doi:10.1155/2011/958602
Research Article
Dynamics of a Rational System of Difference
Equations in the Plane
Ignacio Bajo,
1Daniel Franco,
2and Juan Per ´an
21Departamento de Matem´atica Aplicada II, E.T.S.E. Telecomunicaci´on,
Universidade de Vigo, Campus Marcosende, 36310 Vigo, Spain
2Departamento de Matem´atica Aplicada, E.T.S.I. Industriales, UNED, C/ Juan del Rosal 12,
28040 Madrid, Spain
Correspondence should be addressed to Juan Per´an,[email protected]
Received 10 December 2010; Accepted 21 February 2011
Academic Editor: Istvan Gyori
Copyrightq2011 Ignacio Bajo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.
1. Introduction
In recent years, rational difference equations have attracted the attention of many researchers for varied reasons. On the one hand, they provide examples of nonlinear equations which are, in some cases, treatable but whose dynamics present some new features with respect to the linear case. On the other hand, rational equations frequently appear in some biological models, and, hence, their study is of interest also due to their applications. A good example of both facts is Ricatti difference equations; the richness of the dynamics of Ricatti equations is very well-knownsee, e.g., 1,2, and a particular case of these equations provides the classical Beverton-Holt model on the dynamics of exploited fish populations3. Obviously, higher-order rational difference equations and systems of rational equations have also been widely studied but still have many aspects to be investigated. The reader can find in the following books 4–6, and the works cited therein, many results, applications, and open problems on higher-order equations and rational systems.
for systems of equations of the type
xn1
α1β1xnγ1yn
A1B1xnC1yn
yn1 Aα2β2xnγ2yn 2B2xnC2yn
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
, n0,1, . . . , 1.1
where the parameters are taken to be nonnegative. As shown in the cited paper, some of those systems can be reduced to some Ricatti equations or to some previously studied second-order rational equations. Further, since, for some choices of the parameters, one obtains a system which is equivalent to the case with some other parameters, Camouzis et al. arrived at a list of 325 nonequivalent systems to which the attention should be focused. They list such systems as pairsk, lwherekandlmake reference to the number of the corresponding equation in their Tables 3 and 4.
In this paper, we deal with the rational system labelled21and 23 in7. Note that, for nonnegative coefficients, such a system is neither cooperative nor competitive, but it has the particularity that denominators in both equations are equal. This allows us to use some of the techniques developed in8to completely obtain the solutions and give a nice description of the dynamics of the system. In principle, we will not restrict ourselves to the case of nonnegative parameters, although this case will be considered in detail in the last section. Hence, we will study the general case of the system
xn1
α1β1xn
yn
yn1 α2yβ2xn
n ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
, n0,1, . . . , 1.2
where the parametersα1, α2, β1, β2 are given real numbers, and the initial conditionx0, y0
is an arbitrary vector ofR2. It should be noticed that whenα
1β2 α2β1 the system can be
reduced to a Ricatti equation or it does not admit any complete solution, which occurs for α2 β2 0 and therefore these cases will be neglected. Since we will not assume
nonnegativeness for neither the coefficients nor the initial conditions, a forbidden set will appear. We will give an explicit characterization of the forbidden set in each case. Obviously, all the results concerning solutions that we will state in the paper are to be applied only to complete orbits. We will focus our attention on three aspects of the dynamics of the system: the boundedness character and asymptotic behavior of its solutions, the existence of periodic orbits in particular, of prime period-two solutions, and the stability of the equilibrium points. It should be remarked that, depending on the parameters, they may appear asymptotically stable fixed points, stable but not asymptotically stable fixed points, nonattracting unstable fixed points, and attracting unstable fixed points.
The paper is organized, besides this introduction, in three sections.Section 2is devoted to some preliminaries and some results which can be mainly deduced from the general situation studied in 8. Next, we study the caseβ2 0 since such assumption yields the
uncoupled globally 2-periodic equationyn1 α2/ynand the system is reduced to a linear
dynamics in the general caseβ2/0. We finish the paper by describing the dynamics in the
particular case where the coefficients and the initial conditions are taken to be nonnegative.
2. Preliminaries and First Results
Systems of linear fractional difference equationsXn1 FXnin which denominators are common for all the components ofFhave been studied in8. If one denotes byqthe mapping given byqa1, a2, . . . , ak1 a1/ak1, a2/ak1, . . . , ak/ak1fora1, a2, . . . , ak1∈Rk1with
ak1/0 and:Rk → Rk1is given bya1, a2, . . . , ak a1, a2, . . . , ak,1, it is shown in such work that the system can be written in the formXn1q◦A◦Xn, whereAis ak1×k1
square matrix constructed with the coefficients of the system. In the special case of our system 1.2one actually has
This form of the system lets us completely determine its solutions in terms of the powers of the associated matrix
Actually, the explicit solution to the system with initial conditionx0, y0is given by
xn1, yn1
tq◦Anx
0, y0,1
t, 2.3
whereMtstands for the transposed of a matrixM. Therefore, our system can be completely solved, and the solution starting atx0, y0is just the projection byqof the solution of the
linear systemXn1 AXn with initial conditionX0 x0, y0,1twhenever such projection
exists.
Remark 2.1. When such projection does not exist, thenx0, y0lies in the forbidden set. Clearly,
this may only happen when, for somen≥1, one has
0,0,1Anx0, y0,1
t
0. 2.4
Therefore, ifain ∈ R, 0 ≤ i ≤ 2 are such thatAn a0nI a1nAa2nA2, then one
immediately obtains that the forbidden set is given by the following union of lines:
The explicit calculation ofain, 0 ≤ i≤ 2 for eachn≥ 3 may be done in several ways. For instance, one has thata0n a1nxa2nx2 is the remainder of the division ofxnby the
characteristic polynomial ofA. Further, by elementary techniques of linear algebra one can also compute them in terms of the eigenvalues ofAan approach using the solutions to an associated linear difference equation may be seen in9.
Remark 2.2. As mentioned in the introduction, all through the paper we will consider that
β2α1/β1α2, 2.6
this is to say that the matrix Ais nonsingularsince the cases with β2α1 β1α2 may be
reduced to a single Ricatti equation. Actually, ifα2 β2 0, then the system does not admit
any complete solution, whereas, forα2/0 orβ2/0, one has that there exists a constantCsuch
thatα1 Cα2andβ1 Cβ2, and hence the first equation of the system may be substituted by
xn1Cyn1and then the second one reduces to the Ricatti equation
yn2
α2β2Cyn1
yn1 , n
0,1, . . . , 2.7
with initial conditiony1 α2β2x0/y0.
Our main goal will be to give a description of the dynamics of the system in terms of the eigenvalues of the associated matrixAgiven in2.2. We begin with the following result concerning 2-periodic solutions which is the particularization to our system of the analogous general result given in Theorem 3.1 and Remark 3.1 of8.
Proposition 2.3. Consider the system1.2withα1β2/α2β1. One has the following:
1Ifβ2/0, then there are exactly as many equilibria as distinct real eigenvalues of the matrix
A. More concretely, for each real eigenvalueλ, one gets the equilibriumλ2−α
2/β2, λ.
2Whenβ20, one finds that:
aifα2<0, then there are no fixed points,
bif0 < α2/β21, then there are two fixed points at α1/√α2−β1,√α2and−α1/
√α2β1,−√α2,
cifα2β21andα1/0, then the only equilibrium point is−α1/2β1,−β1,
difα2β21andα10, then there is an isolated fixed point0,−β1and a whole line of
equilibriax0, β1.
3There exist periodic solutions of prime period 2 if and only ifα1β20.
Proof. As stated in 8, a point a, b ∈ R2 is an equilibrium if and only if a, b,1 is an
eigenvector of the associated matrixA. Whenβ2/0, it is straightforward to prove that, for
each real eigenvalueλ, the vectorλ2−α
2/β2, λ,1is an eigenvector. In the caseβ2 0, the
equilibrium points can be easily computed directly from the equationsα2y2, α1β1xxy.
obviously implies that the trace ofAis also an eigenvalue. Hence,β1 is an eigenvalue, but
this is only possible ifα1β20. Ifα10, then the initial condition0, y0gives a prime period
2 solution whenevery2
0/α2whereas, ifα1/0 andβ2 0, a direct calculation shows that the
solution with initial conditions0,−β1is periodic of prime period 2.
We now study the stability of fixed points in some of the cases. Recall that a fixed point of our systemx∗, y∗always verifiesy∗ λfor some real eigenvalueλof the matrixA. We will say in such case that the fixed pointx∗, y∗isassociatedtoλ.
Proposition 2.4. Consider the system1.2withα1β2/α2β1. LetρAbe the spectral radius of the
matrixAgiven in2.2, and letλbe an eigenvalue ofA.
1If|λ|< ρA, then the associated equilibrium is unstable.
2If |λ| ρAand all the eigenvalues of A whose modulus isρAare simple, then the associated fixed point is stable. Further, if in this case λis the unique eigenvalue whose modulus isρA, then it is asymptotically stable.
Proof. The Jacobian matrix of the mapFx, y α1β1x/y,α2β2x/yat a fixed point
x∗, y∗is given by
DFx∗, y∗
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
β1
y∗ −x∗/y∗
β2
y∗ −1
⎞ ⎟ ⎟ ⎟ ⎟
⎠. 2.8
Consider an eigenvalue λ of A, and let λ2, λ3 be the other nonnecessarily different
eigenvalues ofA. Let us show that the eigenvalues of the Jacobian matrix at a fixed point associated toλare justλ2/λandλ3/λ. The result is trivial whenβ2 0 since the eigenvalues
ofAareβ1and±√α2and fixed points are always associated to one of the eigenvalues±√α2.
Ifβ2/0, thenx∗ λ2−α2/β2andy∗λand, therefore, one obtains
traceDFx∗, y∗ β1−λ
λ λ2λ3
λ
detDFx∗, y∗ −β1λλ
2−α 2
λ2
detA
λ3
λ2λ3
λ2 ,
2.9
showing that the eigenvalues of DFx∗, y∗are as claimed. Now, the first statement follows at once since, if|λ|< ρA, then at least one of the eigenvalues of DFx∗, y∗lies outside the unit circle. Moreover, when|λ|ρAand it is the unique eigenvalue with such property, then the eigenvalues of DFx∗, y∗are inside theopenunit ball, and, hence, the equilibriumx∗, y∗ is asymptotically stable, which proves the second part of2.2.
For the proof of the first part of2.2, let us recall that ifx∗, y∗is a fixed point of1.2
associated to the real eigenvalueλ, thenX∗ x∗, y∗,1tis a fixed point of the linear system
Xn1 1/λAXn. The eigenvalues of the matrixM 1/λAare obviously 1, λ2/λandλ3/λ.
3. Case
β
20
Recall that, since we are assuming that inequality2.6holds, we haveβ1α2/0. In this case,
the forbidden set of the system reduces to the liney0. Sinceβ2 0, the second equation of
the system becomes the uncoupled equation
yn1 yα2
Substituting such values in the first equation of the system, we obtain a first-order linear difference equation with 2-periodic coefficients whose solution is given byx1 α1β1x0/y0
Hence, we have proved the following.
From the proposition above, one can easily derive the following result which completely describes the asymptotic behaviour of the solutions to the system.
Corollary 3.2. Considerβ20andβ1α2/0.
1Whenβ21α2one finds that
aif α1/0, then every solution to the system is unbounded except those with initial
conditionx0,−β1, which are 2-periodic,
bifα10, the system is globally 2-periodic.
2Ifβ2
1 −α2, then the system1.2is globally 4-periodic. Further, the solution corresponding
with the initial conditionx0, y0is of prime period 2 if and only if2β21x0α1β1y0 0.
3Ifβ2
1/|α2|, then the solutions with initial conditionα1β1y0/α2−β21, y0are
period-two solutions. Moreover,
aifβ2
1>|α2|, then any other solution to the system1.2is unbounded,
bifβ21<|α2|, then any other solution of 1.2is bounded and tends to one of the
period-two solutions described above.
Proof. The proof is a straightforward consequence of the explicit formulas for xn and yn given in Proposition 3.1. It should, however, be mentioned that the globally periodicity of the system in the case β2
1 −α2 can be easily seen since the associated matrixAgiven by
2.2in such case verifiesA4β4
1I, whereIstands for the identity matrix. Actually, a simple
calculation proves that the solution starting atx0, y0is the 4-cycle
x0, y0
,
α1β1x0
y0 ,
−β2 1
y0
,
−x0−
α1
β1y0
β2 1
, y0
,
−β2
1x0α1y0
β1y0 ,
−β2 1
y0
, 3.6
which is obviously 2-periodic if and only ifx0−x0−α1β1y0/β21.
From the above result andProposition 2.4, one easily get the following information about the stability of the fixed points.
Corollary 3.3. Considerβ20andβ1α2/0.
1Ifβ2
1α2, then
aforα1/0, the unique fixed point of 1.2is unstable,
bforα10, every fixed point of 1.2is stable but not asymptotically stable.
2Ifβ2
1/α2>0, then
aforβ2
1 > α2, both fixed points of 1.2are unstable,
bforβ2
4. Case
β
2/
0
Proposition 4.1. Supposeβ2/0andx0, y0is an initial condition not belonging to the forbidden
setF. In such case, the solution of system1.2is given by
xn vvn1 n−1
1
β2 −
α2
β2, yn
vn
vn−1, 4.1
wherevnis the unique solution of the linear difference equation
vn3−β1vn2−α2vn1
β1α2−β2α1
vn0, 4.2
with initial conditionsv−11, v0y0, andv1 β2x0α2.
Proof. As we have seen inSection 2, the solution to System1.2starting at a pointx0, y0not
belonging to the forbidden set is just the projection byqof the solution of the linear system un1, vn1, wn1tAun, vn, wntwith initial conditionx0, y0,1t, whereAis given by2.2.
Since the third equation of such linear systems readswn1vn, it can be reduced to the planar linear system of second-order equations
un1β1unα1vn−1,
vn1β2unα2vn−1,
4.3
and hence, ifun, vnis the solution to4.3obtained for the initial conditionsu0, v0, v−1
x0, y0,1, then the solution of our rational system for the initial valuesx0, y0will be
xn1 vun
n−1, yn1
vn
vn−1. 4.4
It is clear that forβ2/0, we have thatun can be completely determined by4.3in terms of
vn1andvn−1, and hence it suffices to solve the third-order linear equation
vn3−β1vn2−α2vn1
β1α2−β2α1
vn0 4.5
trivially deduced from4.3and substitute the corresponding values in 4.4 to obtain the result claimed.
In the following results, we will discuss the behavior of the solutions to 1.2 by usingProposition 4.1. We shall consider three different cases depending on the roots of the characteristic polynomial of the linear equation 4.2. Recall that such roots are also the possibly complexeigenvalues of the matrixAgiven in2.2.
FromProposition 4.1, we see that the asymptotic behavior of the solutions of System 1.2will depend on the asymptotic behavior of the sequencesvn/vn−1,vn being solutions of the linear difference equation4.2. The theorem of Poincar´e2, Theorem 8.9establishes a general result for the existence of limn→ ∞vn/vn−1. In our case, since4.2has constant
4.1. The Characteristic Polynomial Has No Distinct
Roots with the Same Module
Letλ1, λ2, andλ3 be the three roots of the characteristic polynomial of the linear difference
equation4.2in this case. A condition on the coefficients for this case can be given by
2/3β1α2−β2α1−2/27β31
2
2
≤
α2 1/3β21
3
3
, 4.6
withα1/0 orα2 ≤0. Recall that we assume here thatβ2α1/β1α2andβ2/0.
Ifλ1is the characteristic root of maximal modulus, we will denote byLthe line
Lx, y:β2x
β1−λ1
yλ1
. 4.7
Proposition 4.2. Suppose that β2/0 and every root of the characteristic polynomial of the linear
difference equation4.2is real and no two distinct roots have the same module. Whenx0, y0is not
in the forbidden set, one finds the following:
1If|λ1|>|λ2|>|λ3|, then
aSystem1.2admits exactly the three equilibriaλ2
i −α2/β2, λi, i1,2,3,
bthe fixed pointλ2
1−α2/β2, λ1attracts every complete solution starting on a point
x0, y0which does not belong to the lineL,
cthe corresponding solution to the system with initial condition x0, y0/ λ23 −
α2/β2, λ3andx0, y0∈Lconverges toλ22−α2/β2, λ2.
2If|λ1|>|λ2|andλ1has algebraic multiplicity 2, then
aSystem1.2admits exactly the two equilibriaλ2
i −α2/β2, λi, i1,2,
bthe fixed point λ2
1−α2/β2, λ1attracts every complete solution except the other
fixed point.
3If|λ1|>|λ2|andλ2has algebraic multiplicity 2, then
aSystem1.2admits exactly the two equilibriaλ2
i −α2/β2, λi, i1,2,
bthe fixed pointλ2
1−α2/β2, λ1attracts every complete solution starting on a point
x0, y0which does not belong to the lineL,
cthe corresponding solution to the system with initial conditionx0, y0∈Lconverges
toλ2
2−α2/β2, λ2.
4Ifλ1has multiplicity 3, then
aSystem1.2has a unique equilibriumλ2
1−α2/β2, λ1,
Proof. In all the cases, the equilibrium points are directly given by Proposition 2.3. The assertions concerning the asymptotic behaviour can be derived as a consequence of Case 1 in2, page 240, bearing in mind that
xn vvn1 n−1
1
β2 −
α2
β2, yn
vn
vn−1, 4.8
and thatvnis the solution to the linear equation4.2with initial conditionsv−11,v0 y0,
andv1β2x0α2.
4.2. The Characteristic Polynomial Has Two Distinct Real
Roots with the Same Module
It is easy to check that this case occurs whenβ1/0, β2/0, α10 andα2>0. Thus, the roots
of the characteristic polynomial of the linear difference equation4.2areβ1and±√α2.
Proposition 4.3. Supposeβ1/0, β2/0, α10andα2 >0. Assume also thatx0, y0is not in the
forbidden set.
1Ifβ2
1α2, then
athere are two equilibrium points0,±β1,
bthe equilibrium point0, β1attracts every complete solution not starting on a point
of the linex0,
cthe solutions starting on a point x0, y0 of the linex 0 are prime period-two
solutions except the two equilibrium points0,±β1.
2Ifβ2
1> α2, then
athere are three equilibrium pointsβ2
1−α2/β2, β1and0,±√α2,
bthe equilibrium pointβ2
1−α2/β2, β1attracts every complete solution not starting
on a point of the linex0,
cthe solutions starting on a point x0, y0 of the linex 0 are prime period-two
solutions except the two equilibrium points0,±√α2,
3Ifβ2
1< α2, then
athere are three equilibrium pointsβ2
1−α2/β2, β1and0,±
√α
2,
bthe solutions starting on a point of the linex0are prime period-two solutions except the two equilibrium points0,±√α2,
cthe solutions starting on a point of the linesβ2x α2−β21/β1y0orx β21−
α2/β2are unbounded with the only exception of the fixed pointβ21−α2/β2, β1,
dthe solutions starting on any other pointx0, y0are bounded and each tends to one
of the two-periodic solutions.
When β21 α2, the roots are β1, with algebraic multiplicity two, and −β1. By
Proposition 4.1, we know that any solution of the system can be written as
β2xn n1P1P2P3−1 n1
n−1P1P2P3−1n−1
β2 1−β21,
yn nP1P2P3−1 n
n−1P1P2P3−1n−1
β1,
4.9
whereP1, P2, andP3actually satisfy
P1P2−P3
β2x0β21
β1 , P2P3y0, −P1P2−P3β1.
4.10
IfP1/0, thenxn, ynobviously tends to0, β1. From4.10, we see thatP1 0 if and only
ifx0 0 and, in such case,xn 0 andyn takes alternatively the values Aβ1andA−1β1with
A P2P3/P2−P3. Notice thaty0/0 guarantiesP2P3/0 and, sinceβ1/0, we can not
haveP10 andP2−P30. This completes the proof of1.2.
In the caseβ21/α2, byProposition 4.1, we can write the general solution of the system
as
β2xn
P1
P2P3−1n1
√
α2/β1
n1
P1
P2P3−1n−1 √α2/β1
n−1β 2 1−α2,
yn P1 !
P2P3−1n
"√α
2/β1
n
P1
P2P3−1n−1
√
α2/β1
n−1β1,
4.11
whereP1,P2, andP3satisfy
P1β1 P2−P3√α2β2x0α2,
P1P2P3y0,
P1β−11 P2−P3
#
α−1
2 1.
4.12
Whenβ2
1 > α2, one immediately gets the results of statement2.2with an argument similar
to that of the previous case. Therefore, we will focus our attention on the caseβ2
1 < α2. The
conditionx00 is, according to4.12, equivalent toP10, and, in such case, one getsxn0 andyntakes alternatively the valuesK√α2andK−1√α2withK P2P3/P2−P3 y0/α2.
Now, ifP1/0 and the initial conditions are taken such thatP2P3/0/P2−P3, thenxn, yn
tends obviously to the 2-cycle{0, K√α2,0, K−1√α2}whereK P2P3/P2−P3. On the
contrary, if eitherP2P30 orP2−P30and only one of both equalities holds, then both
sequencesxnandynare unbounded. From System4.12, one gets thatP2−P3 0 if and only
ifx0 β21−α2/β2and thatP2P30 is equivalent toβ2x0 α2−β12/β1y00. This shows
4.3. The Characteristic Polynomial Has Complex Roots
Now, we consider the case in which the characteristic polynomial of the linear difference equation has a couple of complex rootsρe±iθ, with sinθ >0. Letλ /0 be the real root. It can be easily shown that
β1λ2ρcosθ, α2−
$
2λρcosθρ2%, β2α1λρ2β1α2, 4.13
and that this situation occurs when
ByProposition 2.3, we know that the unique equilibrium isλ2−α
2/β2, λ. Denote byLthe
Theorem 4.4. Supposeβ2/0and the characteristic polynomial of the linear difference equation have
complex roots and assume thatx0, y0is not in the forbidden set.
1The solutions starting on the line L remain on it, and they are either all periodic or all unbounded.
2If|λ|> ρ, then the unique equilibrium attracts all the solutions not starting onL. 3If|λ|< ρ, then every nonfixed bounded subsequence of a solution accumulates onL. 4If|λ|ρ, then every complete solution (neither starting on the fixed point nor onL) lies on
a nondegenerate conic, which does not contain the equilibrium.
Proof. Assume thatx0, y0is not the fixed point. UsingProposition 4.1, we have
Let us consider the sequences
σn2 $ρ
λ
%n
cosanθ, τn2 $ρ
λ
%n
sinanθ. 4.18
It can be easily proved that
α2β2xnλ2PPσσn1
n−1, ynλ
Pσn
Pσn−1,
4.19
λσn1ρσncosθ−ρτnsinθ, ρσn−1λσncosθλτnsinθ. 4.20
As a consequence,λ2σ
n1−2λρσncosθρ2σn−10, and then
α2β2xn2ρyncosθ−ρ2Pλ
2−2ρλcosθρ2
Pσn−1 ,
4.21
which is equivalent to
β2xn−β1−λ
ynλPλ
2−2ρλcosθρ2
Pσn−1 .
4.22
Using4.17, one has thatx0, y0∈Lif and only ifP 0, and, from4.22, we then get that
xn, yn∈Lfor alln≥1.
Furthermore, by4.19, we see that ifx0, y0∈L, then the solutionxn, ynis periodic wheneverθ/π is a rational number and unbounded otherwise.
Assume now that the solutionxn, yndoes not start onL, this to say,P /0. We will now distinguish the three cases:|λ|> ρ,|λ|< ρ, and|λ|ρ.
If|λ|> ρ, then by4.19, one immediately hasxn → λ2−α2/β2andyn → λ.
Suppose now that|λ|< ρ. Ifxnk, ynkis a subsequence satisfying that infk|cosank− 1θ|>0, then one obviously hasσnk−1 → ∞. Using the definition ofσn, one easily gets that
σnk/σnk−1is bounded. Then,xnk, ynkis a bounded subsequence, and4.22shows that it is attracted by the lineL.
On the other hand, if cosa nk−1θ → 0, then the left equation in4.20leads us to |σnkλ/ρnk| → 2 sinθ >0. Thus,σnk → ∞and, using4.20once more, we getσnk/σnk−1 →
∞. Therefore,xnk, ynkis an unbounded subsequence.
Finally, let us supposeρ|λ|. If we consider the change of variables
x$β2xα2−ρ2
% λ
2λcosθ−2ρ −
y−λλρλcosθ
cosθ−ρ,
y$β2xα2−ρ2
% 1
2 sinθ −
y−λ
λρcosθ
sinθ ,
then one may deduce from 4.20 that xn ρλσn−1/P σn−1, yn ρλτn−1/P σn−1.
Therefore, one immediately gets that
xn2yn24
ρλ2
Pσn−12
, xn−ρλ2P2
ρλ2
Pσn−12
, 4.24
which clearly shows thatxn, ynlies in the conicx2y2 4/P2x−ρλ2, having its focus in0,0, its directrix in the linex ρλand eccentricity 2/P. Further, one immediately sees that the fixed pointλ2−α
2/β2, λis transformed by the change of variables above in0,0
and, hence, it does not belong to the conic.
Remark 4.5. In the case |λ| < ρ of this last theorem, one might conjecture that every subsequence of a solutioneven a nonbounded oneactually approaches the lineL, but this is not the case. Let us take, for example, the system withα11, andβ13, α2−4, β2−10, in
which the characteristic roots of the associated polynomial are given byλ1 and√2eiπ/4and
consider the solution starting onx0, y0 −11/20,3/2. We then have thata0, P 1, and
σ24k0 for allk≥0. One may use4.22to show that all the points of the formx34k, y34k lay on the line 10x2y30 while the lineLis given by 10x2y20. Note, however, that the subsequencesx4k, y4k,x14k, y14k, andx24k, y24kare all bounded and converge respectively to−3/5,2,−2/5,1, and−1/5,0, which do belong toL.
It should also be noticed that the fixed point lays on the line 10x2y3 0. This is also the case in the general setting. It follows from4.22that wheneverσnk−1 0 then the
pointxn−k, yn−kis on the line containing the fixed point which is parallel toL.
Remark 4.6. Notice that, according to the results in8, when|λ| ρand the argument θof the complex root is a rational multiple ofπ, the system is globally periodic.
4.4. Stability of Fixed Points
We finish this section with the complete study of the stability of the fixed points in the case
β2/0.
Theorem 4.7. Suppose thatβ2/0, let λ be a real eigenvalue of the matrix A given in 2.2. Let
λ2−α
2/β2, λbe the associated fixed point and denote byρAthe spectral radius ofA.
1If|λ|< ρA, then the associated fixed point is unstable.
2If|λ|ρA, then the associated equilibrium is stable if and only if every eigenvalue whose modulus isρAis a simple eigenvalue. Moreover, the stability is asymptotic if and only if
λis a simple eigenvalue and it is the unique eigenvalue ofAwhose modulus isρA.
Proof. The first statement was already proved inProposition 2.4. Besides, in such proposition, we have shown that if every eigenvalue whose modulus isρAis simple then the associated equilibrium is stable. Let us prove the converse.
According to the results of the previous subsections, the only cases in which one has a nonsimple eigenvalue of maximal modulus are the cases treated inProposition 4.21and 4and the first case ofProposition 4.3. We will see that in such cases the equilibrium points associated to eigenvalues of maximal modulus are unstable.
We begin with the case of an eigenvalueλ1 of maximal modulus with multiplicity 2.
For eachN ∈ N, N > 1, one may consider the solution with initial conditions x0, y0
λ2
given byvn λn11N21−Nn−N/N21, which cannot vanish sinceN > 1. For this
solution, one has|yN−λ1||λ1|N, proving that the equilibriumλ21−α2/β2, λ1is unstable.
Similarly, ifAhas a unique eigenvalueλof multiplicity 3 then, for eachN∈N, N /0 let us considerx0, y0 λ2−α2/β2−2λ2/β2N2, λ. The corresponding solution to4.2
is given byvn N2−n−n2λn1/N2. It is not difficult to see thatvn/0 for alln≥1, and then the solution to our System1.2is complete. Further, sinceyn vn/vn−1, one gets that
|yN−λ|2|λ|. Therefore, the fixed pointλ2−α2/β2, λis not stable.
Whenα1 0, β21 α2/0, there are two equilibrium points associated to eigenvalues
of maximal modulus: 0,±β1. The fixed point 0,−β1 is, according to the result of
Proposition 4.3, unstable since the other equilibrium attracts all the solutions not starting on the line x 0. To see that 0, β1is also unstable, let us choose, for each odd N ∈ N,
the solution starting atx0, y0 −2β21/Nβ2, β1. Then, using4.10and the expression foryn given just above such equation, we havevnβn11P1nβn1ifnis even andvn βn11P1n1βn1
ifnis odd, whereP1−β1/N. SinceNis odd, we see thatynexists for alln∈Nand, further, we get that|yN−β1|2|β1|, which clearly implies that0, β1cannot be stable.
Finally, it only remains to prove that whenAhas distinct simple eigenvalues whose modulus equalρA, then the fixed point is not asymptotically stable, but this situation can only happen if either one has the situation described inProposition 2.34or the one given inProposition 4.33. In the case of complex eigenvalues, we had seen that all the orbits lie on conics not going through the fixed point, and, hence, it cannot be asymptotically stable. In the other case, it is clear that the fixed points0,±√α2are not attracting, since every solution
starting on the linex0 is 2-periodic.
Remark 4.8. It is interesting to notice that, in the three cases in which there is an eigenvalue of maximal modulus with multiplicity larger than 1, the corresponding fixed point is attracting but unstable.
5. Nonnegative Solutions to the System with
Nonnegative Coefficients
When the coefficients of our System 1.2 are nonnegative and we restrict ourselves to nonnegative initial conditions, many of the cases studied in the previous sections cannot appear. Further, in such case, one may describe which kind of orbits appear and their asymptotic behaviour without the previous calculation of the characteristic roots.
It should be noticed that whenever the coefficients in System1.2are nonnegative and
α1β2/α2β1, every initial conditionx0, y0withx0 ≥0, y0 >0 gives rise to a complete orbit
except forα20 where the conditionx0>0 is also necessary.
It will be convenient to independently study the case α1β2 0. The next result is a
simple summary of the results inSection 3andProposition 4.3, and, hence, we omit its proof.
Corollary 5.1. Consider that the coefficients in System1.2are nonnegative andα1β20/α2β1.
1Ifβ20, one has the following.
aWhenα2 ≤β21, there are no nonnegative periodic orbits and all nonnegative solutions
are unbounded, with the only exception of the caseα2β21, α10, which is globally
2-periodic.
bWhenα2> β21, there exists a nonattractive fixed pointα1/√α2−β1,√α2and the
whole lineα2−β21x0 α1β1y0of 2-periodic solutions. Every other nonnegative
2Ifβ2/0α1, then every nonnegative solution is bounded and the ones starting in the line
x00are 2-periodic. Moreover,
awhen α2 < β21, there are two nonnegative fixed points: β21 −α2/β2, β1, which
attracts all nonperiodic nonnegative solutions, and0,√α2.
bWhenα2 β21, there is a unique nonnegative equilibrium 0, β1which attracts all
nonperiodic nonnegative solutions.
cWhen α2 > β21, the unique nonnegative equilibrium is 0,
√α
2 which is not an
attractor. Every nonnegative solution converges to one of the periodic solutions.
The remaining cases are jointly treated in the following result. All the definitions and results on nonnegative matrices, which are used in its proof, may be found in10, Chapter 8.
Proposition 5.2. Suppose that System1.2has nonnegative coefficients and thatα1β2/0.
1Ifα2/0orβ1/0, then there is a unique nonnegative (actually, positive) stable equilibrium
which attracts all nonnegative solutions.
2Ifα2β10, the system is globally 3-periodic with a unique equilibrium.
Proof. Let us consider A as in 2.2. A simple calculation shows that AI2 is positive and, therefore, Ais irreducible. Then, the spectral radiusρAis a strictly positive simple eigenvalue ofA.
If there exists another eigenvalueλsuch that|λ|ρAthen, sinceAis nonnegative and irreducible, the eigenvalues of A should be λk1 ρAeikπ/3 where k 0,1,2 and,
consequently,A3 ρA3I. The direct computation ofA3 shows that this is possible if and
only ifα2β1 0 and, hence, in that case, the system is 3-periodic, and the only equilibrium
is the one associated to the real eigenvalueρA.
In the remaining cases, λ1 ρA is a dominant eigenvalue and, according to our
results of Propositions2.4,4.2andTheorem 4.4, the corresponding fixed point is stable and attracts all complete solutions except those starting on the line
Lx, y:β2x
β1−λ1
yλ1
. 5.1
Sinceλ1is the largest eigenvalue ofA, one has that detA−μI<0 for allμ > λ1. However,
detA−β1I α1β2>0, showing thatβ1 < λ1. Thus, for everyx0≥0 andy0 >0, one obtains
β2x0≥0 andβ1−λ1y0λ1<0, which proves thatx0, y0∈/L.
The equilibrium associated to the eigenvalueλ1 ρAisλ21−α2/β2, λ1, which is
positive since, as before, one sees that detA−√α2I α1β2>0 and henceλ1 >√α2.
Acknowledgments
The authors want to thank Professor Eduardo Liz for his useful comments and suggestions. This paper was partially supported by MEC Project MTM2007-60679.
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