids for Reaction Kinetics2004. Physica A, V.333, pp.106-154

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PeopleDoctor Iliya Karlin ETH, Zurich

Professor Alexander GorbanUniversity of Leicester, UK

European conference on complex systems, Paris, November 2005