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Critical manifold of an oscillatory reaction model with more than one fast variable

Critical manifold of an oscillatory reaction model with more than one fast variable. Ž. D. Čupić 1 , A. Z. Ivanović 1 , S. R. Anić 1 ,   G.Schmitz 3 , V. M. Markovi ć 2 and Lj. Z. Kolar-Anić 2.

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Critical manifold of an oscillatory reaction model with more than one fast variable

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  1. Critical manifold of an oscillatory reaction model with more than one fast variable Ž. D. Čupić1, A. Z. Ivanović1, S. R. Anić1,   G.Schmitz3, V. M. Marković2and Lj. Z. Kolar-Anić2 1Institute of Chemistry, Technology and Metallurgy, University of Belgrade, Department of Catalysis and Chemical Engineering, Njegoševa 12, Belgrade, Serbia (zcupic@nanosys.ihtm.bg.ac.rs) 2Faculty of Physical Chemistry, University of Belgrade, Studentski trg 12-16, Belgrade, Serbia 3Facultédes Sciences Appliquées, UniversitéLibre de Bruxelles, CP165/63, Av. Roosevelt 50,1050 Bruxelles, Belgium

  2. Contents • Bray-Liebhafsky (BL) Oscillatory Reaction • Critical manifolds of the BL oscillatory reaction in CSTR • Mechanism of mixed mode oscillations in the BL oscillatory reaction

  3. BRAY-LIEBHAFSKY Oscillatory Reaction Bray, W. C, A Periodic Reaction in Homogeneous Solution and its Relation toCatalysis, J. Am. Chem. Soc. 43 (1921) 1262-1267. Bray, W. C, Liebhafsky, H. A, Reaction involving Hydrogen Peroxide, Iodine and Iodate Ion. I. Introduction, J. Am. Chem.Soc. 53 (1931) 38-44.

  4. Experimental measurements in BL system

  5. Reaction mechanism Schmitz, G, J. Chim. Phys. 84 (1987) 957-965. Kolar-Anić, Lj, Schmitz, G, J. Chem. Soc. Faraday Trans. 88 (1992) 2343-2349. Kolar-Anić, Lj, Mišljenović, Đ, Anić, S, Nicolis, G, React. Kinet. Catal. Lett. 54 (1995) 35-41. Lj. Kolar-Anić, Ž. Čupić, S. Anić, G. Schmitz, J. Chem. Soc. Faraday Trans., 93, (1997),2147-2152. Ž. Čupić and L. Kolar-Anić, Journal of Chemical Physics, 110, (1999), 3951-3954. A. Z. Ivanovic, Z. D. Cupic, M. M. Jankovic, Lj. Z. Kolar-Anic, S. R. Anic, PCCP, 10 (2008), 5848 - 5858.

  6. Reaction dynamics equations in batch reactor

  7. Numerical simulations Schmitz, G, Cinetique de la reaction de Bray, J. Chim. Phys. 84 (1987) 957-965.

  8. Critical manifolds of the BL reaction - Nullclines in batch reactor dynamics FAST FAST FAST FAST SLOW

  9. Schmitz, G, Kolar-Anić, Lj, Anić, S, Čupić, Ž, J. Chem. Educ. 77 (2000) 1502-1505.

  10. Additional steps describing flow in the CSTR H2O2 is intermediary now !!! Second slow variable

  11. CSTR experiment: (KIO30 = 4,74×10-2M, H2SO40 = 4,79×10-2M, H2O20 = 1,55×10-1M) kf = 3,24×10-3 min-1; (a) T = 60,0 C, (b) T = 58,8 C, (c) T = 57,5 C, (d) T = 55,6 C, (e) T = 54,4 C, (f) T = 52,8 C, (g) T = 50,3 C, (h)T = 49,8 C, (i) T = 49,3 C, (j) T = 48,8 C, (k) T = 47,8 C, (l) T = 47,6 C, 179. Vukojević, V, Anić, S, Kolar-Anić, Lj, Phys. Chem. Chem. Phys. 4 (2002) 1276-1283.

  12. CSTR numerical simulations • j0 = 4.70×10-3 min-1 • j0 = 4.90×10-3 min-1 • j0 = 5.00×10-3 min-1 • j0 = 5.10×10-3 min-1 G. Schmitz, Lj. Kolar-Anić, T. Grozdić, V. Vukojević, J. Phys. Chem. A, 110, 10361 (2006).

  13. For the slow-fast systems with two slow variables see: P. Szmolyan, M. Wechselberger,Relaxation Oscillations in R3, J Differential Equations (2004), Vol. 200, 69-104. For multidimensional dynamical systems with mixed mode oscillations: M. Wechselberger, A propos de canards (apropos canards), to appear in Transactions of the American Mathematical Society (2011). Available at: www.maths.usyd.edu.au/u/wm/

  14. Critical manifolds of the BL reaction in CSTR FAST FAST FAST FAST SLOW SLOW

  15. Critical manifold of the BL reaction in CSTR

  16. Simple relaxation oscillation on critical manifold

  17. Critical manifolds of the BL reaction in CSTR Reduced subsystem

  18. Critical manifolds of the BL reaction in CSTR Layer subsystem

  19. Mixed mode oscillations in slow-fast systems with two slow variable are usually connected with folded node singularity and generated canard solutions

  20. Canards in R3 P. SzmolyanandM. Wechselberger, JournalofDifferentialEquations 177 (2001)419–453

  21. Desingularized form • (n-2) zero eigenvalues • 2 eigenvalues with non-zero real part • λ1/2 Complex conjugate and Re(λ1/2)<>0 – Folded focus • Im(λ1/2)=0 and λ1 λ1<0 – Folded saddle • Im(λ1/2)=0 and λ1 λ1>0 – Folded node

  22. Iodine nullcline

  23. Hydrogen peroxide nullcline

  24. For the singular Andronov Hopf bifurcation see also: M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. Osinga, M. Wechselberger, Mixed-mode oscillatons with multiple time-scales,to appear in SIAM Review (2011). Available at http://hdl.handle.net/1983/1594

  25. Tourbillion arise when fast subsystem goes through Andronov Hopf bifurcation For the tourbillion see again: M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. Osinga, M. Wechselberger, Mixed-mode oscillatons with multiple time-scales,to appear in SIAM Review (2011). Available at http://hdl.handle.net/1983/1594

  26. Thank you very much for your attention

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