MATHEMATICAL MODELING AND THE STABILITY STUDY OF SOME CHEMICAL PHENOMENA

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1 HENRI COND IR FORCE CDEM ROMNI INTERNTIONL CONFERENCE of SCIENTIFIC PPER FSES rasov -6 Ma GENERL M.R. STEFNIK RMED FORCES CDEM SLOVK REPULIC MTHEMTICL MODELING ND THE STILIT STUD OF SOME CHEMICL PHENOMEN Olivia FLORE Monica.P. PURCRU Facult of Mathematics and Computer Sciences Transilvania Universit of raşov Romania bstract: In this paper is studied a chemical phenomenon an eample of an autocataltic reaction. Usin the stabilit in first approimation and the theor of bifurcations is studied the stabilit the autocataltic reaction. The mathematical modellin was made usin the Maple software. Mathematics Subject Classification : D85 Kewords: dnamical sstems asmptotic stabilit Hopf ifurcation. INTRODUCTION Reaction diffusion sstems are mathematical models which eplain how the concentration of one or more substances distributed in space chanes under the influence of two processes: local chemical reactions in which the substances are transformed into each other and diffusion which causes the substances to spread out over a surface in space. This description implies that reaction diffusion sstems are naturall applied in chemistr. However the sstem can also describe dnamical processes of non-chemical nature. Eamples are found in biolo eolo and phsics and ecolo. The nonlinear reactiondiffusion sstems are the eneral form: a b f l t a b f l t The russelator is a theoretical model for a tpe of autocataltic reaction. The russelator model was proposed b Ila Prioine and his collaborators at the Free Universit of russels. The russelator is oriinall a sstem of two ordinar differential equations as the reaction rate equations for an autocataltic oscillatin chemical reaction []. In man autocataltic sstems comple dnamics are seen includin multiple stead states periodic orbits and bifurcations. The elousov - Zhabotinsk reaction [] is a eneric chemical reaction in which the concentrations of the reactants ehibit somewhat oscillatin behaviour. To obtain the russelator model in sstems we denote b: a b a b f l where and are positive constants. In particular the russelator model describes the case in which the chemical reactions follow the scheme:

2 E D where D E and are chemical compounds. Let t and t be the concentrations of and and assume that the concentrations of the input compounds and are held constant durin the reaction process.. THE DNMICL SSTEM The dnamical sstem which models these processes is: We are proposin to stud the stabilit and the eistence of limit ccles for this dnamical sstem makin a discussion about the real positive parameters and. The main aim is to evaluate what are the values that lead us to obtain an attractor solution. For this it is a must to find the equilibrium point of the sstem b computin the followin sstem: We ll find that the equilibrium point is: P P. We are interestin about the behaviour of the null solution. For the stabilit stud of the solution we have to make the translation to arrive in the oriin so:. The new form of the sstem is:. ecause is the solution of sstem is obtained the followin sstem: 5 Which has the equilibrium point in oriin. The stabilit stud is made usin the method in the first approimation. So we have: f f J The Jacobi matri in the equilibrium point O is J H. This is equivalent with the linear homoeneous sstem: 6 The characteristicall polnomial is: det I H P We ll stud the solutions stabilit takin into account the tpe of the characteristicall polnomial s solutions. ecause the product of these two roots is which is alwas a positive number the stud is made for the discriminant and the trace of the matri H : [ ][ ] H Tr S Δ The characteristicall equation roots are: [ ][ ] ±. STILIT NLISS

3 HENRI COND IR FORCE CDEM ROMNI INTERNTIONL CONFERENCE of SCIENTIFIC PPER FSES rasov -6 Ma GENERL M.R. STEFNIK RMED FORCES CDEM SLOVK REPULIC We consider the followin cases: Case. If < < implies that: Δ >. In this case the roots are real TrF neative and different: R. Results that the equilibrium point is an attractive node non deenerate the sstem is asmptoticall stable. The phase portrait is: Case. If implies that: Δ. TrF > In this case the roots are real positive and equal: R >. Results that the equilibrium point is rejector node non deenerate after the line the sstem is unstable. The phase portrait is: Case. If > implies that: Δ >. TrF > In this case the roots are real positive and different: R >. Results that the equilibrium point is a rejector node non deenerate the sstem is unstable. The phase portrait is: Case. If implies that: Δ. TrF In this case the roots are real neative and equal: R. Results that the equilibrium point is an attractor node non deenerate after the line the sstem is asmptoticall stable. The phase portrait is:

4 Case 5. If < < implies that Δ. In this case the roots are comple TrF > with the imainar part positive C \ R Re >. Results that the equilibrium point is a rejector focus. The phase portrait is: Case 6. If < < implies that Δ. In this case the roots are comple TrF with the imainar part neative C \ R Re. Results that the equilibrium point is an attractive focus. The phase portrait is: Case 7. If implies that Δ. In TrF this case the roots are comple with the imainar part null C \ R Re. We have: μ iω where μ and > dμ ω and. Results d that the Hopf theorem s conditions [5] are fulfilled for these values of parameter. The new matrices form of the sstem 5 is: With h For the eienvalue i we have the i vector: q q q q q and for the eienvalue i we have the vector: i p p p p p where q p are random numbers. choice for the vector q is i q. Takin into account that q p from the relation: q p q p q p it is obtained p i. The vector p has the new form: i p i i. We introduce a new variable usin the diffeomorphic transformation: q q. In this case it is obtained: i i. Compute the new function in the new variables:

5 HENRI COND IR FORCE CDEM ROMNI INTERNTIONL CONFERENCE of SCIENTIFIC PPER FSES rasov -6 Ma GENERL M.R. STEFNIK RMED FORCES CDEM SLOVK REPULIC h p O For we evaluate the epression: C The first coefficient of Liapunov for is a real number and it is definin b Re C L α. ω Usin the affirmations above if sn L sn Re i ω results that in the vicinit of the sstem admits a stable limit ccle that means a supercritical bifurcation; sn L > results that in the vicinit of the sstem admits an unstable limit ccle that means a subcritical bifurcation. ecause L the sstem admits an unstable limit ccle in the vicinit of. The phase portrait is: The trajector is a rotatin ellipse. Net we ll determine the equation of the trajector correspondin to the sstem described above. Makin the ratio between and from the linear sstem we obtain the differential total eact equation: [ ] d d interation it is obtained the equation of trajector: C REFERENCES. Neamţu M. The stud of Differential Sstems usin eometrical structures PhD Thesis. Vernic C. naltical methods for the stud of biodnamical sstems Ed. uusta Timisoara p ISN unchen ou Global Dnamics Of The russelator Equations Dnamics Of Pde Vol. No I.R. Epstein Comple Dnamical ehavior In Simple Chemical Sstems J. Phs. Chemistr O. Florea C. Ida Practical pplications of dnamical sstems in enineerin Ed. LuLibris rasov