Asymptotic Behaviour of Solutions of Certain Third Order Nonlinear Differential Equations via Phase Portrait Analysis

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1 Applied Mathematics, 6, 7, ISSN Olie: ISSN Prit: Asymptotic Behaviour of Solutios of Certai Third Order Noliear Differetial Equatios via Phase Portrait Aalysis Roselie Ngozi Okereke, Sadik Olaiyi Maliki Departmet of Mathematics, Michael Okpara Uiversity of Agriculture, Umudike, Nigeria How to cite this paper: Okereke, RN ad Maliki, SO (6) Asymptotic Behaviour of Solutios of Certai Third Order Noliear Differetial Equatios via Phase Portrait Aalysis Applied Mathematics, 7, Received: July 7, 6 Accepted: December, 6 Published: December 4, 6 Copyright 6 by authors ad Scietific Research Publishig Ic This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY 4) Ope Access Abstract The global phase portrait describes the qualitative behaviour of the solutio set for all time I geeral, this is as close as we ca get to solvig oliear systems The questio of particular iterest is: For what parameter values does the global phase portrait of a dyamical system chage its qualitative structure? I this paper, we attempt to aswer the above questio specifically for the case of certai third order oliear differetial equatios of the form + a + g ( ) + c = The liear case where g ( ) = b is also cosidered Our phase portrait aalysis shows that uder certai coditios o the coefficiets as well as the fuctio g, we have asymptotic stability of solutios Keywords Phase Portrait, Trajectory, Flow, Homeomorphism, Asymptotic Stability, MathCAD Itroductio A lie coectig the plotted poits i their chroological order shows temporal evolutio more clearly o the graph The complete lie o the graph (ie the sequece of measured values or list of successive iterates plotted o a phase space graph) describes a time path or trajectory [] A trajectory that comes back upo itself to form a closed loop i phase space is called a orbit [] A orbit for a system usually idicates that the dyamical system uder cosideratio is coservative We also ote that each plotted poit alog ay trajectory has evolved directly from the precedig poit As we plot each successive poit i phase space, the plotted poits migrate aroud Orbits ad trajectories therefore reflect the DOI: 436/am67883 December 4, 6

2 movemet or evolutio of the dyamical system Thus, a orbit or trajectory moves aroud i the phase space with time The trajectory is a eat, cocise geometric picture that describes part of the system s history Whe draw o a graph, a trajectory must ot always be smooth; istead, it ca zigzag all over the phase space, mostly for discrete data [3] [4] [5] The phase space plot is a world that shows the trajectory ad its developmet Depedig o various factors, differet trajectories ca evolve for the same system The phase space plot ad such a family of trajectories together are a phase space portrait, phase portrait, or phase diagram A phase space with plotted trajectories ideally shows the complete set of all possible states that a dyamical system ca ever be i The Flow Defied by a Differetial Equatio We et describe the otio of the flow of a system of differetial equatios We begi with the liear system = A, =, () ( ) The solutio to the iitial value problem associated with () is give by At ( ) = t e The set of mappigs e At : may be regarded as describig the motio of poits alog trajectories of () This set of mappigs is called the flow of the liear system () Remark The mappig φ e At t = satisfies the followig basic properties for all ) φ ( ) = ; ) φs( φt( ) ) = φ s + t( ), for all st, ; φ φ = φ φ = for all t 3) t ( t ( )) t ( t ( )) For the oliear system ( ) : = f () we defie the flow φ t ad show that it satisfies the above basic properties Subsequetly we itroduce the otio of maimal iterval of eistece ( ab, ) of the solutio φ ( t, ) of the iitial value problem = f ( ), ( ) =, (3) by I( ) sice the ed poits a ad b of the maimal iterval geerally depeds o Defiitio Let E be a ope subset of f C E For E φ t, be the solutio of the iitial value problem (3) defied o its maimal iterval of eistece I( ) The for t I( ), the set of mappigs φ t defied by ad let ( ), let ( ) 35

3 ( ) φ(, ) is called a flow of the differetial Equatio () vector field f ( ) 3 Remark φt = t (4) φ t is also referred to as the flow of the ) We ca thik of the iitial poit as beig fied ad let I = I( ), the the mappig φ (, ) : I E defies a solutio curve or trajectory of the system () through the poit E Naturally the mappig is idetified simply with its graph i I E ad a trajectory is visualized as a motio alog a curve Γ through the poit E of the phase space as varyig throughout K (Figure (a)) O the other had, if we thik of the poit E, the the flow of the differetial Equatio (), φt : K E ca be viewed as the motio of all poits i the set K (Figure (b)) Figure (a) A trajectory Γ of the system (); (b) The flow of the system () ) If we thik of the differetial Equatio () as describig the motio of a fluid, the a trajectory of () describes the motio of a idividual particle i the fluid while the flow of the differetial Equatio () describes the motio of the etire fluid 3) It ca be show that the basic properties (i)-(iii) of liear flows are also satisfied by oliear flows [6] 4) The followig theorem, provides a method of computig derivatives i coordiates 4 Theorem Give f : is differetiable at, the the partial derivatives fi j, i, j =,,, all eist at ad for all, f ( ) Df ( ) = j (5) j= j φ t 36

4 Thus, if f is a differetiable fuctio, the derivative Df is give by the Jacobia matri 5 Defiitio Df f i = j A equilibrium of the system = f ( ) the Jacobia Df ( ) have o-zero real part 6 The Hartma-Groβma Theorem (6) is called hyperbolicif all eigevalues of The Hartma-Groβma Theorem [7] is aother very importat result i the local qualitative theory of ordiary differetial equatios The theorem shows that ear a hyperbolic equilibrium poit, the oliear system = f (7) has the same qualitative structure as the liear system = A (8) with A = Df ( ) Throughout this sectio we shall assume that the equilibrium poit has bee traslated to the origi 7 Defiitio Two autoomous systems of differetial equatios such as (7) ad (8) are said to be topologically equivalet i a eighborhood of the origi or to have the same qualitative structure ear the origi if there is a homeomorphism Φ mappig a ope set U cotaiig the origi oto a ope set V cotaiig the origi which maps trajectories of (7) i U oto trajectories of (8) i V ad preserves their orietatio by time i the sese that if a trajectory is directed from to i U, the its image is directed from Φ ( ) to Φ ( ) i V If the homeomorphism Φ preserves the parameterizatio by time, the the systems (7) ad (8) are said to be topologically cojugate i a eighborhood of the origi ( ) Figure (a) Phase portrait of = A ; (b) Phase portrait of y = By 37

5 8 Eample 3 Cosider the liear systems = A ad y = By with A=, B = 3 4 Let Φ ( ) = R, where R =, R = The oe ca easily check that B = RAR, ad lettig y =Φ ( ) = R or = R y t = = It the follows that if ( ) At Bt ( ) ( ( )) ( ) e e y RAR y By = e At is a solutio of the first system through, the y t =Φ t = R t = R = R is the solutio of the secod system passig through R I other words Φ maps trajectories of the first system oto trajectories of the secod system ad preservig the parametrizatio, sice At Bt Φ e = e Φ (9) that the mappig ( ) The phase plae portraits of the two systems are show i Figure It clearly shows Φ = R is simply a rotatio through 45 ad thus it is a homeomorphism 9 Theorem (Hartma-Groβma) Let E be a ope subset of cotaiig the origi, suppose f C ( E), ad φ t = Df the flow of the oliear system = f ( ) Let f ( ) = ad the matri A ( ) has o eigevalue with zero real part The there eists a homeomorphism Φ of a ope set U cotaiig the origi oto a ope set V cotaiig the origi such that for each U, there is a ope iterval I cotaiig zero such that for all I ad t I Φ φ e At t = Φ () ( ) ( ) ie Φ maps trajectories of the oliear system = f ( ) = A ear the origi ad preserves the parametrizatio by time 3 Mai Results oto trajectories of I [8] Okereke demostrated very clearly the veracity of the Hartma-Groβma theorem by cosiderig the simulatio of the oliear ad liearized system of ordiary differetial equatios i terms of their phase portrait aalysis Cosider the oliear system; =, = 3 The equilibria of the above system is obtaied by settig = = to get =, 3 = 3, To obtai the liearizatio at the origi, we begi by computig the Jacobia: Df ( ) = 3 Solvig the above equatios we obtai the equilibria as (, ) ad ( ) 38

6 Evaluatig the Jacobia at the first equilibrium gives Df (,) = 3, ad therefore the liearizatio of our system at (, ) is = 3 Sice Tr ( A) 3, A 3 = = < = <, we immediately see that the origi is a saddle for the liearized system Evaluatig the Jacobia at the secod equilibrium gives Df ( 3, ) 6 =, 3, is ad therefore the liearizatio of our system at ( ) Here Tr ( A), A 3 system 6 = = = >, thus the equilibrium poit is a cetre for the liearized I the simulatio which follows we will cosider oly the otrivial equilibrium 3, poit ( ) 3 MathCAD Simulatio a) The give oliear system =, = 3 ca be recast i MathCAD [9] format as follows Solutio matri is give i Figure 3 Y Y D( ty, ) : = Y Y 3 Y Vector of derivatives t : 5 = Iitial value of idepedet variable t : 5 = Termial value of idepedet variable 95 Y : = 46 Vector of iitial values Figure 3 Solutio matri for the system =, = 3 39

7 N : = 5 Number of solutio values i [ t,t ] S : = R kadap t Y, t, t, N, D Solutio matri ( ) t: = S Idepedet variable values : : = S First solutio fuctio values = S Secod solutio fuctio values The solutio profiles are depicted i Figures 4(a)-(c) (a) (b) Figure 4 (a) Trajectory of ( t ) ; (b) Trajectory of ( ) ( 3, ) (c) t ; (c) Phase portrait of system ear, is ow cosidered The liearized system = 6, = ca be recast i MathCAD format b) The phase portrait of the liearized system ear the origi ( ) as follows Solutio matri is give i Figure 5 6 Y D( ty, ) : = 5 Y Vector of derivatives t : 5 = Iitial value of idepedet variable t : 5 = Termial value of idepedet variable 95 Y : = 46 Vector of iitial values N : = 5 Number of solutio values i [ t,t ] S : R Y, t, t, N, D Solutio matri = kadapt ( ) t: = S Idepedet variable values : : = S First solutio fuctio values = S Secod solutio fuctio values 33

8 Figure 5 Solutio matri for the system = 6, = The solutio profiles are depicted i Figures 6(a)-(c) (a) (b) Figure 6 (a) Trajectory of ( t ) of liearized system; (b) Trajectory of ( ) system; (c) Phase portrait of liearized system ear (, ) (c) t of liearized 3 Observatio The phase portraits of the oliear system ear ( ) 3, ad liearized system about the origi, show stability but ot asymptotic stability This is because the graph is a cetre, ad as a result, we coclude that the system is coservative I each case we see that the phase portraits for the oliear ad liearized system are topologically the, respectively same ear the equilibrium poit ( 3, ) ad ( ) 4 Phase Portrait Aalysis for Stability of Third Order ODE I this sectio we cosider a third order liear equatio 33

9 which is equivalet to the system = = 3 = a b c where a, b, c are all positive costats + a + b + c = () 3 3 We study the asymptotic properties of the above system with the help of MathCAD ab c > simulatio The costats a, b, c are chose such that ( ) 4 Simulatio a : = 4 b : = 6 c : = Y D( ty, ) : = Y ay by cy Vector of derivatives t : 4 = Iitial value of idepedet variable t : 5 = Termial value of idepedet variable Y : = 3 Vector of iitial values N : = 5 Number of solutio values i [ t, t ] S : = R kadap t Y, t, t, N, D Solutio matri ( ) t: = S Idepedet variable values : : 3 : = S First solutio fuctio values = S Secod solutio fuctio values 3 = S Third solutio fuctio values The solutio matri for the above system is give i Figure 7, while the solutio profiles are depicted i Figures 8(a)-(d) () Figure 7 Solutio matri for 3 rd order ODE 33

10 Figure 8 (a) Trajectory of ( t ) ; (b) Trajectory of ( t ) ; (c) Trajectory of 3 ( ) 4 The Geeral Noliear Third Order ODE t ; (d) Phase portrait of + a + b + c = We ow cosider the more geeral oliear third order ODE give by + h( ) + µ ( ) + k( ) = p( t) where h, µ, k C(, ), p C( [, ], ) We have the followig theorem 43 Theorem Give that ) k( ) > ; ) h( y) > a > ; aµ y k (3) 3) ( ) ( ( )) ; 4) p( t) d t < The every solutio t ( ) of Equatio (3) satisfies t ( ) t ( ) t ( ) lim =, lim =, ad lim = t t t 333

11 The proof follows that give by Omeike [] h y a, µ y b k Fially, whe ( ) = ( ) = ad ( ) reduces to the liear equatio = c are all costats, Equatio (3) + a + b + c = p ( t) (4) We have the followig result followig immediately from the above theorem 44 Corollary Give that ) a >, b >, c > ; ) ab > c ; 3) p( t) dt < The every solutio t ( ) of Equatio (3) satisfies t ( ) t ( ) t ( ) 45 Remark lim =, lim =, ad lim = (5) t t t ) We ote that () ad () are the well kow Routh-Hurwitz coditios [6] for the asymptotic stability of the followig third-order homogeeous liear differetial equatio + a + b + c = ) For the third order differetial equatio =, the coditios of corollary 44 are clearly satisfied ad from the simulatio (Figures 8(a)-(c)) we ca see the truth i the limit coditios (5) Figure 8(d) depicts a spiral sik i the simulatio, ad this further stresses the asymptotic ature of the solutios 5 Coclusio I this study, we ivestigated the stability aalysis of certai third order liear ad oliear ordiary differetial equatios We employed the method of phase portrait aalysis We showed, usig simulatio that the Hartma-Groβma Theorem is verified, for a secod order liearized system as a eample, approimates the oliear system preservig the topological features I the case of the third order oliear system + a + g ( ) + c =, we stated appropriate theorems guarateeig asymptotic stability of solutios For the liear case where g ( ) = b, our phase portrait aalysis shows that uder certai coditios o the coefficiets as well as the fuctio g, we have asymptotic stability of solutios Refereces [] Maliki, SO ad Nwoba, PO (4) Stability Aalysis of a System of Coupled Harmoic Oscillators Pelagia Research Library Advaces i Applied Sciece Research, 5, 95-3 [] Ogudare, BS (9) Qualitative ad Quatitative Properties of Solutios of Ordiary Differetial Equatios Uiversity of Fort Hare Alice South Africa [3] Ogbu, HM, Okereke, RN ad Aliyu, BY () The Three Equivalet First Order Systems of a Third Order Scalar Equatio ad Its Applicatio o Stability of Solutios of Or- 334

12 diary Differetial Equatios IJAPS, 4 [4] MathSoft, Ic (4) MathCAD 4 User s Guide [5] Omeike, M () New Results o the Asymptotic Behaviour of a Third Order Noliear Differetial Equatio Joural of Differece Equatios ad Applicatios,, [6] Maliki, SO ad Okereke, RN (6) A Note o Differetial Equatio with a Large Parameter Applied Mathematics, 7, [7] Maliki, SO () Aalysis of Numerical ad Eact Solutios of Certai SIR ad SIS Epidemic Models Joural of Mathematical Modellig ad Applicatio,, 5-56 [8] Okereke, RN (6) Lyapuov Stability Aalysis of Certai Noliear Ordiary Differetial Equatios Upublished MSc Thesis, Michael Okpara Uiversity of Agriculture Umudike (MOUAU), Nigeria [9] Cartwright, ML ad Littlewood, JE (947) O Noliear Differetial Equatios of the Secod Order II Aals of Mathematics, 48, [] Perko, L () Differetial Equatios ad Dyamical Systems 3rd Edito, Spriger, Berli Submit or recommed et mauscript to SCIRP ad we will provide best service for you: Acceptig pre-submissio iquiries through , Facebook, LikedI, Twitter, etc A wide selectio of jourals (iclusive of 9 subjects, more tha jourals) Providig 4-hour high-quality service User-friedly olie submissio system Fair ad swift peer-review system Efficiet typesettig ad proofreadig procedure Display of the result of dowloads ad visits, as well as the umber of cited articles Maimum dissemiatio of your research work Submit your mauscript at: Or cotact am@scirporg 335