Global Dynamics of Cubic Second Order Difference Equation in the First Quadrant

We investigate the global behavior of a cubic second order diﬀerence equation n 0,1, ... , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of period-two solutions. We then use this result to give global behavior results for special ranges of the parameters and determine the basins of attraction of all equilibrium points. We give a class of examples of second order diﬀerence equations with quadratic terms for which a discrete version of the 16th Hilbert problem does not hold. We also give the class of second order diﬀerence equations with quadratic terms for which the Julia set can be found explicitly and represent a planar quadratic curve.


Introduction and preliminaries
In this paper we study the global dynamics of the following polynomial difference equation: x n+ = Ax  n + Bx  n x n- + Cx n x  n- + Dx  n- + Ex  n + Fx n x n- + Gx  n- + Hx n + Ix n- + J, () where the parameters A, B, C, D, E, F, G, H, I, J are nonnegative numbers with condition A + B + C + D >  and initial conditions x - and x  are arbitrary nonnegative numbers. The condition A + B + C + D >  is necessary in order to avoid the quadratic case, which was completely studied in []. Polynomial difference equations and corresponding maps have been studied in both the real and the complex domain and many results have been obtained; see [-]. One of the major problems in the dynamics of polynomial maps is determining the basin of attraction of the point at ∞ and in particular the boundary of that basin known as the Julia set. In [] we precisely determined the Julia set for a second order quadratic polynomial equation, that is, () where A + B + C + D = , and we obtained the global dynamics in the interior of the Julia set, which includes all the point for which solutions are not asymptotic to the point at ∞. It turned out that the Julia set for (), where A + B + C + D = , is the union of the stable manifolds of some saddle equilibrium points or nonhyperbolic equilibrium points or period-two points. The asymptotic formulas for these manifolds were obtained in []. The advantage of our results is that these manifolds are continuous decreasing functions of which the parametrization is simple and so their asymptotic formulas can easily be derived by using the method of undetermined coefficients.
In this paper we restrict our attention to nonnegative initial conditions and nonnegative parameters, which will make our results more special but also more precise and applicable. Our results are based on a number of theorems which hold for monotone difference equations, which will be described in the next section. Our principal tool is the theory of monotone maps, and in particular cooperative maps, which guarantee the existence and uniqueness of the stable and unstable manifolds for the fixed points and periodic points. Our results can be extended to () to hold in the whole plane, when B = C = E = F = G = . The curve H separates the first quadrant into two regions: the region below the curve H is the basin of attraction of E  (, ) and the region above the curve is the basin of attraction of point at infinity. Thus, the curve H is the Julia set for () in this case. This result shows that the discrete version of the th Hilbert problem does not hold, which is the problem if there exists a quadratic system of difference equations in the plane with an infinite number of periodic solutions. It is well known that in the case of quadratic systems of differential equations the number of periodic solutions is finite; see [, ]. In this paper we give the explicit formula for the Julia set for a whole class of difference equations with cubic terms. The Julia set consists of an infinite number of period-two solutions and thus provides the whole class of examples of the second order difference equations with the cubic terms with an infinite number of a period-two solutions; see Theorem .

Remark 
The rest of this section presents some results as regards monotone difference equations in the plane. The second section presents the local stability analysis of the equilibrium solutions. The third section describes the local stability analysis of the period-two points in all cases. The fourth section gives the global dynamics, which includes the basins of attraction of all equilibrium points and the period-two points. Some Mathematica outputs are given in the Appendix.
Consider the difference equation x n+ = f (x n , x n- ), n = , , . . . , where f is a continuous and increasing function in both variables.
Here we list some of the results that will be needed in this paper. The first result was obtained in [] and it was extended to the case of higher order difference equations and systems in [, ].

Theorem  Let f : [a, b] × [a, b] → [a, b] be a continuous function satisfying the following properties:
(i) f (x, y) is nondecreasing in each of its arguments, i.e. x → f (x, y) is nondecreasing for every y and y → f (x, y) is nondecreasing for every x; (ii) () has a unique equilibrium x ∈ [a, b]. Then every solution of () converges to x.
The following result was obtained in []. (iii) One of them is monotonically increasing and the other is monotonically decreasing.
As a consequence of Theorem  every bounded solution of () approaches either an equilibrium solution or period-two solution or the singular point on the boundary and every unbounded solution is asymptotic to the point at infinity in a monotonic way; see []. Thus the major problem in the dynamics of () is the problem of determining the basins of attraction of three different types of attractors: the equilibrium solutions, periodtwo solution(s), and the point(s) at infinity. The following two results can be proved by using the techniques of the proof of Theorem  in [].
Theorem  Consider () where I ⊆ R is an interval (finite or infinite) and f ∈ C[I × I, I] is an increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E  (x  , y  ) and E  (x  , y  ) are two consecutive equilibrium points in North-East ordering that satisfy (x  , y  ) ne (x  , y  ) and that E  is a local attractor and E  is a saddle point or a nonhyperbolic point with second characteristic root in the interval (-, ), with the neighborhoods where f is strictly increasing. Then the basin of attraction B(E  ) of E  is the region below the global stable manifold W s (E  ). More precisely The basin of attraction B(E  ) = W s (E  ) is exactly the global stable manifold of E  . The global stable manifold extends to the boundary of the domain of (). If there exists a periodtwo solution, then the end points of the global stable manifold are exactly the period-two solution.
Also, we will use the following theorem from [].
Theorem  Consider () where I ⊆ R is an interval (finite or infinite) and f ∈ C[I × I, I] is an increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E  (x  , y  ), E  (x  , y  ), and E  (x  , y  ) are three consecutive equilibrium points in North-East ordering that satisfy and that E  and E  are saddle points with the neighborhoods where f is strictly increasing, and E  is a local attractor. Then the basin of attraction B(E  ) of E  is the region between the global stable manifolds W s (E  ) and W s (E  ). More precisely The basins of attraction B(E  ) = W s (E  ) and B(E  ) = W s (E  ) are exactly the global stable manifolds of E  and E  .
The following result gives the necessary and sufficient condition for the local stability of () when f is nondecreasing in all its arguments; see [].
be the standard linearization about the equilibrium x of () Then the equilibrium x of () is one of the following: (a) locally asymptotically stable if p  + p  < , (b) nonhyperbolic and locally stable if Let f  (x) := a n x n + a n- x n- + · · · + a  x + a  and g  (x) := b m x m + b m- x m- + · · · + b  x + b  be two polynomials of degrees n and m, respectively. Their resultant (see [-]) Res(f  , g  ) is the determinant of the (m + n) × (m + n) Sylvester matrix given by a n a n- · · · a  a   · · ·   a n a n- · · · a  a  · · ·  . . .


· · · a n a n- a n- The following theorem is from [].
Theorem  Assume that f  , g  ∈ R[x] and n, m ≥ . Then the following hold: The discriminant of f  is given by and only if the discriminant is = .
be the resultant of f and g with respect to the variable y. Then: (a) f and g have a nontrivial common factor if and only if r is identically zero.
(b) If f and g are co-prime (do not have a common factor), the following conditions are equivalent: We list some results when solution {x n } of () tends to the point at infinity in a monotonic way.
Theorem  If H + I > , then every solution {x n } of () satisfies lim n→∞ x n = ∞.
Proof If {x n } is a solution of (), then {x n } satisfies the inequality x n+ ≥ Hx n + Ix n- , n = , , . . . , which in view of the result on difference inequalities, see Theorem .. in [], implies that x n ≥ y n , n ≥ , where {y n } is a solution of the initial value problem y n+ = Hy n + Iy n- , y - = x - and y  = x  n = , , . . . . Consequently, where λ , = H ± √ H  + I  and λ  < λ  and λ  > , which implies lim n→∞ x n = ∞.
Theorem  If J ≥ , then every solution {x n } of () satisfies lim n→∞ x n = ∞.
Proof Let J ≥  and set A + B + C + D + E + F + G + H + I = α > . If {x n } is a solution of (), then x n > J for n ≥ , and by using the fact that the function is increasing in both variables we obtain the following result: . . .

Continuing in this way we obtain
x n+ > ( + α) n J and x n+ > ( + α) n J for all n ≥ , which implies x n+ → ∞ and x n+ → ∞.
 is a part of the basin of attraction of the point at infinity of ().

Proof Set
Clearly, the function f (x, y) is increasing in both variables. If x, y ∈ [, ∞), then Since the initial conditions x - , x  ∈ [, ∞), An immediate application of mathematical induction yields x n- ≥ k n + k n- + k n- + · · · +  J, x n > k n + k n- + k n- + · · · +  J for all n ≥ . If k > , then k n → ∞ and if k =  then k n- + k n- + · · · +  → ∞. In both cases we get x n- → ∞ and x n → ∞.
Proof If {x n } is a solution of (), then {x n } satisfies the inequality In view of the results for () in [], see Lemma , there is no prime period-two solution when (H + I -)  ≤ J(E + F + G) or E ≥ G or H + I > . So if there is no equilibrium point and there is no prime period-two solution of (), then every solution {y n } of () satisfies lim n→∞ y n = ∞, which implies lim n→∞ x n = ∞.
Remark  Theorems , , , and  describe the parametric region which is the part of the basin of attraction of the point at ∞ called the escape region. The remaining part of the escape region will be described more precisely in the next sections. We will precisely describe the boundary of the escape region in all situations when the equilibrium points and period-two points are hyperbolic. In particular, we will describe the manifold that solutions will be following on their way to ∞. In general, it is clear from the proof of Theorem  that the escape region of () is a subset of the escape region of (). In the subsequent sections we will consider global dynamics of () in the complement of the parametric region described by Theorems , , , and .

Local stability analysis of equilibrium solutions
The equilibrium solutions of () are the nonnegative solutions of the equation Define the function g(x) such that so the nonnegative solutions of g(x) =  are the nonnegative equilibrium solutions of ().

The case J > 0
Since g(-∞) = -∞ and g() = J > , we always have one negative root, which means that there are at most two nonnegonnegative equilibrium solutions. The first derivative of g(x) is Denote by the discriminant of quadratic equation g (x) = , that is, We have: (i) If < , then g (x) >  for all x and the function g(x) is monotonically increasing, which implies that there is no nonnegative root of g(x) = .
(ii) If = , then Thus, g(x) >  for all x ≥ , which implies that () has no positive solutions.
(iii) If > , then for the roots x , = -(E+F+G)± √ (A+B+C+D) (x  < x  ) of g (x) =  we get the following: which means that at least x  is negative, and From () we see that the following statements hold: • If H + I > , then () has no positive solutions.
• If H + I = , then x  =  and g(x) ≥ g(x  ) = J >  so there is no positive solution of g(x) = .
This case is possible, for example when the values of parameters are (c) If g(x  ) <  there are two positive solutions x  ∈ (, x  ) and x  ∈ (x  , +∞) of g(x) = . The calculation of g(x  ) gives After a straightforward calculation we get Also, a straightforward calculation yields

The case J = 0
In this case () becomes so we always have the zero equilibrium. Denote by  discriminant of quadratic equation that is The following statements are immediate: (i) If  < , then the zero equilibrium is the only equilibrium.
(ii) If  = , then a solution of the quadratic equation () is given by .
If E + F + G >  there is no positive solution. E + F + G =  implies H + I = , and again the zero equilibrium is the only equilibrium.
(iii) If  > , then for equilibriums we have x  <  and Now, we see that the following statements hold: where where g(x) is given by (). Let x  and x  (x  < x  ) be the roots of g (x) = . By applying Theorem  and the local stability theorem from [] we obtain the following result on the local stability of the positive equilibrium(s).
Proposition  The positive equilibrium solution of () is one of the following: The next theorems will describe the local stability for the positive equilibrium(s) in more detail.
Proof (a) When x ∈ (, x  ), then g (x) < , and for x  ∈ (, x  ) we get so by applying Theorem  we see that x  is locally asymptotically stable. When x ∈ (x  , +∞) then g (x) > , and for x  ∈ (x  , +∞) we get so by applying Theorem  we see that x  is unstable. By using the Proposition  we obtain the following result: It remains to determine qp for equilibrium x  : (b) In this case we have and Theorem  implies that x  is nonhyperbolic and locally stable.
Theorem  If J =  and H +I <  and  is given by (), then the only positive equilibrium so by applying Theorem  we see that x + is unstable. By using Proposition  we obtain the following result: It remains to determine qp for equilibrium x + :

Example  It has been shown in [] that the difference equation
has one positive equilibrium x + = √   , which is a saddle point. The difference equation such that D > A + B + C has one positive equilibrium x + =  (A+B+C+D) . As the positive equilibrium is a repeller.

Local stability of period-two solutions
Period-two solutions . . . , , , , , . . . satisfy the system: It is clear that if there is a period-two solution it has to be I ≤ . Also, for H + I >  or J ≥  or (H + I -)  < J(E + F + G) we have seen that x n → ∞ (see Theorems , , , and ) so interesting dynamics for our system happens in the parametric region H + I ≤ , J <  and (H + I -)  ≥ J(E + F + G), which will be our main objective in the sequel. As the system (), () is symmetric, which means that if ( , ) is a solution of the system (), (), then also ( , ) is a solution of the system (), (); then we assume without a loss of generality that  ≤ < . By subtracting () and (), and using the fact that -> , we get By summing () and () we get If we set + = x >  and = y ≥ , then () and () yields another system One can see that the special cases when the period-two solutions do not exist are This immediately leads to the following result.
Here g(x) and are defined by () and (), respectively, and x  is the greater root of equation implies that the period-two solutions do not exist. If ≤  or >  and g(x  ) ∈ (, J), then there is no positive solution of g(x) = , so there are no positive equilibriums of (). As a consequence of Theorem  every bounded solution of () approaches either an equilibrium solution or a period-two solution and every unbounded solution is asymptotic to the point at infinity in a monotonic way. Hence, every solution {x n } of () satisfies lim n→∞ x n = ∞.
then a period-two solution does not exist, so if there exists an unstable positive equilibrium of () we obtain which implies that the positive equilibrium is a saddle point. In this case a global result follows from Theorem .
Set u n = x n- and v n = x n , n = , , . . . and write () in the equivalent form: Let the function f (u, v) be defined by () and let T be a map on The period-two solution is locally asymptotically stable if the eigenvalues of the Jacobian matrix J T  , evaluated at ( , ), lie inside the unit disk. By definition .
By computing the partial derivatives of g and h and by using the fact that g( , ) = f ( , ) = , we find the following identities: and By applying the linearized stability theorem, the period-two solution ( , ) is locally asymptotically stable if |S| <  + D and D < , that is, which implies the following lemma.
Lemma  The eigenvalues of the Jacobian matrix of J T  at a period-two solution are nonnegative numbers.
In order to solve system of equations () and (), we consider two different cases.
If D ≤ A and G ≤ E, then the period-two solutions do not exist, so at least one of the following inequalities is true: Also, H + I <  implies I -H - < -H ≤ . The solutions of the quadratic equation () are given by is the only positive solution of ().
(b) If D < A, then () can be written as It is clear that E < G. As which is impossible for positive , . For positive parameter H we have H + I > H +  > , so there is no minimal period-two solution. If G = E, then In all the above cases, when (E + G -F) + (A + D -B -C)x = , y is given by and, finally, from () we get the following cubic equation: where Remark  Equation () shows that () can have at most three period-two solutions. Lemma  gives an upper bound of the number of period-two solutions of equation () in some special cases. Equation () can be solved but its solutions are very complicated and would depend on  parameters. In the remaining part of the paper we will work under the assumptions that () has between zero and three period-two solutions and we will present the global dynamics in all possible cases. In particular, Theorem  describes a global dynamics in the case when () has one or three period-two solutions, while Theorem  gives a global dynamics in the case when () has zero or two period-two solutions.
The existence of at least one period-two solution is guaranteed by Theorem .

Global behavior
In this section we present the global dynamics of () in different parametric regions.

The case that there exists a minimal period-two solution on the coordinate axes ( = 0 and > 0)
In this section we consider the special case when =  and > . Equation () implies Since > , we get A = E = H = J = . Now, from () we obtain If I = , then we must have D = G =  and ∈ R + . If I < , then is a positive solution of (), that is, Also, if D =  and I < , then   ) where t ∈ R + is a period-two solution of the difference equation All other minimal period-two solutions ( , ) are solutions of the system One can show that this system has no other positive solutions except ( , ) = (, t) where Since bc = (a -)(d -) or S =  + D, the period-two solution is nonhyperbolic. The global behavior of this equation is described in Section  and, in this case, every solution goes to infinity. ) is the period-two solution of the difference equation All other minimal period-two solutions ( , ) are solutions of the system Since H + I < , there are locally stable zero equilibrium and positive unstable equilibrium.
It is useful to note that if B = D and F = G, then ( , ) = (, ) is also a solution of the equation In this case every point on the curve is a period-two solution of () except the equilibrium point (x + , x + ).
One can see that for ( , ) = (, After some calculation we find and for the positive equilibrium x + we obtain Then  det V = B  -C  and by the well-known result as regards the identification of the quadratic curves, we obtain the following statements: We avoid the cases when U is one point set or empty set. As a consequence of I <  and the fact that the curve () has endpoints in the first quadrant we conclude that the point (, ) is always below the curve (). It remains to describe global behavior of () when B = D and F = G. If an initial point (x - , x  ) is below the curve (), then Thus the point (x  , x  ) is below the curve () and Continuing in this way, we obtain (, ) ne · · · ne (x  , x  ) ne (x  , x  ) ne (x - , x  ). Hence, both of the subsequences {x n } and {x n+ } are decreasing, which implies x n →  and x n+ → . If we start at the point (x - , x  ) above the curve (), then Now the proof is very similar and will be omitted. In this case both of the subsequences {x n } and {x n+ } are increasing, which implies x n → ∞ and x n+ → ∞.
If B = C, then B x  + y  + Bxy + F(x + y) + I = , Since F  + B( -I) > , U is the union of two parallel lines. If we want to obtain two intersecting lines it has to be B < C and a straightforward calculation gives another condition F  + (B + C)( -I) = , which is not true. Therefore, As x n+ = (Cx n x n- + F(x n + x n- ) + I)x n- , we have Thus the point (x  , x  ) is below J  too and Continuing in this way, we obtain (, ) ne · · · ne (x  , x  ) ne (x  , x  ) ne (x - , x  ). Hence, both subsequences {x n } and {x n+ } are decreasing, which implies x n →  and x n+ → . If we start at the point (x - , x  ) above J  , then which, in a similar way as above implies (x - , x  ) ne (x  , x  ) ne . . . . In this case both subsequences {x n } and {x n+ } are increasing, which implies that x n → ∞ and x n+ → ∞.
The special cases studied in Theorem  leads to the following general result.
Theorem  Consider the difference equation where P is a symmetric polynomial function with nonnegative coefficients. If we assume that P(, ) < , then the curve P(x, y) =  is the Julia set and separates the first quadrant into two regions: the region below the curve is the basin of attraction of E  (, ) and the region above the curve is the basin of attraction of a point at infinity.
Proof Equilibrium points of () are solutions of the equation which implies that () has zero equilibrium. Let h(x) = P(x, x) -. Then and we see that the equation P(x, x) =  has exactly one positive equilibrium x + . As a consequence of the symmetry we have ∂P ∂x (x, y) = ∂P ∂y (y, x) and where p and q denote partial derivatives of function g(x, y) = yP(x, y) evaluated at the equilibrium x. For the zero equilibrium we get and by applying Proposition , the zero equilibrium is locally asymptotically stable. Similarly, in the case of the positive equilibrium x + , we get Since > , we have P( , ) =  and P( , ) = P( , ) = . Hence, every point of the set J  = {(x, y) : P(x, y) = } is a period-two solution of () except the equilibrium point (x + , x + ). If we start at the point (x  , x - ) below the curve J  , then P(x  , x - ) <  and from the fact that P(x, y) is an increasing function in both variables we obtain the following: Therefore the point (x  , x  ) is also below J  and Continuing in this way, we obtain (, ) ne · · · ne (x  , x  ) ne (x  , x  ) ne (x - , x  ). Hence, both subsequences {x n } and {x n+ } are decreasing, which implies x n →  and x n+ → . If we start at the point (x - , x  ) above J  , then The proof of the remaining case is similar and will be omitted. In this case both subsequences {x n } and {x n+ } are increasing, which implies x n → ∞ and x n+ → ∞.
.. The Case: J = , H + I ≥  Next we consider the case where J =  and H + I ≥ . In this case () has exactly one equilibrium which is the zero equilibrium, which in view of Proposition  is unstable.
The following result describes a global dynamics of () in this case. (b) Assume that {x n } is not a minimal period-two solution of (). Then {x n } is eventually monotone or the subsequences {x n } and {x n+ } are eventually monotone. If {x n } is eventually decreasing, then x n < x n- for all n ≥ K , which implies x n+ ≥ Hx n +Ix n- > (H +I)x n = x n , which is a contradiction. If the subsequences {x n } and {x n+ } are eventually monotone, then without loss of generality we can assume that {x n } is eventually nondecreasing and {x n+ } is eventually non-increasing. In this case x n → ∞, which would imply that x n+ → ∞, which is a contradiction. Thus the remaining possibility is that {x n } is eventually increasing, which implies that x n → ∞ as n → ∞. Another way of proving the global behavior in the case when the equilibrium point E is nonhyperbolic was used in [, ].

Theorem 
To complete this we will find the image of E + tv, where t >  and v is the eigenvector that corresponds to the eigenvalue , under the map T. Since E + tv = (t, t), we have By using the condition H + I - = , we have which implies E +tv ne T(E +tv) for every t > . This shows that every point in u ∈ (x, ∞)  is a supersolution for the map T, that is, u ne T(u), see [], and so every solution tends to ∞.
(c) In view of I >  + H from () we have x n+ > Ix n- , which implies x n+ > I k x - or x n+ > I k x  for some k such that k → ∞ as n → ∞. Consequently every solution {x n } of () satisfies lim n→∞ x n = ∞.

The case of two equilibrium points and a finite number of hyperbolic minimal period-two solutions
In this section we present the global dynamics of () in the parametric regions where there exist two distinct equilibrium points E -(x -,x -) and E + (x + ,x + ), such that Ene E + , which holds if and only if and a finite number of period-two solutions which are hyperbolic. In this case, we prove that the Julia set is the union of the stable manifolds of some saddle period-two points and separates the first quadrant into two regions: the region below the curve is the basin of attraction of Eand the region above the curve is the basin of attraction of the point at infinity.
Let T  (x, y) = (g(x, y), h(x, y)) where g(x, y) = f (y, x) and h(x, y) = f (f (y, x), y). Then the period-two curves, that is, the curves of which the intersection is a period-two solution, are given by Let P i , i = , , ,  be the polynomials in the Appendix. The following lemma gives us information as regards the number of minimal period-two solutions.
Lemma  Assume that Res x (F,G) = P  (y)P(y) ≡ . Then there exist at most deg(P)/ isolated minimal period-two solutions. Let PS denote the number of isolated minimal period-two solution. The following statements are true: The following lemma gives the necessary conditions under which an isolated period-two solution is nonhyperbolic.

Proof
(i) Suppose that ( , ) is a nonhyperbolic minimal period-two solution. This implies thatF( , ) = ,G( , ) = . By Theorem  we have P  ( )P( ) = . Since is not an equilibrium point, we obtain P  ( ) =  and P( ) = . Taking derivatives of g(x, y) = x and h(x, y) = y with respect to x we get y) .
One can see that where p(μ) is the characteristic equation of the matrix By Lemma  we have μ  , μ  ≥ . Since ( , ) is a nonhyperbolic minimal period-two solution, we have μ  =  or μ  = . This implies y F ( )y G ( ) = . Since P ≡ , by Theorem  the curves CF and CG have no common components. In view of Lemmas  and  from [], the curves C F and C G intersect tangentially at ( , ) (i.e. y F ( )y G ( ) = ) if and only if is zero of P  (y)P(y) of multiplicity greater than one. Since P  ( ) = , is a zero of P(y) of multiplicity greater than one. By Theorem , P(y) has zero of multiplicity greater than one if and only if the discriminant Dis(P) is equal to zero, from which the proof follows.
(ii) Suppose thatx is a nonhyperbolic equilibrium point. This implies thatF(x,x) = , G(x,x) = . Similarly, as in (i), we have P  (x)P(x) = . Since J T  (x,x) = (J T (x,x))  , we see that all eigenvalues that correspond to J T  (x,x) are nonnegative. Sincex is a nonhyperbolic equilibrium point, we have μ  =  or μ  = . This implies y F (x)y G (x) = . Similarly, as in (i), we see that Dis(P  · P) = Dis(P  ) Dis(P) Res(P  , P)  = .
In view of Theorem  it is easy to see that {T n (x  , y  )} is either asymptotic to (∞, ∞) or converges to a period-two solution, for all (x  , y  ) ∈ R = [, ∞)  . In view of Lemma  we can suppose that Res x (F,G) = P  (y)P(y), P ∈ R[y]. If P ≡  and Dis(P  · P) = Dis(P  ) × Dis(P) Res(P  , P)  = , then by Theorem  and Lemma  we see that E + (x + ,x + ) is a repeller or a saddle point and all minimal period-two solutions are hyperbolic. By Lemma  we see that int(Q  (E + )) ⊂ B(x -,x -) and int(Q  (E + )) ⊂ B(∞, ∞). Let S  denote the boundary of B(x -,x -) considered as a subset of Q  (E) and S  denote the boundary of B(x -,x -) considered as a subset of Q  (E + ). It is easy to see that E + ∈ S  , E + ∈ S  and T(R) ⊂ int(R).
The proof of the following lemma for a cooperative map is the same as the proof of Claims  and  [] for a competitive map, so we skip it.
Lemma  Suppose that P ≡  and Dis(P  · P) = . Let S  and S  be the sets defined as above. Then Corollary  Suppose that P ≡  and Dis(P  · P) = . If E + (x + ,x + ) is a repeller, then int(Q  (E + )) ∪ int(Q  (E + )) contains one or three distinct minimal period-two solutions. If T has one minimal period-two solution {(  ,  ), (  ,  )}, then it is a saddle point and (  ,  ) se E + se (  ,  ). If T has three minimal period-two solutions {( i , i ), ( i , i )}  i= , then they are ordered in the South-East ordering. If (  ,  ) se (  ,  ) se (  ,  ) se E + se (  ,  ) se (  ,  ) se (  ,  ), then odd indexed period-two points are saddles and even indexed period-two points are repellers.
Proof By Lemma  all equilibrium points and minimal period-two solutions are hyperbolic. In view of Theorem  we see thatF andG have no common component. From Lemma  the number of minimal period-two solutions is at most three. In view of Theorem , T has at least one minimal period-two solution, which is a saddle point. Assume that T has two minimal period-two solutions {(  ,  ), (  ,  )} and {(  ,  ), (  ,  )}. Assume that {(  ,  ), (  ,  )} is a saddle point and (  ,  ) se E + se (  ,  ). Further, suppose that (  ,  ) se (  ,  ) se E + se (  ,  ) se (  ,  ). The map T  satisfies all conditions of Theorem , which yields the existence of the global stable manifolds W s ({(  ,  ), (  ,  )}), the union of two curves W s (  ,  ) and W s ((  ,  )) that have a common endpoint E + . Then W s (  ,  ) has the second endpoint at (  ,  ) and W s (  ,  ) has the second endpoint at (  ,  ). Furthermore, the minimal period-two solution {(  ,  ), (  ,  )} is a repeller. Since the global stable manifold is unique, the set (S  ∩ Q  (  ,  )) ∪ (S  ∩ Q  (  ,  )) is invariant under T. Similarly, as in Theorem , one can prove that int(Q  (  ,  )) ∪ int(Q  (  ,  )) contains exactly one minimal period-two solution, which is a saddle point. Hence, if T has two minimal period-two solutions, then there exists a third minimal period-two solution. This proves the lemma. If (  ,  ) se (  ,  ) se E + se (  ,  ) se (  ,  ) the proof is similar and will be omitted.
Corollary  Assume that P ≡  and Dis(P  · P) = . If E + (x + ,x + ) is a saddle point, then int(Q  (E + )) ∪ int(Q  (E + )) contains either zero or two minimal period-two solutions {( i , i ), ( i , i )}, i = , , which are ordered to the South-East ordering. If there exist two minimal period-two solutions such that (  ,  ) se (  ,  ) se E + se (  ,  ) se (  ,  ), then an even indexed period-two point is a saddle and an odd indexed period-two point is a repeller.
Proof The proof is similar to the proof of Corollary  and it will be omitted.
Theorem  Suppose that P ≡  and Dis(P  · P) = . If E + (x + ,x + ) is a repeller, then int(Q  (E + )) ∪ int(Q  (E + )) contains one or three minimal period-two solutions {( i , i ), ( i , i )} n+ i= , where n =  or n = , such that ( i+ , i+ ) se ( i , i ) se E + and E + se ( i , i ) se ( i+ , i+ ), and ( i , i ) = T( i , i ). Furthermore, the odd indexed period-two points are saddles and the even indexed period-two points are repellers and the following hold: (i) If there exists one minimal period-two solution {(  ,  ), (  ,  )}, the global stable manifolds is given by where W s (  ,  ) and W s (  ,  ) are the graphs of a continuous strictly decreasing functions with common endpoint at E + and W s (  ,  ) = T(W s (  ,  )). The Julia set is the curve

The global unstable manifold is given by
where W u (  ,  ) and W u (  ,  ) are the graphs of continuous strictly increasing functions such that W u (  ,  ) = T(W u (  ,  )), with common endpoint at E -. See Figure (a) for visual illustration.
Proof In view of Corollary  the set int(Q  (E + )) ∪ int(Q  (E + )) contains an odd number of minimal period-two solutions {( i , i ), ( i , i )}, i = , . . . , n +  (n =  or n = ), such that ( i+ , i+ ) ne ( i , i ) ne E + and E + ne ( i , i ) ne ( i+ , i+ ) and ( i , i ) = T( i , i ). Furthermore, the odd indexed period-two points are saddles and the even indexed period-two points are repellers. The map T  satisfies all conditions of Theorems , , and  [], which yields the existence of the global stable and unstable manifolds with the above properties. In view of Theorems , , and  [] for (x  , y  ) ∈ W -∩ R there exists n  >  such that n > n  , and for (x  , y  ) ∈ W + ∩ R there exists n  >  such that which completes the proof.
Theorem  Suppose that P ≡  and Dis(P  · P) = . If E + (x + ,x + ) is a saddle point, then either T has no minimal period-two solution or int(Q  (E + )) ∪ int(Q  (E + )) contains two distinct minimal period-two solutions and the following hold: (a) If T has no minimal period-two solution, then there exist two continuous curves W s (E + ) and W u (E + ), both passing through the point E + (x + ,x + ), such that W s (E + ) is a graph of a decreasing function and W u (E + ) is a graph of an increasing function. The first quadrant of the initial condition Q  = {(x - , x  ) : x - ≥ , x  ≥ } is the union of three disjoint basins of attraction, namely where B(E + ) = W s (E + ), and In addition, for every (x - , x  ) ∈ Q  \ W s (E + ) every solution is asymptotic to W u (E + ). See Figure  that separates R into two components Wand W + , which are the basins of attraction of (x -,x -) and (∞, ∞), respectively, where Further, we have The global unstable manifold of {(  ,  ), (  ,  )} is given by where W u (  ,  ) and W u (  ,  ) are the graphs of continuous strictly increasing functions such that W u (  ,  ) = T(W u (  ,  )), with endpoints at (x -,x -) and (∞, ∞). The global unstable manifold W u (E + ) of E + is the graph of a continuous strictly increasing function such that W u (  ,  ) and W u (  ,  ), and W u (E + ) have a common endpoint at (x -,x -).
Proof The proof is similar to the proof of Theorem  and it will be omitted. Appendix: Values of coefficients p i for i = 0, . . . , 6, and Dis(P 1 ) P  (y) = p  y  + p  y  + · · · + p  y + p  ,