STABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT

International Journal of Bifurcation Chaos Vol 7 No 2007 43 52 c World Scientific Publishing Company STABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT E J JANOWSKI M R S KULENOVIĆ Department of Mathematics University of Rhode Isl Kingston RI 0288-086 USA kulenm@mathuriedu Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only Z NURKANOVIĆ Department of Mathematics University of Tuzla 75000 Tuzla Bosnia Herzegovina Received May 0 2005; Revised January 30 2006 We first investigate the Lyapunov stability of the period-three solution of Todd s equation with a period-three coefficient: x n x n x n p n x n2 n 0 α for n 3l p n β for n 3l γ for n 3l 2 l 0 α β γ positive Then for k 2 3 we extend our stability result to the k-order equation x n x n x nk2 n 0 p n x nk p n is a periodic coefficient of period k with positive real values x k x x 0 0 Keywords: Invariant; Lyapunov function; Lyapunov stability; Lyness; periodic coefficient Introduction The systematic analysis of difference equations with periodic coefficients was initiated very recently in [Elaydi & Sacker 2005a 2005b; Franke & Selgrade 2003] There are several recent related papers such as [Henson & Cushing 997] [Kulenović et al 2004] a book [Grove & Ladas 2004] the global behavior of some specific equations with periodic coefficients was considered Our goal is to study the effect of the periodization of the coefficient on the stability of the equilibrium of an equation with an invariant In this paper the notion of stability refers to the stability in the sense of Lyapunov see [Kocic & Ladas 993] [Kulenović & Merino 2002] Consider the kth order equation y n p y n y nk2 y nk n 0 Author for correspondence 43

44 E J Janowski et al Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only k 2 3 When k 2Eqisknown as the Lyness equation which has been studied intensively by many authors using methods ranging from algebraic projective geometry [Bastien & Rogalski 2004a 2004b; Beukers & Cushman 998; Zeeman 996] to hard analyis [Barbeau et al 995; Kocic et al 993; Kulenović 2000] Also several authors have studied the case when k 3known as Todd s equation obtained some results concerning invariants the stability of the equilibrium [Kocic & Ladas 993; Kocic et al 993; Kulenović 2000; Kulenović & Merino 2002] as well as the periodic solutions of certain periods [Cima et al 2006] Cima Gasull Manosa used dynamical systems methods to obtain precise results about attainable periods for Todd s equation [Cima et al 2006] By using the substitution y n px n Eqcan be reduced to the form: x n x n x nk2 n 0 2 px nk k 2 3 Suppose p is periodic with period k Then Eq 2 becomes x n x n x nk2 n 0 3 p n x nk k 2 3 p n is a periodic coefficient of period k We will show that Eq 3 has an invariant we will use this invariant to investigate the stability of the period-k solution of Eq 3 In this paper we will assume that all parameters initial conditions are positive real numbers In order to illustrate the method used in the general case see Sec 4 we will first extend the result on the stability of the equilibrium for Todd s equation which was proven in [Kulenović 2000] to the case the coefficient p n is periodic of period 3 In particular we consider the following difference equation x n x n x n p n x n2 n 0 4 α for n 3l p n β for n 3l γ for n 3l 2 l 0 This equation can be reduced to the following three equations: x 3l x 3l x 3l αx 3l2 5 x 3l2 x 3l x 3l βx 3l 6 x 3l3 x 3l2 x 3l γx 3l l 0 7 By setting u l x 3l v l x 3l w l x 3l2 8 we obtain the following system of difference equations u l u l v l βv l βu l v l βv 2 l l αu l w l αβv l w l αβγu l v l w l v l u l v l l αu l w l αβv l w l w l u l v l l 0 9 l We assume that at least two of the parameters α β γ are not equal We will also need the notion of an invariant [Kulenović & Merino 2002] Definition Consider the difference equation x n fx n n 0 0 x n is in R k f : D D is continuous for D R k We call a nonconstant continuous function I : R k R an invariant for system 0 if Ix n Ifx n Ix n for every n 0 Invariants are a powerful tool for finding solutions of difference equations in exact form investigating the short-term long-term behavior of solutions [Kulenović & Merino 2002 Chap 4; Sedaghat 2003] By using the well-known invariant of the Lyness equation Bastien Rogalski found very precise results about the global behavior of that equation See also [Gardini et al 2003] for some related results In the case of nonlinear systems of difference equations we present the following result which establishes a connection between invariants Lyapunov functions the stability of the equilibrium points This result has been used extensively to prove the stability of several difference equations Theorem Discrete Dirichlet Theorem Consider the difference equation 0 x n is in R k f : D D is continuous for D R k Suppose that I : R k R is a continuous invariant of Eq 0 IfI attains an isolated local minimum or maximum value at the equilibrium point x

Stability of the kth order Lyness Equation with a Period-k Coefficient 45 Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only of this system then there exists a Lyapunov function equal to ±Ix Ix sotheequilibriumx is stable Theorem is a discrete analogue of the Dirichlet theorem for differential equations [Kulenović & Merino 2002 p 208] 2 Nonhyperbolic Equilibrium Solution The equilibrium solution u vw of system 0 satisfies the system of equations u v u v β v β u v β v 2 αu w u v αu w w u v which implies γu 2 v w β v 2 u w 2 u v Notice γu 2 u β v 2 v 2 w u v w 2 which implies that u v if only if γ β Similarly v w if only if α β u w if only if α γ This shows that the equilibrium of 0 is a periodic solution of Eq 4 with minimal period three Lemma 2 System has a unique solution Proof Without loss of generality we can assume that γ min{α β γ} Consider the part of Eq 2 β v 2 v γu 2 u as a quadratic equation in v By solving this equation for v we obtain v 4βγu 2 u 3 2β Similarly by solving the part of Eq 2 2 w γu 2 u as a quadratic equation in w weobtain w 4αγu 2 u 4 2α By substituting Eqs 3 4 in the first equation of we obtain the following equation: 2αβγu 2 2αβ α β β 4αγu 2 u α 4βγu 2 u 0 5 Furthermore u satisfies the polynomial equation 2αβγx 2 2αβ α β 2 α 2 4βγx 2 x β 2 4αγx 2 x 2 2α2αβγx 2 2αβ α β 2 4βγx 2 x 0 Set fx 2αβγx 2 2αβ α β α 4βγx 2 x β 4αγx 2 x x 0 Clearly f0 2αβ < 0 lim x fx In addition in view of γ min{α β γ} the function f is concave up for all x 0since f β γ x 4αβ γ 4βγx 2 4βx 3 α γ 4αγx 2 4αx 3 > 0 Consequently f has a unique positive zero that depends continuously on α β γ Inotherwords there is unique positive u that depends continuously on α β γ Byusingthisu in Eqs 3 4 we obtain the unique positive v w that depend continuously on α β γ this completes the proof of lemma that is Observe that u v β v β u v β v 2 s αu w αu w Similarly u v β v β u v β v 2 αu w > 6 > 7 αγu w> 8

Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 46 E J Janowski et al > 9 Theorem 2 The equilibrium E u vw of system 0 is a nonhyperbolic point with two complex conjugate roots on the unit circle the third root equal to Proof System 0 can be written in the form u l fu l v l w l v l gu l v l w l w l hu l v l w l l 0 fu v w gu v w u v βv βuv βv 2 αuw αβvw αβγuvw u v αuw αβvw hu v w u v The Jacobian matrix of the map T u v w fu v w gu v w hu v w evaluated at the equilibrium point E u vw has the form: f u E f v E f w E J T E g u E g v E g w E h u E h v E h w E f u u vw v β v β v2 αβγu 2 v w f v u vw α αu α 2 f w u vw β v αγu w γu g u u vw g v u vw u v β v 2 u αu w αβ u v 2 β v g w u vw β v h u u vw h v u vw u v h w u vw 2 2 2 The characteristic equation of the Jacobian matrix is λ 3 rλ 2 sλ t 0 r f u g v h w s g u f v f u h w h u f w g v h w g w h v f u g v t f u g v h w h u f w g v g u f v h w f u g w h v h u f v g w g u f w h v Clearly f u u v w < 0g v u v w < 0 h w u vw < 0 which implies that r<0 Furthermore βv r βv 3 > 3 so 3 <r<0 20 Now we also have that βv s β v αγu w γu β v βv 22 βv α 2 β 2 γu v 2 w 2 α 2 w 2 3 r α 2 β 2 γu v 2 w 2 β v β v

Stability of the kth order Lyness Equation with a Period-k Coefficient 47 Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only t β v ] βv ] [ β v [ αγu w γu [ ] β v β v αβ 2 γu v 2 w βv αγu w αβ 2 γu v 2 w αβ 2 γu v 2 w αγu w Hence the characteristic equation becomes with roots λ λ 3 rλ 2 rλ 0 λ 23 r ± i D 2 in view of 20 D r 2 4r r 3 < 0 Thus λ 23 2 4 [r 2 4 r 2 ] Therefore all three roots of the characteristic equation are on the unit circle 3 Invariant Corresponding Lyapunov Function Our first result gives the existence of an invariant for Eq 4 Theorem 3 Equation 4 has an invariant of the form: Ix n x n x n2 p n x n2 for n 0 p n x n p n2 x n x n x n x n2 2 Proof By taking into account that p n is a perodic sequence of period 3 we get that: Ix n x n x n p n x n p n x n x n x n x n p n x n p n2 x n p n x n xn x n x n2 x n x n p n x n2 p n2 x n p n x n x n x n x n2 Ix n x n x n2 x n x n p n x n2 Similarly we can show that the corresponding system has an invariant of the form: Iu l v l w l l βv l γu l u l v l w l for l 0 22 By using this invariant we can obtain the following result Theorem 32 The period-three solution of Eq 4 is stable The Lyapunov function for Eq 4 has the form: V x y z Ix y z Iu vw

Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 48 E J Janowski et al Proof First we will find the minimizer of Ix y z The necessary conditions for the existence of critical points gives: I x y z αz βy γx 2 0 I y x z αz γx βy 2 0 I z x y βy γx αz 2 0 which is equivalent to γx 2 y z βy 2 x z αz 2 x y Thus we conclude that the only positive critical point of Ix y s the equilibrium point E u vw Next the Hessian matrix evaluated at the equilibrium point E u vw has the form: a a 2 a 3 H a 2 a 22 a 23 a 3 a 23 a 33 a 2 I u vw x2 2 u β v 2 2 αβ u v 2 a 22 2 I u vw y2 2 v γu 2 2 αγu 2 v a 33 2 I u vw z2 2 w β v γu 2 2 w βγu 2 v 2 a 2 2 I u vw x y u v w γu 2 β v 2 2 αβγu 2 v 2 a 3 2 I u vw x z u v w β v γu 2 2 2 αβγu 2 v 2 a 23 2 I u vw y z u v w γu β v 2 2 2 αβγu 2 v 2 Next we will show that the principal minors of the Hessian matrix are positive Clearly A deta in view of 6 4 4 A 22 α 2 βγu 3 v 3 > 0 Finally we have Thus 4 α 2 βγu 3 v 3 2 2 αβ u v 2 > 0 4 α 2 β 2 γ 2 u 4 v 4 [ 4 det H A 33 a a 22 a 33 2a 2 a 3 a 23 a a 2 23 a 2 2a 33 a 2 3a 22 A 33 8 6 w 6 6 α 2 β 2 γ 2 u 5 v5 2 α 3 β 3 γ 3 u 6 v6 2 α 3 β 3 γ 2 u 5 v 6 2 6 w 6 α 2 β 3 γ 3 u 6 v6 2 α 3 β 2 γ 3 u 6 v 5 6 2 α 2 β 2 γ 2 u 5 v 5 4w α αβ v w αγu ]

Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 6 2 α 2 β 2 γ 2 u 5 v 5 w w [ w α Stability of the kth order Lyness Equation with a Period-k Coefficient 49 w αβ v w ] αγu 6 2 α 2 β 2 γ 2 u 5 v 5 α αβ v w βγuv αγu w αγu in view of 6 9 is positive Therefore the Hessian matrix H is positive definite at E which implies that the invariant Ix y z attains an isolated minimum at E u vwthat is min{ix y z :x y z 0 3 } Iu vw By Theorem there exists a Lyapunov function V x y z Ix y z Iu v w which shows that the equilibrium E u vw of system 0 is stable In other words the periodthree solution of Eq 4 is stable 4 Higher Order Generalization We now establish the stability of the period-k solution of the kth order Lyness equation 3 k 2 3 p n is a periodic coefficient of period k thatis: p k for n kl p k2 for n kl p n p for n kl k 2 p 0 for n kl k l 0 Equation 3 can be reduced to the following system of equations: x kn x kn x kn x knk2 p k x knk x kn2 x kn x kn x knk3 p k2 x knk2 23 x knk x knk2 x knk3 x kn p x kn x knk x knk x knk2 x kn p 0 x kn By setting w 0 n x kn w n x knw k n x knk 24 we obtain the following system of difference equations n w0 n w n wk2 n p k w k w k wn k2 wk n w0 n n p k2 w k2 n wk3 n 25 wn w2 n w3 n w0 n p wn wn 0 w n w2 n wk n p 0 wn 0 We assume that at least two of the parameters p i k arenotequal The equilibrium equations of system 25 have the form: p k w k 2 w 0 w w k2 p k2 w k2 2 w k w 0 w k3 26 p w 2 w 0 w 2 w k p 0 w 0 2 w w 2 w k Equations 26 imply w i p i w i w 0 w w k i 0 k 27 By using a similar technique as in Lemma 2 we can prove that system 26 has a unique positive solution defined for all p 0 p p k Our next result gives the existence of an invariant for Eq 3 Theorem 4 Equation 3 has an invariant of the form: Ix n x n x nk p n x nk p n x n for n 0 k 2 3 p nk x nk2 x n x nk 28

Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 50 E J Janowski et al Proof Observe that p n is a perodic sequence of period k: Ix n x n x nk2 p n x nk2 x n x nk2 x n x nk2 p n x nk p nk x nk2 x n x nk Ix n x n x nk p n x n p n x n p n x nk Now by using this invariant we can present the following result Theorem 42 For k 2 3 the period-k solution of Eq 3 is stable The Lyapunov function for Eq 3 has the form: V z 0 z z k Iz 0 z z k Iz 0 z z k Proof First we will find the minimizer of Iz 0 z z k The necessary conditions for the existence of critical points gives: I p i zi 2 k j0j i p j z j [ p i z 0 z z k ] 0 for i 0 k so p i z 0 z z k 0 i 0 k 29 Thus the critical point of I satisfies the equilibrium equation 27 this implies that the critical point is exactly the equilibrium point w 0 w k Now we check the Hessian matrix at the positive critical point w 0 w k The second order derivatives of I with respect to are 2 I z 2 i 2 [z 0 p 0 z 0 ] k k p j zj 2 j0j i 2 I [z 0 p 0 z 0 ] k z l k p j zj 2 j0 for i l 0 k i l Letw i for i 0k Then the second order derivatives evaluated at the equilibrium point z 0 z k are 2 I z 0 z k 2p i B z 2 i 2 I z l z 0 z k B B [z 0 p 0 z 0 ] k k p j z 2 j j0 Thus the Hessian matrix takes the form 2p 0 z 0 B B B B 2p z B B H k B B 2p k z k B Now for m 2kthemth order principal minor H m of the Hessian matrix is given by: det H 2p 0 z 0 B>0 det H 2 B 2 4p 0 p z 0 z > 0 det H m B m Q m 2p 0 z 0 2p z Q m 2p m2 z m2 2p m z m By manipulating the rows of Q m we get that Q m 2p m z m Q m 2p m2 z m2 T m

Stability of the kth order Lyness Equation with a Period-k Coefficient 5 Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 2p 0 z 0 2p z T m 2p m3 z m3 m3 2p i Thus for m 2 3 Q m satisfies the following first order difference equation: m2 Q m 2p m z m Q m 2p i 30 Q 2p 0 z 0 whose solution is m m Q m 2p i 2p j z j Hence j0 m det H m B m 2p i m 2p j0 j z j > 0 Indeed by using 27 we obtain: m m 2p i < m m m m m 2p i z 2 i 2p i z 2 i 2 p i z 2 i 2 z 0 z m 2 z 0 z m z 0 z m z 0 z m z 0 z m z 0 z m < Thus the Hessian matrix H k is positive definite so the invariant I attains an isolated minimum at the equilibrium point z 0 z k Thus min{iz 0 z z k :z 0 z z k 0 k } Iz 0 z k so by Theorem V z 0 z z k Iz 0 z z k Iz 0 z k which shows that z 0 z k is stable Therefore the period k solution of Eq 3 is stable Fig The bifurcation diagram of the solution of Eq 4 for β 5γ 2α 00 6 with the initial point 2 4 Fig 2 The bifurcation diagram of the solution of Eq 4 for β 5γ 2 the initial point 2 4 focussed around the value α 38

Int J Bifurcation Chaos 20077:43-52 Downloaded from wwwworldscientificcom by Dr Zehra Nurkanovi on //3 For personal use only 52 E J Janowski et al Note that for k 2k 3wegetLyness Todd s equations respectively The dynamics of Todd s equation are very rich [Cima et al 2006] we believe that the dynamics of Eq 4 are even more complex As an illustration we provide the following bifurcation diagrams in the region of parameter α while β γ are kept fixed see Figs 2 The plots have been obtained by using the software package Dynamica [Kulenović & Merino 2002] References Barbeau E Gelbord B & Tanny S [995] Periodicity of solutions of the generalized Lyness recursion J Diff Eqs Appl 29 306 Bastien G & Rogalski M [2004a] Global behavior of the solutions of Lyness difference equation u n2 u n u n a J Diff Eqs Appl 0 977 003 Bastien G & Rogalski M [2004b] On some algebraic difference equations u n2 u n gu n related to families of conics or cubics: Generalization of the Lyness sequences J Math Anal Appl 300 303 333 Beukers F & Cushman R [998] Zeeman s monotonicity conjecture J Diff Eqs 43 9 200 Cima A Gasull A & Manosa A [2006] Dynamics of some rational discrete dynamical systems via invariants Int J Bifurcation Chaos 6 63 645 Clark C A Janowski E J & Kulenović M R S [2005] Stability of the Gumowski-Mira equation with period-two coefficient J Math Anal Appl 307 292 304 Elaydi S & Sacker R [2005a] Global stability of periodic orbits of nonautonomous difference equations population biology J Diff Eqs 208 258 273 Elaydi S & Sacker R [2005b] Global stability of periodic orbits of nonautonomous difference equations in population biology the Cushing Henson conjectures Proc Eighth Int Conf Difference Equations Applications Chapman & Hall/CRC Boca Raton FL pp 3 26 Feuer J Janowski E & Ladas G [996] Invariants for some rational recursive sequences with periodic coefficients J Diff Eqs Appl 2 67 74 Franke J E & Selgrade J F [2003] Attractors for discrete periodic dynamical systems J Math Anal Appl 286 64 79 Gardini L Bischi G I & Mira C [2003] Invariant curves focal points in a Lyness iterative process Int J Bifurcation Chaos 3 84 852 Grove E A & Ladas G [2004] Periodicities in Nonlinear Difference Equations CRC Press Boca Raton Florida Henson S M & Cushing J M [997] Two effect of periodic habitat fluctuations on a nonlinear insect population model J Math Biol 36 20 226 Kocic V L & Ladas G [993] Global Behavior of Nonlinear Difference Equations of Higher Order with Applications Kluwer Academic Publishers Dordreht/Boston/London Kocic V L Ladas G & Rodrigues I W [993] On the rational recursive sequences J Math Anal Appl 73 27 57 Kulenović M R S [2000] Invariants related Liapunov functions for difference equations Appl Math Lett 3 8 Kulenović M R S & Merino O [2002] Discrete Dynamical Systems Difference Equations with Mathematica Chapman Hall/CRC Boca Raton London Kulenović M R S Ladas G & Overdeep C B [2004] On the dynamics of x n p n x n /x n witha period-two coefficient J Diff Eqs Appl 0 95 928 Sedaghat H [2003] Nonlinear Difference equations Theory with Applications to Social Science Models Mathematical Modelling: Theory Applications Vol 5 Kluwer Academic Publishers Dordrecht Zeeman E C [996] Geometric unfolding of a difference equation preprint Hertford College Oxford