Predmet:Nove fizičkohemijske metode ispitivanja dinamike složenih reakcionih sistema
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1 Predmet:Nove fizičkohemijske metode Tema: Specifične metodem ispitivanja dinamike složenih reakcionih sistema Predavači: Ljiljana Kolar-Ani Anić ć i Željko Čupić Sadržaj I čas 1. Složeni reakcioni sistemi 2. Dinamičke strukture složenih reakcionih sistema i samoorganizacija neravnotežnih sistema 3. Metode ispitivanja dinamike složenih reakcionih sistema Sadržaj II časa Analiza vremenskih serija Analiza atraktora Bifurkaciona analiza Interaktivne metode analize
2 Sadržaj II časa Analiza vremenskih serija Analiza atraktora Bifurkaciona analiza Interaktivne metode analize
3
4 Spektri snage metoda za analizu oscilatornih procesa Spektar snage je kvadrat modula furijeove transformacije signala. Prilikom udvajanja perioda dolazi do pojave subharmonika u spektru snage. Slika 1: Period-2 oscilacije kada je j 0 = min -1 ; (a) vremenska serija, i (c) spektar snage. Slika 2: Haos Kada je j 0 = min -1 ; (a) vremenska serija, i (c) spektar snage. Guy Schmitz, Ljiljana Kolar-Anić, Slobodan Anić, Tomislav Grozdić, Vladana Vukojević J. Phys. Chem. A, 110 (2006) Slika 1 Slika 2
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6 Vejvletna transformacija za signale sa vremenski zavisnom komponentom Xue-Guang Shao, Alexander Kai-Man Leung, and Foo-Tim Chau, Wavelet: A New Trend in Chemistry, Acc. Chem. Res. 2003, 36, Signal se transformiše iz vremenske dimenzije u vremensko frekventnu, tako da se svakom vremenskom intervalu pripisuju udeli procesa na datoj frekvenciji skali. Za razliku od Furijeove analize ovde su talasni paketi vejvleti- lokalizovani u vremenu.
7 Poređenje Furijeove i vejvletne transformacije David Labat, Recent advances in wavelet analyses: Part 1. A review of concepts, Journal of Hydrology 314 (2005)
8 Sadržaj II časa Analiza vremenskih serija Analiza atraktora Bifurkaciona analiza Interaktivne metode analize
9 Fazni prostor i atraktor Periodične promene u vremenu su posledica kretanja dinamičkog sistema po zatvorenoj putanji u faznom prostoru. Haotičnoj dinamici odgovara otvorena putanja po ograničenom delu faznog prostora Atraktor je trajektorija dinamičkog sistema u faznom prostoru posle prolaska tranzijentnog perioda H2O I I x 10-8
10 x 10-8 Slučaj 1 (stabilnost): Atraktor je STACIONARNO STANJE 1.4 (I-) time x H2O x I I x 10-8
11 1.4 x Slučaj 2: Atraktor je GRANIČNI KRUG (I-) time H2O H2O x I I I I x 10-8
12 1.4 x 10-7 (I-) Slučaj 3 (haos): Atraktor je ČUDNI ATRAKTOR (fraktal otvorena linija) 8 x time (I-) time H2O x I I H2O I I x 10-8
13 Rekonstrukcija atraktora Rekonstrukcija atraktora Takens je dokazao da umesto 2n+1 generičkih signala, za prekrivanje n- dimenzionalnog atraktora može biti dovoljna konstrukcija sa vremenskim kašnjenjem izvedena iz samo jednog generičkog signala.
14 Takensov postupak t X X, Y, Z X (t), X (t+dt), X (t+2dt)
15 L. Moniz, J.M. Nichols, C.J. Nichols, M. Seaver, S.T. Trickey, M.D. Todd, L.M. Pecora, L.N. Virgin A multivariate, attractor-based approach to structural health monitoring Journal of Sound and Vibration Volume 283, Issues 1?
16 Matrica trajektorije Razlaganje po singularnim vrednostima postupak sličan razlaganju na svojstvene vrednosti, ali primenljiv i na nekvadratne (pravougaone) matrice. Singular value decomposition (SVD) A= Y Y. Y d 2 3. d a. N N 1. N d N 1... A 11. AN1 V 11. VN S1 0 0 U 11. U1d x... x S d U dxd d1. U dd A. A V. V 1d Nd Nxd 1d Nd Nxd Postupak SVD obezbeđuje dobijanje singularnih vrednosti u formi opadajućeg intenziteta. U idealnom slučaju samo nekoliko singularnih vektora odgovara singularnim vrednostima koje daju značajan doprinos, dok ostalima odgovaraju nule. Postupak se koristi i za eliminaciju šuma iz signala. T dxd
17 Primeri rekonstruisanih atraktora eksperimentalno snimljenih signala elektrodnog potencijala u oscilatornoj reakciji BL A. Z. Ivanović, Ž. D. Čupić, M. M. Janković Lj. Z. Kolar-Anić and S. R. Anić, The chaotic sequences in the Bray Liebhafsky reaction in an open Reactor, Phys. Chem. Chem. Phys., 2008, 10,
18 Poenkareovi preseci Dimenzionalnost dinamičkog sistema se smanjuje i Kontinualni dinamički sistem se diskretizuje
19 Periodični sistemi imaju diskretan mali broj tačaka u Poenkareovom preseku Haotični sistemi imaju neograničen broj tačaka u Poenkareovom preseku H2O H2O I I x I I x 10-8
20 Periodični sistemi imaju diskretan mali broj tačaka u Poenkareovom preseku Haotični sistemi imaju neograničen broj tačaka u Poenkareovom preseku [ ],10-8 mol x dm (c) [I - ], 10-8 mol x dm [ ], 10-4 mol x dm -3 [ ], 10-4 mol x dm
21 H2O I I x
22 Poenkareovi preseci atraktora BL reakcije u dinamičkom stanju x [I - ], 10-8 mol x dm -3 (c) Time, min [ ] [H 2 O 2 ] Poenkareov presek
23 Poenkareovi preseci atraktora BL reakcije u dinamičkom stanju [I - ],10-8 mol x dm [I - ],10-8 mol x dm Vreme, min x [ ], 10-4 mol x dm -3 [ ] [H 2 O 2 ] Poenkareov presek
24 Iteracione mape Iteracione mape nam daju mogućnost da prikažemo Poenkareov presek u formi diskretizovanog dinamičkog sistema. [ ] x [H 2 O 2 ] [I2], n+1, mol x dm x [I2]n, mol x dm-3 Poenkareov presek Poenkareova iteraciona mapa
25 [ ] / 10-4 M [ ] n [I2 ]n ] / M 10-7[I 8x10-8 6x10-8 4x10-8 2x Time / min Time / min 1.2 [I - ],10-8 mol x dm [ ],10-4 mol x dm -3
26 1.408 x x 10-4 [I2], n+1, mol x dm [I2]n+1, mol x dm [I2]n, mol x dm [I2]n, mol x dm-3 Periodika Haos
27 Udvajanje perioda scenario nastanka haosa Prikaz atraktora u faznom prostoru
28 [ ] n+1, 10-4 mol x dm (a) [ ] n, 10-4 mol x dm -3 Udvajanje perioda scenario nastanka haosa [ ] n+1, 10-4 mol x dm (b) [ ] n, 10-4 mol x dm -3 Prikaz Jednodimenzionih Mapa Poenkareovog preseka atraktora [ ] n+1, 10-4 mol x dm (c) [ ] n, 10-4 mol x dm -3
29 6.00E E-008 [I - ] / M 4.00E E E [ ] / M
30 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
31 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
32 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
33 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
34 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
35 x 10-4 I - =4.8e-8M Velike i Male ^ (n+1) (n)
36 1.41 x 10-4 I - =4.8e-8M Velike i Male (4^1)x (n+1) (n)
37 1.41 x 10-4 I - =4.8e-8M Velike i Male (4^1)x (n+1) (n)
38 1.41 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
39 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
40 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
41 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
42 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
43 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
44 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
45 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
46 1.41 x 10-4 I - =4.8e-8M Velike i Male ^1 3^ (n+1) (n)
47 1.41 x 10-4 I - =4.8e-8M Velike i Male ^1 3^ (n+1) (n)
48 1.41 x 10-4 I - =4.8e-8M Velike i Male ^1 3^ (n+1) (n)
49 1.41 x 10-4 I - =4.8e-8M Velike i Male (4^1 3^1)x (n+1) (n)
50 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
51 1.409 x 10-4 I - =4.8e-8M Velike i Male (n+1) (n)
52 x I - =4.8e-8M Velike i Male (3^1)x (n+1) (n)
53 6 Čudni atraktor [I - ], 10-8 mol x dm [ ],10-4 mol x dm x 10-4 Nestabilna fiksna tačka [I2]n+1, mol x dm [I2]n, mol x dm-3
54 [I - ], 10-8 mol x dm [I - ], 10-8 mol x dm [ ],10-4 mol x dm -3 [ ], 10-4 mol x dm x 10-4 [I2]n+1, mol x dm [I2]n, mol x dm-3
55 Parametrizacija haotičnih dinamičkih sistema Određivanje Ljapunovljevih eksponenata Ljapunovljev eksponent mera brzine razdvajanja dveju inicijalno veoma bliskih tačaka na atraktoru haotičnog dinamičkog sistema B 1old D A 1 B 1 old B DAB A A 1 B 1new D A 1 B 1 new Shematski prikaz procedure određivanja Ljapunovljevih eksponenata iz vremenskih serija Postoje procedure koje Ljapunovljeve eksponente određuju kao svojstvene ili singularne vrednosti operatora dinamike linearizovanog lokalno, u datoj tački na atraktoru. L. Dieci, C. Elia, Mathematics and Computers in Simulation 79 (2008)
56 BELOUSOV-ŽABOTINSKI (BZ) OSCILATOR Deo tipične vremenske serije haotične evolucije tipa 1 1 (a) snimljene Pt-elektrodom spregnutom sa srebrosrebrohloridnom referentnom elektrodom i fazni portret (b) BZ sistema u zatvorenom reaktoru
57 Zavisnost T i i λ L, od temperature za BZ sistem
58 Zavisnost Ljapunovljevog eksponenta od recipročne vrednosti temperature za BZ sistem: 80kJ/mol<E λ <130kJ/mol
59 Spektar Ljapunovljevih eksponenata Osetljivost na početne uslove i na promene vrednosti parametara zavisi od veličine Ljapunovljevih eksponenata. Ako je vrednost bar jednog najvećeg Ljapunovljevog eksponenta pozitivna, u vremenskoj evoluciji se javlja haos.
60 Zavisnost najvećeg Ljapunovljevog eksponenta od vrednosti kontrolnog parametra
61 Sadržaj II časa Analiza vremenskih serija Analiza atraktora Bifurkaciona analiza Interaktivne metode analize
62 Bifurkaciona analiza Bifurkacija je kvalitativna promena dinamike do koje dolazi pri kontinualnoj promeni vrednosti kontrolnog parametra (npr. Brzina protoka kroz reaktor) Bifurkaciona tačka je vrednosti kontrolnog parametra pri kojoj dolazi do bifurkacije dinamičkog sistema Bifurkacioni dijagram je grafik bifurkacionih tačaka u parametarskom prostoru
63 Klasifikacija mehanizma oscilatora na osnovu bifurkacionih dijagrama (SNA) - Polazna ideja J. Phys. Chem. 1989, 93, Use of Bifurcation Diagrams as Fingerprints of Chemical Mechanisms Zoltan Noszticzius, William D. McCormick, and Harry L. Swinney
64 Matematički aparat SNA Stehiometrijska mrežna analiza Clarke, B. L. Adv. Chem. Phys. 1980, 43, 1. Clarke, B. Cell Biophysics 1988, 12, 237. Realizacija Tim Chevalier, Igor Schreiber, and John Ross, Toward a Systematic Determination of Complex Reaction Mechanisms, J. Phys. Chem. 1993,97, Metoda dopušta identifikaciju tipa mehanizma kome odgovara dati bifurkacioni dijagram oscilatorne reakcije
65 BL S BET = m 2 /g S BET = m 2 /g E/mV chaos E/mV chaos E/mV chaos T/ o C T/ o C T/ o C 48.0 T c / 0 C S BET [m 2 /g]
66 Symposium Nonlinear Dynamics Milutin Milanković CL-35. ID-75. TYPES OF BIFURCATION POINTS IN BRAY-LIEBHAFSKY OSCILLATORY REACTION, B. Stanković, Ž. Čupić and Lj. Kolar-Anić
67 Sadržaj II časa Analiza vremenskih serija Analiza atraktora Bifurkaciona analiza Interaktivne metode analize
68 Metoda WeiW Prater dc1 k k k k C dt dc 2 dt dc3 Metoda omogućava određivanje svih konstanti brzina modela mreže reakcija pseudo-prvog reda. Zasniva se na jednostavnoj relaciji između konstanti brzina i svojstvenih vrednosti operatora dinamike. dt za linearne sisteme k k k k C k k k k C Atraktor linearnog sistemaje jedna tačka - ravnoteža k31 k21 k12 k13 X1 X1 k k k k X X k31 k32 k13 k 23 X 3 X 3 J. Wei, C. Prater, The structure and analysis of complex reaction systems, Advances in Catalysis, 13, 1962, 203.
69 Metoda prigušenja oscilacija Metoda prigušenja oscilacija omogućava određivanje konstanti brzina modela mehanizma oscilatorne reakcije. Kao i u metodi Wei Prater određuju se svojstvene vrednosti operatora dinamike, ali u ovom slučaju to je linearizovani operator u ustaljenom stanju Jakobijan. Hynne F, Sørensen PG. Quenching of Chemical Oscillations. (1987) J Phys Chem. 91, Vukojević V, Sørensen PG, Hynne F. Quenching Analysis of the Briggs-Rauscher Reaction. (1993) J Phys Chem. 97, Atraktor je granični krug
70 Upravljanje haosom H2 + O2 Davies,M.L.; Halford-Maw,P.A.; Hill, J.; Tinsley,M.R.; Johnson,B.R.; Scott,S.K.; Kiss,I.Z.; Gáspár,V.: Control of Chaos in Combustion Reactions, J. Phys. Chem. A, 2000, 104,
71 Hvala na pažnji. Apstrakte na jednoj strani slati na adresu:
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