Some Dynamical Behaviours of a Two Dimensional Nonlinear Map

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1 Iteratioal Joural of Moder Egieerig Research (IJMER) Vol.2, Issue.6, Nov-Dec. 202 pp ISSN: Some Dyamical Behaviours of a Two Dimesioal Noliear Map Tarii Kumar Dutta, Debasish Bhattacharjee 2, Basistha Ram Bhuya 3 Departmet of Mathematics; Gauhati Uiversity; Guwahati 7804; INDIA 2 Departmet of Mathematics; B. Borooah College; Guwahati 78007; INDIA 3 Departmet of Mathematics; CB College; Guwahati 780; INDIA Abstract: We cosider the Nicholso Bailey model f(x,y)= ( x e a y, x e ay ) Where ad a are adjustable parameters, ad aalyse dyamical behaviours of the model. It is observed that the steady state occurs whe there is o predator ad prey for a certai rage of the cotrol parameters ad that there exists a certai regio of the cotrol parameters i which the atural equilibrium state ever occurs. I that case a modified versio of the model is cosidered by taig care of the uboudedess of the prey system. It is further foud that the model follows the stability of period-doublig fashio obeyig Feigebaum uiversal costat δ ad at last attais ifiite period doublig route leadig to chaos i the system. The bifurcatio poits are calculated umerically ad after that the accumulatio poit i.e. oset of chaos is calculated based o the experimetal values of bifurcatio poits. Key Words: Period-Doublig Bifurcatio/ Periodic orbits / Feigebaum Uiversal Costat Accumulatio poit 200 AMS Classificatio: 37 G 5, 37 G 35, 37 C 45 I. Itroductio: The Nicholso Bailey model [4] was developed i 930 s to describe populatio dyamics of host-parasite (predatorprey) system. It has bee assumed that parasites search hosts at radom ad that both parasites ad hosts are assumed to be distributed i a o-cotiguous ("clumped") fashio i the eviromet. However the modified versio of the Nicholso- Bailey model has bee discussed may times by may authors [, 2, 6, 9, 0, ad ]. I this preset discussio i sectio.2 we verify the stability ad dyamic behaviour of the model aalytically ad the i sectio.3 the modified form of the model has bee tae which restricts the uboudedess of the model to some extet. The detailed dyamical behaviour of a particular form of its class has bee studied ad it has bee observed that the map follows period doublig bifurcatio route to chaos provig that the atural equilibrium chages its ature from periodic order to chaos.i sectio.4 umerical evaluatios has bee carried out to prove the geometrical behaviour. astly, i sectio.5 the calculatio of the accumulatio poit from where chaos starts has bee evaluated umerically, [3, 5, 6,8,2, ad 3].. Nicholso-Bailey model: The model as discussed by Nicholso ad Bailey is as follows: x + = x e ay y + = x e ay, where x + represets the umber of hosts (or prey) at stage ad y + represets umber of parasites(or predator) at th stage. The differece equatio ca also be writte i the fuctio form as follows: f(x,y)= ( x e a y, x e ay ).2. Steady state of the above system: The fixed poit is give as follows: x e a y = x x e ay = y Clearly (0,0) is oe of the fixed poits. et x 0 the e ay = i. e. ay = log From (.2..2) we have x = a log ( ) a i.e x = log Thus the fixed poits are ( a log i. e. y = a log ( ) from (.2..) (.2..2) for ay value of x. Hece ay (x,0) is a fixed poit for =. (.2..) (.2..2), a log ( )) ad (0,0). However at =,(.2..) gives y=0 ad it automatically satisfy.3.2 Stability of the equilibrium poits: Now the Jacobia matrix is give by e ay axe ay e ay axe ay The eigevalues of which are: 4302 Page

2 Iteratioal Joural of Moder Egieerig Research (IJMER) Vol.2, Issue.6, Nov-Dec. 202 pp ISSN: e ay + ax 4ae ay x + + ax 2 ad 2 e ay + ax 4ae ay x + + ax 2 For fixed poit (0,0), the eigevalues are 0,. This shows that (0,0) is a stable solutio till =. However for other fixed poits say (x,y), we have e ay =, hece the eigevalues become 2 + ax 4a2 x + + ax 2 ad 2 + ax + 4a2 x + + ax 2 I particular for =, the eigevalues are ax,. Thus if ax< oe of the eigevalues become less tha.that is why at = the trajectory coverges to (x,0) such that ax<. Now for the period- doublig bifurcatio poit, 2 + ax + 4a2 x + + ax 2 = i.e. + ax + 4a 2 x + + ax 2 =-2 i.e. 3 + ax = 4a 2 x + + ax 2 i.e. 3 + ax 2 = 4a 2 x + + ax 2 i.e ax + 4a 2 x = 0 i.e.2 + ax + ax = 0 i.e. = ax Puttig x= 2+ax a log From eq (.2.2.) we have, = i.e. t = t (2+ t ) 2 t +log (t) i (.2.2.) we have ax = t i.e. t = 2(t ) (.2.2.) i.e. logt t + = 2(t ) if t> the l.h.s. is egative ad r.h.s. is positive. If t< the l.h.s. is positive but r.h.s. is egative. Hece there is o solutio i.e. period doublig bifurcatio does ot occur. + ax + 2 4a2 x + + ax 2 = i.e. + ax + 4a 2 x + + ax 2 =2 i.e. + ax = 4a 2 x + + ax 2 i.e. + ax 2 = 4a 2 x + + ax 2 i.e. 4ax + 4a 2 x = 0 i.e. ax + ax = 0 i.e. = (.2.2.2) Agai we cosider the stability of the other fixed poit ( a log, a log ( )) Now we cosider the expressio 4a 2 x + + ax 2 for the fixed poit ( is 2( +) 2 g ( +2)+2og []( + 2 og []) ( +) 3 2 =g() a log, a log ( )).The simplified expressio Fig.2.2.a : Abcissa represets the cotrol parameter ad ordiate represets g() Clearly,g() is egative for > Hece magitude of the eigevalues become 4303 Page

3 Iteratioal Joural of Moder Egieerig Research (IJMER) Vol.2, Issue.6, Nov-Dec. 202 pp ISSN: ax 4a2 x + + ax 2 = + ax ax 2 + 4a 2 x log = ax = =h()(say). For > ad for large value of, the above expressio shows the graph as h Fig.2.2.b: Abcissa represets the cotrol parameter ad ordiate represets h() Hece the fixed pit is ustable, ad this shows that the model has bee made to fulfill the fact that equilibrium stage ever occurs for predator system i ature. We ow tae a modified versio of the Nicolso Bailey model, i.e. we tae x + = x e ay x 2 y + = x e ay The additioal term e ( x 2 ) with x + helps to restrict the ulimited growig of host(or prey)..3 Dyamical behaviour of the map eepig a costat: We ow fix the parameters say a ad eep varyig to aalyse the detailed dyamical behaviours of the map. et us tae a=0.. O ispectio it ca be see that (0,0) is a fixed poit of the model satisfyig the equatio f(x,y)=(x,y) =(xe ay x2, x e ay ) ay x^2 i.e. x= xe y= x e ay (.3.) Usig Mathematica software we geerate the bifurcatio diagram for the observatio of the whole dyamical behaviour of the map as is varied. Fig.3.a: The figure is geerated usig poits of which the last 300 poits are tae at every parameter value of, ad plotted the x coordiate of the poit (x,y) vs.. The eige values of the liearised form are as follows: 2 e x 2 2ay (e ay + ae x 2 +ay x 2e ay x 2 ± ( e ay ae x 2 +ay x + 2e ay x 2 ) 2 4e x 2 +2ay (ae ay x 2ax 3 ), which ca be re- writte as 2 (eay + ax 2e ay x 2 ± ( e ay ax + 2e ay x 2 ) 2 4e ay (ae ay x 2ax 3 ) The diagram shows that the model follows period doublig route to chaos o icreasig the cotrol parameter. For (0,0) the eigevalues are 0,, which says that (0,0) loses stability at =. et (x 0,y 0 ) be a fixed poit of the map f where either of x 0, y 0 are equal to zero. The fixed poit is stable till both the eige values at x 0, y 0 are less tha i modulus. However the first bifurcatio poit ca be obtaied from the equatios (.2.) ad mi{λ,λ 2 }=-.If we ow begi to icrease the value of exceedig the bifurcatio poit, the fixed poit (x 0,y 0 ) loses its stability ad there arises aroud it two poits, say, (x 2 (), y 2 ()) ad (x 22 (), y 22 ()) formig a stable periodic trajectory of period 2. O icreasig the value of oe of the eige values starts decreasig from positive values to egative ad whe we reach a certai 4304 Page

4 Iteratioal Joural of Moder Egieerig Research (IJMER) Vol.2, Issue.6, Nov-Dec. 202 pp ISSN: value of, we fid that oe of the eigevalues of the Jacobia of f 2 becomes -, idicatig the loss of stability of the periodic trajectory of period two. Thus, the secod bifurcatio taes place at this value 2 of. We ca repeat the same process, ad fid that the periodic trajectory of period 2 becomes ustable ad a periodic trajectory of period 2 + appears i its eighbourhood for all =,2,3,., [ 5,6,8,0 ]..4 Numerical Method for Obtaiig Bifurcatio Poits: We have used Newto-Raphso method to obtai the periodic poits which has bee proved to be worthy for sufficiet accuracy ad time savig. The Newto Recurrece formula is x + = x Df x f(x ), where = 0,,2, ad Df(x ) is the Jacobia of the map f at the vector x = (x, x 2 )(say). We see that this map f is equal to f -I i our case, where is the appropriate period. The Newto formula actually gives the zero(s) of a map, ad to apply this umerical tool i our map oe eeds a umber of recurrece formulae which are give below. et the iitial poit be ( x 0, y 0 ) ad let M(x,y)= xe ay x2,n(x,y)= x e ay, et A 0 = M (x 0,y 0 ), B 0 = M ad A = x M x (x,y ) N x (x,y ) (x 0,y 0 ), C 0 = N x y M y (x,y ) N y (x,y ) A (x 0,y 0 ), D 0 = N y (x 0,y 0 ) B C D Sice the fixed poit of the map f is a zero of the map F(x,y) = f(x,y)-(x,y), the Jacobia of F () is give by J A I C B D. Its iverse is C A J D B I where =(A -)(D -)-B C, the Jacobia determiat. Therefore, Newto s method gives the followig recurrece formula i order to yield a periodic poit of F x y x y (D )(x x ) B (y y ) ( C )(x x ) (A )(y y ) where F ( x ) ( x, y ).4. Numerical Methods for Fidig Bifurcatio Values: As described above for some particular value of = say, the fixed poit of f is calculated ad hece the eigevalues of J ca be calculated at the fixed poit. et (x,y ),(x 2,y 2 ),. (x,y ) be the periodic poits of f at. et λ, λ 2 be the two eige values of J at, let I(, )= mi{λ, λ 2 }, where =2 is the period umber.the we search two values of say ad 22 such that (I(, )+)(I(, 22 )+) < 0.The the existece of th bifurcatio poit is cofirmed i betwee ad 22. The we may apply some of the umerical techiques viz. Bisectio method or Regula Falsi method o ad 22 for sufficiet umber of iteratios to get such that I(,)=-. Our umerical results are as follows: Table.4..a: Bifurcatio poits calculated with the above umerical procedure are give as follows: Period Bifurcatio poit = = = = = = = = = = = = The Feigebaum uiversal costat is calculated usig the experimetally calculated bifurcatio poit usig the followig formula δ = A A,where A A + A represets th bifurcatio poit. The values of δ are as follows Page

5 δ = δ 2 = δ 3 = δ 5 = δ 6 = δ 7 = δ 8 = δ 9 = δ 0 = δ = Iteratioal Joural of Moder Egieerig Research (IJMER) Vol.2, Issue.6, Nov-Dec. 202 pp ISSN: It may be observed that the map obeys Feigebaum uiversal behaviour as the sequece {δ } coverges to δ as becomes very large..5 Accumulatio Poit: The accumulatio poit ca be calculated by the formula A =(A 2 -A )/(δ-), where δ is Feigebaum costat. But it has bee observed that {δ } coverges to δ as.therefore a sequece of accumulatio poit { A, }is made usig the formula A, =(A + -A )/(δ-)[8]. From the above experimetal values of bifurcatio poits ad usig δ= the sequece of values is costructed as follows: A, = A,2 = A,3 = A,4 = A,5 = A,6 = A,7 = A,8 = A,9 = A,0 = A, = It may be observed that the sequece coverges to the poit After which chaotic regio starts. Refereces:. Beddigto, J.R., Free, C.A., awto, J.H., Dyamic Complexity i Predator-Prey models framed i differece equatios, Nature, 225(975), Comis,H.N., Hassel,M.P., May,R., The spatial dyamics of host-parasitoid systems,joural of Aimal Ecology,vol-6(992),pp Falcoer, K.J., Fractal Geometry: Mathematical Foudatios ad Applicatios, Joh Wiley publicatio, Feigebaum, M.J., Qualitative Uiversility for a class of o-liear trasformatios, J.Statist.Phys,9:(978), Feigebaum, M.J., Uiversility Behavior i o-liear systems, os Alamos Sciece,.(980), Hassel,M.P.,Comis,H.N.,May,R., Spatial structure of chaos i isect populatio dyamics,nature,vol- 353(99),pp Heo, M., A two dimesioal mappig with a strage attractor, Comm. Math. Phys. ett.a 300(2002), Hilbor, R.C., Chaos ad No-liear dyamics,oxford Uiv.Press Hoe, A.N.W., Irle, M.V.,Thurura, G.W., O the Neimar-Sacer bifurcatio i a discrete predetor-prey system, Kujetsov, Y., Elemets of Applied Bifurcatio Theory, Spriger(998).. May, R.M., Simple Mathematical Models With Very Complicated Dyamics, Nature,Vol.26(976), Murray, J.D., Mathematical Biology :A Itroductio,Third Editio, Spriger Murray, J.D., Mathematical Biology II: Spatial Models ad Biomedical Applicatios, Spriger, Nicholso,A.J.,Bailey V.A., The Balace of Aimal Populatios-Part-,Proceedigs of the Zoological Society of odo,vol-05,issue-3(935),pp Page

Math 312 Lecture Notes One Dimensional Maps

Math 312 Lecture Notes One Dimensional Maps Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,