Razvoj metoda za ispitivanje stabilnosti neravnotežnih stacionarnih stanja složenih reakcionih sistema

Abstract

U okviru ovog rada urađena je detaljna analiza, prilagođavanje i dopuna različitih metoda za ispitivanje stabilnosti stacionarnih stanja sa ciljem njihove efikasnije primene u istraživanju dinamičkih stanja složenih reakcionih sistema. Klasična analiza stabilnosti, analiza stehiometrijskih mreža, metode numeričke kontinuacije i analiza dinamičkih stanja primenjene su na različite probleme koji se sreću prilikom izvođenja analize stabilnosti, sa ciljem da se pokaže kako se dobijeni rezultati što bolje mogu iskoristiti radi dobijanja korisnih informacije o dinamičkim stanjima i načinu funkcionisanja pomenutih sistema. U zavisnosti od problema koji je obrađivan korišćeni su model autokatalatora, modeli oscilatornih reakcija Belousov- Zhabotinsky i Bray-Liebhafsky, kao i model HPA sistema. Primenom klasične analize stabilnosti, pristupa koji se zasniva na određivanju analitičkih izraza za svojstvene vrednosti nalaženjem nula karakterističnog polinoma, na modelu autokatalatora pokazano je da je primena ovog pristupa ograničena na one reakcione sisteme koji imaju najviše dve intermedijerne vrste. U slučaju analize stehiometrijskih mreža (SNA), obrađen je niz problema koji se javljaju pri primeni ove metode u ispitivanju stabilnosti neravnotežnih stacionarnih stanja modela složenih reakcionih sistema i predložena su rešenja za njihovo prevezilaženje. Za potrebe analize napisani su programi u MATLAB programskom paketu, koji omogućavaju brzo izračunavanje matrice ekstremnih struja E primenom opisanih algoritama, kao i efikasno određivanje negativnih dijagonalnih minora matrice brzine struja V(j). Pored navedenog, objašnjeno je kako izabrati intermedijerne vrste bitne za izvođenje analize stabilnosti, odrediti i pojednostaviti dobijene uslove nestabilnosti. Pokazano je i kako odrediti funkcionalne delove modela odgovorne za nastanak bifurkacija sedlasti-čvor i Andronov-Hopf, što je urađeno na modelima HPA sistema i BL reakcije. Takođe je ispitana i važnost brzina reakcija vss i brzina struja j za...

In this doctoral thesis detailed analysis of the various methods for stability examinations of stationary states was carried out with the main goal to improve their efficiency in examination of the dynamical states of complex reaction systems. Different approaches such as: classical stability analysis, a stoichiometric networks analysis, methods of numerical continuation and analysis of dynamical states were applied to the different problems encountered when performing stability analysis with aim to show how obtained results can be utilized in the most useful way in order to acquire informations about dynamical states and the way analyzed systems functioning. Depending on the problem which was processed different models such as: autocatalator, models of the Belousov-Zhabotinsky and Bray-Liebhafsky oscillating reactions and the model of the hypothalamic-pituitary-adrenal (HPA) axis were analyzed. Classical stability analysis, an approach that is based on the determination of the analytical expression for the eigenvalues through finding the zeros of the characteristic polynomial, was applied to the model of autocatalator and it was demonstrated that the application of this approach is limited to those reaction systems that have at most two intermediate species. In the case of stoichiometric network analysis (SNA) a number of problems that arise when this method is applied in examination of the stability of non-equilibrium steady state of the models of complex reaction systems were analyzed and their solutions were proposed. For the purpose of the analysis, programs which perform fast calculation of the matrix of extreme current E by applying the algorithms described in this thesis and also allow efficient determination of the negative diagonal minors of the matrix V(j) were written in MATLAB program package. In addition, it was explained how to choose the intermediate species essential for performing stability analysis, determine and simplify obtained instability conditions. It was also shown on the models of HPA system and BL reactions how to determine the functional parts of the analyzed...