Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 Dynamical Sysems, Phase Plan Analysis- Linearizaion Mehod Alfred DACI Deparmen of Mahemaics, Mahemaical Physical Engineering Faculy, Polyechnic Universiy of Tirana Tirana, Albania alfreddaci@gmail.com Absrac Linear sysems of differenial equaions are of ineres because hey play a crucial role in he classificaion of he fixed poins of nonlinear sysems. Mos problems which arise in he real world are nonlinear, in mos cases nonlinear sysems canno be solved. One mehod o find approximae soluion for he nonlinear sysems is linearizaion mehod. Since mos models reduce down o wo or more such equaions, since only wo variables can easily be drawn, we concenrae more on a sysem of wo equaions. The objecive of his paper is o briefly describe he fundamenal definiions of he differenial equaions sysems abou fixed poins, sabiliy, classificaion of fixed poins, phase porrai plane, as well as some applicaions of hose sysems o populaion dynamics. Phase plane analysis is one of he mos imporan echniques for sudying he behaviour of nonlinear sysems, since here is usually no analyical soluion for a nonlinear sysem. Anoher approach shown in his paper is ha of he geomeric one which leads o he qualiaive undersing of he behaviour, insead of deailed quaniaive informaion. Consequenly, he analysis of he nonlinear differenial equaions is much undersable han before. Keywords dynamical sysems, fixed poins, sabiliy, linearizaion, phase plane I. INTRODUCTION A lo of phenomena of differen fields of science echnology are modeled mahemaically by linear nonlinear differenial equaions. I is well-known ha second-order linear differenial equaions can be solved analyically mainly hose of consan coefficien, while he nonlinear differenial equaions can be solved analyically only in rare cases. For hese sysems we will see he linearizaion mehod. The aim of his paper is o learn how o inerpre qualiaively he soluions of he equaions of differenial sysems, wihou passing hrough he analyic process of finding he soluions. This is achieved by combing he analyic mehod he geomeric inuiion. The objec of his sudy is dynamical sysems. A dynamical sysem is any mahemaical model which describes he sae of a sysem in ime. For example, mahemaical models which describe he oscillaion of he mahemaical pendulum, he flow of he waer in he ube, he number of populaion in a meropolis, ec are dynamical sysems. In his paper we will rea wo linear nonlinear dimensional sysems. Things have changed dramaically during he las hree decades. We can easily find compuers everywhere, a lo of sofware packes are a our disposal. They can be used o approximae he soluions of he differenial equaions see he resuls graphically. As a resul, he analysis of he nonlinear differenial equaions is airily undersood. II. GENERAL KNOWLEDGE Le us invesigae he dynamical sysem in he plane, x f ( x, y) y g( x, y) We know he soluion of he sysem (1), in he ime inerval T, is any pair of he funcions x( ), y( ) for which he sysem equaions (1) are ransformed ino ideniies T. [1], [] The plo of any soluion x( ), y( ) in he sysem Oxy is called he rajecory of he sysem (1). A any poin ( x, y) of he area, where he funcions f are deermined, he sysem (1) defines a direcion, se by he vecor x ', y ' f ( x, y), g( x, y). g Thus, in he area, he sysem (1) defines a direcion field (vecors). (1) www.jmes.org JMESTN4351755 5495
So, a rajecory of he sysem (1) is any line a w, which has he qualiy: a any poin of is, whose direcion of he angen fis he direcion field. Plane Fixed Poins is he phase space of he sysem (1). Definiion. The poin is called a fixed poin (equilibrium poin) of he sysem (1) if f ( x, y ) g( x, y ) ( x, y ) Any fixed poin ( x, y ) is a rajecory. If he phase poin is found in a momen in he fixed poin ( x, y ) ( x, y) hen i is lef all he ime a ha poin.[1] III. LINEAR SYSTEMS The linear dynamical sysem in he plane is modelled by he linear differenial sysem [1], [4],[5] x ax by () y cx dy Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 While for he funcion x () we disinguish some cases. When, he funcion x () decays exponenially, so all he rajecories x e, y e approach he origin of he coordinaes, move away indefiniely when When., he funcion exponenially ending o he sign of when y. ) when Thus, all he rajecories x e, y e x () changes (depending on, end o zero approach he veer away origin of he coordinaes when endlessly when We should now show wheher hese rajecories are concave or convex. For his we only sudy he sign of he second derivaive:. d y y x y x ( ) ( ) ( ) ( ) 3 x() ( y e )( x e ) ( y e )( x e ) y (1 ) e 3 xe xe Noice ha where a, b,, c d are real parameers. Example. I is given he sysem x ' x (3) y y Plo he phase porrai when moves from o. Soluion For, he sysem () has only one fixed poin, which is he origin of he coordinaes O (, ), while for, any poin of he Ox -axis s a fixed poin. The differenial equaions of he sysem are independen from each oher, hus each equaion can be solved separaely. From heir soluion we have x() c e y() c e, 1 The pair ( ), ( ), x y x e y e is he rajecory of he sysem (3) which passes hrough he arbirary poin x y a he iniial poin. The phase, porrais for differen values of he parameer are shown in Figure 1. The funcion y () decays exponenially o zero when, ends o when. d y sgn sgn( 1) y which means d y d y (1) for y we have:, for 1; :, for 1 ;, for. () for y we have:, for 1; :, for 1 ;, for. This is he way he phase porrai is explained in Figure 1, in he cases (a), (c) (e). In he case, he rajecories x e, y e have he form x, y e which means hey are perpendicular half lines x x (Figure 1/d), where x. In he case 1, he rajecories x e, y e have he form x e, y e. For www.jmes.org JMESTN4351755 5496
x y shows x e y kx, he rajecory is, ye shows half Oy, y e Oy, where. For x - axis, for which for y, he rajecories show he line of he form k y x (Figure 1/b). [1],[5] Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 phase plane, ha is why his poin is called globally aracing. In he case (d), i.e in he case when, here is an enire line of nonisolaed fixed poins in he Ox - axis. Every rajecory approaches hese fixed poins along verical lines. In he case (e), mos rajecories veer away endlessly from ; here is an excepion O(; ) only for he rajecories which sar on he poins of he Oy -axis, so he fixed poin is unsable. In he case (e), he fixed poin saddle poin he manifold. Ox O(; ) O(; ) is called a -axis is called an unsable IV. CLASSIFICATION OF THE TRAJECTORIES OF THE LINEAR SYSTEMS Le us invesigae he linear dynamical sysem x ' ax by (3) y cx dy Fig.1 Commens explanaions In he case(a), when, he rajecories approach he origin O (; ), angenially wih he Ox - axis. On he oher side, if we look backwards, which means,he rajecories, moving away endlessly from O (; ), end o become parallel o he Ox -axis. In he case (c), he behaviour of he rajecories is he same wih ha of he case of (a), wih he change ha he place of he Ox -axis in his behaviour is subsiued by he Oy -axis. In he cases(a) (c)he fixed poin O(; ) is called a sable node. In he case(b)he fixed poin O(; ) is called a symmerical node or sar. In he cases (a), (b) (c), he fixed poin O (; ) is called an aracing poin or a sucion poin. All he rajecories ha come ou of he poin around he origin O (; ),approach i when. In fac, he poin O(; ) aracs all he rajecories of he The origin O(; ) is a fixed poin of (3), whaever he coefficiens are a, b, c, d. Bu when a b c d, apar from he poin O, he sysem (3) has a se of infinie oher fixed poins. Le us see how he sysem (1) is solved using he marix calculus. Le us wrie (3) in he form of differenial marix equaion X ' AX (3 ) where x X y, x a b X y A c d. The soluion x( ), y( ) can be now wrien in he form x () of a vecor or column marix X() y () he soluions of he equaion (3 ) in he form. We search X () e V, (4) v1 where is a real or complex consan, V v a nonzero vecor (column marix), which does no depend on he ime. We can now find V. For his we subsiue X () e V in (3 ) : www.jmes.org JMESTN4351755 5497
e V Ae V e V Ae V V AV ( A I) V (5) where he marix is a uni marix. From he algebra course we know ha he values of for which, he equaion (5) has he soluionv O are I called eigenvalues, while he responsive soluions V are called eigenvecors. In exped form, he equaion (5) shows he ( a ) v bv 1 1 homogeneous wih wo unknowns. cv ( d ) v 1 From he algebra course we know ha he sysem (5) has a soluion differen from he zero one only when is deerminan is equal o zero: a c b d based on which we have ( a d) ( ad bc) (6) i is called he characerisic equaion of he A marix. We wrie he equaion (6) shorly in he form (6 ) where a d (he race of he A marix), ad bc (he deerminan of A marix).[1] The eigenvalues are he roos of he equaion (6 ): 1 4 4 This way, afer deermining he eigenvalues he eigenvecors, we find he rajecories of he sysem (3), as well as we sudy heir behaviour around he fixed poins. I is obviously clear ha he behaviour of he rajecories is defined by he numbers ; consequenly even he numbers. 1 The rules of reading he behaviour around he origin If he values are such ha he poin (, ) is found in: Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 (1)he open area resrained by he parabola 4 O a unsable node; -axis (he firs quadran), hen () he open area, resrained by he parabola he O -axis (he second quadran), hen O(; ) is 4 O(; ) is a sable node; (3) he open area, resrained by he parabola 4 he O O(; ) is an unsable spiral; - axis(he firs quadran), hen (4) he open area, resrained by he parabola hen 4 O(; ) he O - axis (he second quadran), is a cenre spiral; (5)he open area below he O - axis (he second he fourh quadrans), hen is a saddle of one of is wo forms; (6)he parabola O(; ) b b c 4 O(; ) (he firs quadran), hen is an unsable sar, when 1 a d c ;,or unsable node when 1 (7) he parabola 4 (he second quadran), hen O (; ) is a sable sar when 1 a d bc, or a sable node when 1 b (8) c ; O - axis, hen he sysem O(; ) is a cenre. (9) O - axis i.e., hen he sysem (3) has an infinie se of fixed poins, nonisolaed from each oher, which fill a spoiling line, or one of he axes of he coordinaes, where he rajecories behave owards as in (Figure 1/d). All he bahaviours of he rajecories of he dynamical sysem (3) around he fixed poin O (; ) as well as heir classificaion, are summarized in he diagram of Figure. The presenaion of all his informaion in he gives us a visual summary of all differen plane linear sysems. There are cerain issues o be aken ino consideraion. Firsly, he plane is a wo- dimensional represenaive of ha which is a four-dimensional space in realiy, since he marices are deermined by four parameers, he coefficiens of he marix. Thus here is an infinie number of differen marices which correspond o any poin in he plan. Alhough all hese marices have he same configuraion of he eigenvalues, here may be delicae disincions in he phase porrais, such as www.jmes.org JMESTN4351755 5498
cenres spirals, or he possibiliy of one or wo independen eigenvecors in he case of repeiive eigenvecors. We also hink of he plane as an analogue of he diagram of bifurcaion for he plane linear sysems. The parabola of he phase porrai submis o a bifurcaion: A huge change happens in he geomery of he phase porrai. In conclusion, we noice ha we can obain a lo of informaion from concerning he sysem, wihou aking ino accoun he eigenvalues. For example, if, we know we have a saddle poin. 4 Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 If we make subsiuion in he sysem (7) y y v I has he form x x u ( x u) f ( x u, y v) (9) ( y v) g( x u, y v) Since x y are consan, heir derivaives are zero, so ha he sysem (3) has he form, Fig. V. NONLINEAR DYNAMICAL SYSTEMS AND THE METHOD OF LINEARIZATION Consider he nonlinear dynamical sysem x ' f ( x, y) y ' g( x, y) Le ( x, y ) be condiions (7) is fixed poin, i.e. i saisfies he f ( x, y ) dhe g( x, y ) (8) Le u x x v y y be he componens of a small disurbance from he fixed poin ( x, y ) o he poin ( x, y ). [1] u f ( x u, y v) (1) v g( x u, y v) If we exp he funcions f ( x u, y v) g( x u, y v) in he Taylor series wih he cenre a he poin ( x, y ), he sysem (1) akes he form: f f u f x y x y u x y v O u v O uv x y g g v g x y x y u x y v O u v O uv x y (, ) (, ) (, ) ( ) O( ) ( ) (, ) (, ) (, ) ( ) O( ) ( ) where Ou ( ), O( v ), O( uv) are he sums of he erms of he Taylor series of he same order of he quaniy respecively u, v, uv. Since f ( x, y ) g( x, y ) (see (8)), he above sysem akes he form www.jmes.org JMESTN4351755 5499
f f u x y u x y v O u v O uv x y g g v x y u x y v O u v O uv x y (11) (, ) (, ) ( ) O( ) ( ) (, ) (, ) ( ) O( ) ( ) Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 VI. APPLICATION Consider he compeiive model beween wo species wihou overcrowding (Loka- Volerra) [] In order o unders wha happens o a rajecory which sars in a poin around he fixed poin ( x, y ),we ake exremely nearby exremely small. x y ( x, y) exremely nearby x, which means u v are Since u v are exremely small, he erms O( v ) O( uv) y Ou ( ) don have o be considered (are negleced), so he sysem (11) can be approximaed o he linear dynamical sysem [1], [5] f ( x, y ) f ( x, y ) u ' u v x y g( x, y ) g( x, y ) v ' u v x y which has u v as is dynamical erms. The marix f ( x, y ) f ( x, y ) x y J ( x, y ) g( x, y ) g( x, y ) x y (1) is called a Jacobian marix in he fixed poin ( x, y ). The essence of he linearizaion mehod In order o sudy he behaviour of he rajecories of he nonlinear sysem (7) nearby he fixed poin ( x, y ), i is fairly enough o sudy he behaviour of he rajecories of he linear sysem (1) around is fixed poin (;). The classificaion of he fixed poin (;) he sysem (1)is done by he Jacobian marix J ( x, y ). The behaviour which he rajecories of he sysem (1) have owards he poin (; ) is he same as he behaviour of he rajecories of he dynamical sysem (7) have owards he poin ( x, y )., Fig. 3 Le s denoe he populaion of he wo species y () respecively. Remember x () populaion of he presen prey a ime x () shows he shows he populaion of he predaor a ime Suppose boh x () y () are nonnegaive. y () One sysem of differenial equaions which can model he changes in he populaion of hese wo species is The erm x x x xy d dy y y xy d in he equaion for d represens he exponenial growh of he prey in he absence of he predaors, he erm corresponds o he xy negaive effec over he prey of he ineracion predaor- prey. The erm y in dy d corresponds o he assumpion ha he predaors die ou if here is no prey o ea, he erm xy corresponds o he poziive effec over he predaors of he ineracion predaor-prey. The coefficiens,-1,-1 1 depend on he included species. x x xy We solve his sysem by aking y y xy find he fixed poins ( x 1, y 1 ) (, ) ( x, y ) (1, ).The fixed poin ( x, y ) (, ) has a perfec meaning 1 1 -if he wo populaions predaor prey exinc, we. www.jmes.org JMESTN4351755 55
surely do no expec he populaion o grow a anylaer ime. The oher soluion ( x, y ) (1, ), means ha, if he populaion of he prey is 1 he populaion of he predaor is, he sysem is in perfec equilibrum. There is enough prey o suppor a consan populaion of predaors of, a he same ime here aren neiher many predaors (which would cause he exincion of he populaion of he prey) nor less (in which case he number of prey would increase). Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 x x xy d dy y y xy d i is imporan o noe ha he change of eiher populaion depends boh on x () on y (). The phase porrai for his sysem for a special soluion is [], [3], [6] The birh rae of each specie is exacly he same as he is deah rae, hese populaions are kep a an indefinie ime. The sysem is in equilibrum. If, he firs equaion in his sysem vanishes. Tha x is why he funcion x ( ) saisfies his differenial equaion wihou considering wha iniial condiion we have chosen for. In his case he second differenial equaion is reduced o dy d y y which we know as an exponenial model of he exincion for he populaion of he predaors.from his equaion we know ha he populaion of he predaors ends o zero in an exponenial manner. This enire scenario for x is reasonable, because if here is no prey for some ime, hen here will never be any prey despie he number of he predaors here may be. Furhermore, wihou food supply he predaors would die ou. In he same way, noice ha he equaion for dy vanishes if y x d, he equaion for d d reduces o which is a model of exponenial growh. This means ha any nonzero prey populaion grows wihou bound under hese assumpions. Once more, hese conclusions make sense because here are no predaors o conrol he growh of he prey populaion. In order o unders all he soluions of his predaor-prey sysem Fig. 4 I is ofen helpful o view a soluion curve for a sysem of differenial equaions no merely as a se of poins in he plane bu, raher in a more dynamic way, as a poin following a curve which is deermined by he soluion of he differenial equaion. In Figure 4 we show a special soluionwhich sars a he poin P. As increases, increases a firs, while y () x () says relaively consan. Near x 3.3, he soluion curve urns significanly upward. Thus he predaor populaion sars increasing significanly. As y () lef. Thus nears x () y y () sared o decrease. As y (), he curve sars heading o he has reached a maximum has increases, he values of x () change as indicaed by he shape of he soluion curve. Evenually he soluion curve reurns o is saring poin P begins is cycle again. We can plo simulaneously a lo of soluion curves on he phase plane using Maple Sofware. In Figure 5 we see he complee phase plane for our predaor- prey sysem. Neverheless, we resric our aenion o he firs quadran since i does no make sense o alk abou he negaive populaions. [3] >wih(deools); phaseporrai([(d(x))() = x()*(- y()), (D(y))() = (x()-1)*y()], [x(), y()], = -1.. 1, [[x() = 1, y() = ], [x() =, y() = 1], [x() = 4, y() = 1], [x() = 1.1, y() = 3.1],[x() =.1, y() = 4.1],[x() =.19, y() = 4.19],[x() =.199, y() = 9.199]], x =.. 5, y =.. 5, sepsize =.5e-1, linecolour = blue, arrows = SLIM, hickness = ); www.jmes.org JMESTN4351755 551
Fig. 5 In his predaor-prey sysem, all oher soluions for which x y yeild oher curves ha loop around he equilibrum poin ( x, y ) (1, ), in a counerclockwise manner. In he end, hey reurn o heir iniial poins, hence his model forecass ha apar from he equilibrum soluion, boh x () y () rise fall in a periodic manner. If we use he linearizaion mehod o classify he fixed poins, we calculae he Jacobian marix : J y y x 1 x For he fixed poin ( x, y ) (, ) we have 1 1 J 1, 1 4 9, 1 1, This fixed poin is a saddle node. J, 1 1, Journal of Mulidisciplinary Engineering Science Technology (JMEST) ISSN: 458-943 Vol. 3 Issue 8, Augus - 16 Nonlinear sysems can be invesigaed wih qualiaive mehods. In wo- dimensional sysems, he analysis of fixed poins linear approximaion near he fixed poins in general allows o unders he sysem. The linearizaion echnique is used o find approximae soluions for he nonlinear sysems, bu is i imporan o emphasize ha linearizaion is valid only in a neighborhood of he fixed poin. In nonlinear sysems, sabiliy near a fixed poin is dependen on iniial condiions, only in specific cases local analysis also give insigh in he global behaviour of he model, ha is, he model dynamics far away from he fixed poin. The sudy of he compeiion among he living species hrough dynamical sysems is he bes mehod ever exising. This influences us on he change of he behaviour owards hem in mainaining he equilibrium he biological ecosysems creaed during he long process of naural evoluion, realized from he compeiion wihin species among species. Nowadays pace of developmen of he compuer graphics enables he numerical graphical presenaion of he phase porrai of he wo dimensional nonlinear dynamical sysem. References [1] Srogaz, S.H. 1994 Nonlinear Sysems Chaos. Perseus Publishing, Cambridge, MA. [] Blanchard, P., Devaney, R.L., Hall, G.R., Boson Universiy 11, Differenial Equaions, Fourh Ediion. [3] Lynch, S., Dynamical Sysems wih Applicaions using Maple TM, Second Ediion, 1. [4] Perko, L., Differenial Equaions Dynamical Sysems, Third Ediion, 1. [5] Branam, J.R., Boyce, W.E., Differenial Equaions, An Inroducion o Modern Mehods Applicaions, Second Ediion, 7. [6] De Roos, A.M., Modeling Populaion Dynamics, 14. For he fixed poin ( x, y ) (1, ) we have J, 1, 1, 4 i 1 This fixed poin is a cenre. IV. CONCLUSIONS There are general formulas for soluions of linear sysems. These general formulas include all soluions. Unforunaely, for nonlinear sysems no such general formulas exis. This means ha i is very difficul, or no even possible, o esablish properies of soluions. www.jmes.org JMESTN4351755 55