WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag Dyamical Behavior of High-order Ratioal Differece System Qiahog Zhag Guizhou Uiversity of Fiace ad Ecoomics School of Mathematics ad Statistics Guiyag Guizhou Chia zqiahog68@63com Wezhua Zhag Guizhou Uiversity of Fiace ad Ecoomics Guizhou Key Laboratory of Ecoomics System Simulatio Guiyag Guizhou Chia zhwezh97@hotmailcom Abstract: This paper is cocered with the boudedess persistece ad global asymptotic behavior of positive solutio for a system of two high-order ratioal differece equatios Moreover some umerical examples are give to illustrate results obtaied Key Words: differece equatios boudedess persistece global asymptotic behavior Itroductio Differece equatio or discrete dyamical system is diverse field which impact almost every brach of pure ad applied mathematics Every dyamical system + = f( 2 k ) determies a differece equatio ad vise versa Recetly there has bee great iterest i studyig differece equatios systems Oe of the reasos for this is a ecessity for some techiques whose ca be used i ivestigatig equatios arisig i mathematical models [] describig real life situatios i populatio biology [2] ecoomic probability theory geetics psychology etc The study of properties of ratioal differece e- quatios [3] ad systems of ratioal differece equatios has bee a area of iterest i recet years There are may papers i which systems of differece equatios have studied Ciar et al [4] has obtaied the positive solutio of the differece equatio system + = m + = p Ciar [5] has obtaied the positive solutio of the differece equatio system + = + = Also Ciar [6] has obtaied the positive solutio of the differece equatio system + = z + = z + = Ozba [7] has ivestigated the positive solutios of the system of ratioal differece equtios + = k + = m m+k Papaschiopoulos et al [8] ivestigated the global behavior for a system of the followig two oliear differece equatios + = A+ p + = A+ q = where A is a positive real umber p q are positive itegers ad x p x x q x are positive real umbers I 22 Zhag Yag ad Liu [9] ivestigated the global behavior for a system of the followig third order oliear differece equatios + = 2 B + 2 + = 2 A + 2 where A B ( ) ad the iitial values x i y i ( ) i = 2 Ibrahim [] has obtaied the positive solutio of the differece equatio system i the modelig competitive populatios + = + α + = + β Although differece equatios are sometimes very simple i their forms they are extremely difficult to uderstad thoroughly the behavior of their solutios I book [] Kocic ad Ladas have studied global behavior of oliear differece equatios of E-ISSN: 2224-288 368 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag higher order Similar oliear systems of ratioal d- ifferece equatios were ivestigated (see [2][3]) Other related results reader ca refer ([4] [5] [6] [7] [8][9][2][2][22][23][24][25]) Motivated by above discussio our goal i this paper is to ivestigate the solutios of the twodimesioal system of ratioal oliear differece e- quatios i the form + = + = B+ r s A+ p q = () where A B ( ) p q r s N + ad the iitial values x max{pq} x max{pq} x ( ); y max{rs} y max{rs} y ( ) Moreover we have studied the local stability global stability boudedess of solutios We will cosider some special cases of () as applicatios Fially we give some umerical examples 2 Prelimiaries Let I x I y be some itervals of real umber ad f : Ix m Iy t I x g : Ix m Iy t I y be cotiuously differetiable fuctios The for every iitial coditios (x i y j ) I x I y (i = m m + ; j = t t+ ) the system of differece equatios for = 2 + = f( m t ) (2) + = g( m t ) has a uique solutio {( )} = max{mt} A poit ( x ȳ) I x I y is called a equilibrium poit of (2) if x = f( x x ȳ ȳ) ȳ = g( x x ȳ ȳ) amely ( ) = ( x ȳ) for all Let I x I y be some itervals of real umbers iterval I x I y is called ivariat for system () if for all > x m x x I x y t y y I y I x I y Defiitio Assume that ( x ȳ) be a fixed poit of (2) The (i) ( x ȳ) is said to be stable relative to I x I y if for every ε > there exists δ > such that for ay iitial coditios (x i y j ) I x I y (i = m m+ ; j = t t+ ) with i= m x i x < δ j= t y j ȳ < δ implies x < ε ȳ < ε (ii) ( x ȳ) is called a attractor relative to I x I y if for all (x i y j ) I x I y (i = m m + ; j = t t + ) lim = x lim = ȳ (iii) ( x ȳ) is called asymptotically stable relative to I x I y if it is stable ad a attractor (iv) Ustable if it is ot stable Theorem 2 [] Assume that X( + ) = F (X()) = is a system of differece e- quatios ad X is the equilibrium poit of this system ie F (X) = X If all eigevalues of the Jacobia matrix J F evaluated at X lie iside the ope uit disk λ < the X is locally asymptotically stable If oe of them has modulus greater tha oe the X is ustable Theorem 3 [2] Assume that X( + ) = F (X()) = is a system of differece e- quatios ad X is the equilibrium poit of this system the characteristic polyomial of this system about the equilibrium poit X is P (λ) = a λ +a λ + + a λ + a = with real coefficiets ad a > The all roots of the polyomial p(λ) lie iside the ope uit disk λ < if ad oly if k > for k = 2 (3) where k is the pricipal mior of order k of the matrix = 3 Mai results a a 3 a 5 a a 2 a 4 a a 3 a Cosider the system () if A < B < system () has equilibrium ( ) ad ( A B) I additio if A < B = the system () has ifiite equilibrium poits ( x ) where x ad if A = B < the system () has ifiite equilibrium poits ( ȳ) where ȳ Fially if A > ad B > ( ) is the uique equilibrium poit Theorem 4 Assume that A < B < The the followig statemets are true (i) The equilibrium ( ) is locally ustable (ii) The uique positive equilibrium ( A B) is locally ustable E-ISSN: 2224-288 369 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag Proof: (i) Let M = max{p q r s} We ca easily obtai that the liearized system of () about the equilibrium ( ) is Φ + = DΦ (4) where Φ = ( M M ) T D = (d ij ) (2M+2) (2M+2) = B A The characteristic equatio of (4) is (5) ( f(λ) = λ 2M λ ) ( λ ) = (6) A B This shows that the roots of characteristic equatio λ = A ad λ = B lie outside uit disk So the u- ique equilibrium ( ) is locally ustable (ii) We ca easily obtai that the liearized system of () about the equilibrium ( A B) is Φ + = GΦ (7) B B i which B are i colum M + r + 2 ad M + s + 2 respectively A are i colum p + ad q + respectively Let λ λ 2 λ 2M+2 deote the 2M + 2 eigevalues of Matrix G Let D = diag(d d 2 d 2M+2 ) d i (i = 2 2M + 2) be a diagoal matrix Clearly D is ivertible Computig DGD we obtaied DGD = where Φ = ( M M ) T G = A A d 2 d dp+ d p d q+ d q d M+ d M A A E-ISSN: 2224-288 37 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag B B d M+3 d M+2 dm+r+2 d M+r+ d M+s+2 d M+s+ It is well kow that G has the same eigevalues as DGD we obtai that max k 2M+2 λ k d 2M+2 d 2M+ = DED { = max d 2 d d M+d M d M+3d M+2 d 2M+2 d 2M+ + 2 A + 2 } B > It follows from Theorem 3 that equilibrium ( A B) is locally ustable Theorem 5 Assume that A > B > The the equilibrium ( ) is globally asymptotically stable Proof: For A > B > from Theorem 4 ( ) is locally asymptotically stable From () it is easy to see that every positive ( ) is bouded i e x y Now it is sufficiet to prove that( ) is decreasig From () we have + = B+ r s B < y + = A+ p q y A < This implies that the sequeces { } ad { } are decreasig Hece lim = lim = Therefore the equilibrium ( ) is globally asymptotically stable Theorem 6 Let A < ad B < The for solutio ( ) of system () followig statemets are true (i) If the (ii) If the 4 Rate of covergece I order to study the rate of covergece of positive solutios of () which coverge to equilibrium poit ( ) of this system first we cosider the followig results that gives the rate of covergece of solutio of a system of differece equatios X + = [A + B()]X (8) where X be m dimesioal vectora C m m is a costat matrix B : Z + C m m is a matrix fuctio satisfyig B() (9) as where be ay matriorm which is associated with the vector orm (x y) = x 2 + y 2 Propositio 7 (Perros Theorem)[26] Suppose that coditio (9) holds If X is ay solutio of (8) the X = for all large or X + ρ = lim X () exists ad is equal to the modulus of oe of the eigevalues of matrix A Propositio 8 [26] Suppose that coditio (9) holds If X is ay solutio of (8) the X = for all large or ρ = lim X + () exists ad is equal to the modulus of oe of the eigevalues of matrix A Let( Y ) be a arbitrary positive solutio of system () such that lim = lim = It follows from () that ad + = + = B + r s = A + p q = B + r s A + p q Let E = E 2 = the we have E + = A E + B E 2 E 2 + = C E + D E 2 where A = B + r s B = E-ISSN: 2224-288 37 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag Moreover C = D = A + p q 8 7 6 lim A = B lim D = A Now the limitig system of error terms ca be writte as ( ) ( ) ( ) E + /B E = /A E 2 + E 2 which is similar to liearized system of () about the equilibrium poit ( ) Usig Propositio 7 ad Propositio 8 we have followig result Theorem 9 Assume that ( ) be a positive solutio of () such that lim = lim = the the error vector E = (E E) 2 T of every solutio of () satisfies the followig asymptotic relatios lim E = λ 2 F j ( ) E + lim E = λ 2F j ( ) where λ 2 F j ( ) = A or B are the characteristic of Jacobia matrix F J ( ) 5 Numerical examples I order to illustrate the results of the previous sectios ad to support our theoretical discussios we cosider several iterestig umerical examples i this sectio These examples represet differet types of qualitative behavior of solutios to oliear differece equatios ad system of oliear differece e- quatios Example If the iitial coditios x = 6 x = 2 x 2 = 8 x 3 = 3 x 4 = 4 y = 6 y = 3 y 2 = 5 y 3 = 2 y 4 = 8 ad A = 2 B = 3 r = s = 3 p = 2 q = 4 we have the followig system + = + = 3 + 3 2 + 2 4 It is clear that A > B > The the equilibrium ( ) is globally asymptotically stable(see Theorem 32 Fig ) & 5 4 3 2 5 5 2 25 3 Figure : The fixed poit () is globally asymptotically stable & 2 8 6 4 2 8 6 4 2 y 5 5 2 25 3 35 4 Figure 2: The fixed poit () ad ( A B) is ustable Example 2 If If the iitial coditios x = 26 x = 72 x 2 = 88 x 3 = 23 x 4 = 84 y = 96 y = 73 y 2 = 55 y 3 = 62 y 4 = 78 ad A = 9 B = 7 r = s = 3 p = 2 q = 4 we have the followig system + = 7 + 3 + = 9 + 2 4 It is clear that A < B < The equilibrium ( ) ad ( A B) are ustable(see Theorem 4 Theorem 6 Fig 2) 6 Coclusio ad future work I this paper we have studied the behavior of positive solutio to system () uder some coditios If A > ad B > the system () has a uique equilibrium ( ) which is globally asymptotically stable If A < ad B < the system () has equilibrium ( ) ad ( A B) ad these equilibriums are ustable We will study the behavior of positive solutio to system uder the coditios A > B < or A = B = i the future E-ISSN: 2224-288 372 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag Ackowledgemets: This work was fiacially supported by the Natioal Natural Sciece Foudatio of Chia (Grat No 362) ad the Scietific Research Foudatio of Guizhou Provicial Sciece ad Techology Departmet([23]J283 [29]J26) ad the Natural Sciece Foudatio of Guizhou Provicial Educatioal Departmet (No284) Refereces: [] MP Hassell ad HN Comis Discrete time models for two-species competitio Theoretical Populatio Biology Vol 9 o 2976 pp 22 22 [2] JE Frake ad AA Yakubu Mutual exclusio versus coexistece for discrete competitive Systems Joural of Mathematical Biology Vol3 o 299 pp 6 68 [3] B Iricai ad S Stevic Some Systems of Noliear Differece Equatios of Higher Order with Periodic Solutios Dyamics of Cotiuous Discrete ad Impulsive Systems Series A Mathematical Aalysis Vol 3 No 3-4 26 pp 499 57 [4] C Ciar I Yalcikaya ad R Karatas O the positive solutios of the differece equatio system + = m + = p J Ist Math Comp Sci Vol8 25 pp35 36 [5] C Ciar O the positive solutios of the differece equatio system + = + = Applied Mathematics ad Computatio Vol58 24 pp 33 35 [6] C Ciar ad I Yalcikaya O the positive solutios of the differece equatio system + = z + = z + = Iteratioal Mathematical Joural Vol5 24 pp 525 527 [7] AY Ozba O the positive solutios of the system of ratioal differece equatios + = y k + = m m k J Math Aal Appl Vol323 26 pp 26 32 [8] G Papaschiopoulos ad CJ Schias O a system of two oliear differece equatios J Math Aal Appl Vol 29 998 pp 45 426 [9] Q ZhagL Yag ad J Liu Dyamics of a system of ratioal third-order differece equatio Advaces i Differece Equatios 22 22: 36 pp 8 [] TF Ibrahim Two-dimesioal fractioal system of oliear differece equatios i the modelig competitive populatios Iteratioal Joural of Basic & Applied Scieces Vol2 o 5 22 pp 3 2 [] VL Kocic ad G Ladas Global behavior of oliear differece equatios of higher order with applicatio Kluwer Academic Publishers Dordrecht 993 [2] MRS Kuleovic ad O Merio Discrete dyamical systems ad differece equatios with mathematica Chapma ad Hall/CRC Boca Rato Lodo 22 [3] K Liu Z Zhao X Li ad P Li More o threedimesioal systems of ratioal differece equatios Discrete Dyamics i Nature ad Society Vol 2 Article ID 78483 2 [4] QZhag WZhag O a system of two highorder oliear differece equatios Advaces i Mathematical Physics Vol 24 (24) Article ID 729273 8 pages [5] Q Zhag WZhag YShao JLiu O the system of high order ratioal differece equatios Iteratioal Scholarly Research Notices Volume 24 (24) Article ID 7652 5 pages [6] Q Zhag J Liu ad Z Luo Dyamical behavior of a system of third-order ratioal differece equatio Discrete Dyamics i Nature ad Society vol 25 pp C6 25 [7] TF Ibrahim ad Q Zhag Stability of a aticompetitive system of ratioal differece equatios Archives Des Scieces Vol 66 o 5 23 pp 44 58 [8] EME Zayed ad MA El-Moeam O the global attractivity of two oliear differece e- quatios J Math Sci Vol 77 2 pp 487 499 [9] N Touafek ad EM Elsayed O the periodicity of some systems of oliear differece e- quatios Bull Math Soc Sci Math Roumaie Vol 2 22 pp 27 224 [2] N Touafek ad EM Elsayed O the solutios of systems of ratioal differece equatios Mathematical ad Computer Modellig Vol 55 22 pp 987 997 [2] S Kalabusic MRS Kuleovic ad E Pilav Dyamics of a two-dimesioal system of ratioal differece equatios of Leslie Gower E-ISSN: 2224-288 373 Volume 6 27
WSEAS TRANSACTIONS o MATHEMATICS Qiahog Zhag Wezhua Zhag type Advaces i Differece Equatios 2 doi:86/687-847-2-29 [22] Q Di Global behavior of a plat-herbivore model Advace si Differece Equatios (25) -2 [23] Q Di K A Kha A Noshee Stability aalysis of a system of expoetial differece equatios Discrete Dy Nat Soc Volume 24 Article ID 37589 pages [24] Q Di Global stability of a populatio model Chaos Solito Fract Vol 59 24 pp 9-28 [25] Q Di T Dochev Global character of a hostparasite model Chaos Solito Fract Vol 54 23 pp -7 [26] M Pituk More o Poicares ad Perros theorems for differece equatios J Diff Eq App Vol 8 22 pp 2-26 E-ISSN: 2224-288 374 Volume 6 27