Invariant grids: method of complexity reduction in reaction networks Andrei Zinovyev Institut Curie, Paris Institut des Hautes Études Scientifiques

2 - European conference on complex systems, Paris, November Stoichiometric equations  s1 A 1 + … +  sn A n   s1 A 1 + … +  sn A n n – number of species, s – number of reactions c1 c2 cncn

3 - European conference on complex systems, Paris, November What is Model Reduction ? 1 Shorten list of species – eliminate some – create “integrated” components 2 Shorten list of reactions – eliminate some – “freeze” fast reactions 3 Decompose motion into fast and slow

4 - European conference on complex systems, Paris, November Approaching steady state

5 - European conference on complex systems, Paris, November Positively Invariant Manifold Steady state fast motion slow motion Why Invariant? once the point on the manifold, the trajectory will stay on it until the equilibrium 

6 - European conference on complex systems, Paris, November Why for do we need invariant manifold? Model reduction: “Macroscopic” system description x  R N – detailed description y  R m – “macroscopic” description. m<<N

7 - European conference on complex systems, Paris, November Why for do we need invariant manifold? Dynamics visualization

8 - European conference on complex systems, Paris, November Other useful non-invariant manifolds Quasy steady-state Fast variables are steady Quasi-equilibrium Manifolds maximizing entropy Intrinsic low-dimensional manifold Decomposition of Jacobian fields

9 - European conference on complex systems, Paris, November Projector P c on (some) manifold induces new (reduced) dynamics J induced dynamics P c J tangent space  = (1-P c )J - invariance defect   xx

10 - European conference on complex systems, Paris, November Quasi-equilibrium manifold is not necessarily invariant entropy S  max macroscopic (reduced) variables

11 - European conference on complex systems, Paris, November Class of dissipative systems c G c eq Lyapunov function

12 - European conference on complex systems, Paris, November Thermodynamic projector J P c J The induced dynamics is dissipative only if

13 - European conference on complex systems, Paris, November Correction of invariance defect C1C1 C3C3 equilibrium invariant manifold initial approximation corrections (1-P c )J = 0 invariance equation Newton iterations

14 - European conference on complex systems, Paris, November Invariant grid EQUILIBRIUM tangent space J invariance defect is corrected for every node independently

15 - European conference on complex systems, Paris, November Invariant grid J TxTx x 0 +ker P C

16 - European conference on complex systems, Paris, November Growing Invariant Flag Phase space

17 - European conference on complex systems, Paris, November Entropic scalar product equilibrium natural parameter entropy

18 - European conference on complex systems, Paris, November Hydrogen burning model reaction 1: H 2  2H 2: O 2  2O 3: H 2 O  H + OH 4: H 2 + O  H + OH 5: O 2 + H  O + OH 6: H 2 + O  H 2 O Conservation laws: 2c H2 +2c H2O +c H +c OH = b H 2c O2 +c H2O +c O +c OH = b O H 2 H O 2 H2OH2O O OH

19 - European conference on complex systems, Paris, November One-dimensional dynamics equilibrium

20 - European conference on complex systems, Paris, November Separation of times is the eigen value of symmetrised matrix

21 - European conference on complex systems, Paris, November Two-dimensional dynamics

22 - European conference on complex systems, Paris, November Visualizing functions: concentration of H “Fast” coordinate

23 - European conference on complex systems, Paris, November Visualizing functions: concentration of H 2 “Slow” coordinate

24 - European conference on complex systems, Paris, November Visualizing functions: concentration of OH

25 - European conference on complex systems, Paris, November Visualizing functions: Entropy and entropy production Entropy Entropy production

26 - European conference on complex systems, Paris, November Visualizing functions: Separation of relaxation times 2 / 1 3 / 2

27 - European conference on complex systems, Paris, November Open system as “closed system in a flow” closed system flow Entropy does not increase everywhere Non-uniqueness of stationary states, auto-oscillations, etc. inertial manifold often exists

28 - European conference on complex systems, Paris, November Zero-order approximation Construct the invariant manifold  for 

29 - European conference on complex systems, Paris, November First-order approximation New invariance equation: “Fast” and “slow” flow:  ’’

30 - European conference on complex systems, Paris, November Conclusions Invariant grids: constructive method for chemical kinetics class of dissipative systems extension to open systems Use of thermodynamics: metrics in the phase space unique thermodynamic projector Possibility to visualize and explore system dynamics globally

31 - European conference on complex systems, Paris, November Papers Gorban A, Karlin I, Zinovyev A. Constructive Methods of Invariant Manifolds for Kinetic Problems Physics Reports 396, pp Gorban A, Karlin I, Zinovyev A. Invariant Grids for Reaction Kinetics Physica A, V.333, pp

32 - European conference on complex systems, Paris, November People Doctor Iliya Karlin ETH, Zurich Professor Alexander Gorban University of Leicester, UK