Global dynamics of cubic second order difference equation in the first quadrant
 Jasmin Bektešević^{1},
 Mustafa RS Kulenović^{2}Email author and
 Esmir Pilav^{3}
https://doi.org/10.1186/s136620150503x
© Bektešević et al. 2015
Received: 28 December 2014
Accepted: 17 May 2015
Published: 11 June 2015
Abstract
We investigate the global behavior of a cubic second order difference equation \(x_{n+1}=Ax_{n}^{3}+ Bx_{n}^{2}x_{n1}+Cx_{n}x_{n1}^{2}+Dx_{n1}^{3}+Ex_{n}^{2} +Fx_{n}x_{n1}+Gx_{n1}^{2}+Hx_{n}+Ix_{n1}+J\), \(n=0,1,\ldots\) , with nonnegative parameters and initial conditions. We establish the relations for the local stability of equilibriums and the existence of periodtwo 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.
Keywords
MSC
1 Introduction and preliminaries
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 (1) to hold in the whole plane, when \(B=C=E=F=G=0\).
Remark 1
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 periodtwo points in all cases. The fourth section gives the global dynamics, which includes the basins of attraction of all equilibrium points and the periodtwo points. Some Mathematica outputs are given in the Appendix.
Here we list some of the results that will be needed in this paper. The first result was obtained in [7] and it was extended to the case of higher order difference equations and systems in [8, 9].
Theorem 1
 (i)
\(f(x,y)\) is nondecreasing in each of its arguments, i.e. \(x\rightarrow f(x,y)\) is nondecreasing for every y and \(y\rightarrow f(x,y)\) is nondecreasing for every x;
 (ii)
(2) has a unique equilibrium \(\overline{x}\in [ a,b]\).
The following result was obtained in [10].
Theorem 2
 (i)
Eventually they are both monotonically increasing.
 (ii)
Eventually they are both monotonically decreasing.
 (iii)
One of them is monotonically increasing and the other is monotonically decreasing.
As a consequence of Theorem 2 every bounded solution of (1) approaches either an equilibrium solution or periodtwo solution or the singular point on the boundary and every unbounded solution is asymptotic to the point at infinity in a monotonic way; see [11]. Thus the major problem in the dynamics of (1) 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 11 in [12].
Theorem 3
Also, we will use the following theorem from [12].
Theorem 4
The following result gives the necessary and sufficient condition for the local stability of (2) when f is nondecreasing in all its arguments; see [13].
Theorem 5
 (a)
locally asymptotically stable if \(p_{0}+p_{1}<1\),
 (b)
nonhyperbolic and locally stable if \(p_{0}+p_{1}=1\),
 (c)
unstable if \(p_{0}+p_{1}>1\).
Theorem 6
 (i)The discriminant of \(f_{1}\) is given byand$$\operatorname{Dis}(f_{1})=(1)^{n(n1)/2}\frac{1}{a_{n}}\operatorname{Res} \bigl(f_{1},f_{1}'\bigr) $$where \(f_{1}'\) is the derivative of \(f_{1}\).$$\operatorname{Dis}(f_{1}\cdot g_{1})= \operatorname{Dis}(f_{1})\operatorname{Dis}(g_{1})\operatorname{Res}(f_{1},g_{1})^{2}, $$
 (ii)
\(\operatorname{Dis}(f_{1}) = 0 \Leftrightarrow f_{1}\) has a double root in \(\mathbb{C}\): Equivalently, \(f_{1}\) has n distinct roots in \(\mathbb{C}\) if and only if the discriminant is ≠0.
For two bivariate polynomials \(f,g \in\mathbb{R}[x,y]\), Theorem 7 in [14, 15] holds.
Theorem 7
 (a)
f and g have a nontrivial common factor if and only if r is identically zero.
 (b)If f and g are coprime (do not have a common factor), the following conditions are equivalent:

\(\alpha\in\mathbb{C}\) is a root of r.

\(f_{n}(\alpha) = g_{m}(\alpha) = 0\) or there is \(\beta\in\mathbb {C}\) with \(f (\alpha,\beta) = 0 = g(\alpha,\beta) = 0\).

 (c)
For all \((\alpha,\beta)\in\mathbb{C}\times\mathbb{C}\): if \(f(\alpha,\beta) = 0 = g(\alpha,\beta)= 0\), then \(r(\alpha) = 0\).
We list some results when solution \(\{ x_{n} \} \) of (1) tends to the point at infinity in a monotonic way.
Theorem 8
If \(H+I>1\), then every solution \(\{ x_{n} \} \) of (1) satisfies \(\lim_{n\rightarrow\infty}x_{n}=\infty\).
Proof
Theorem 9
If \(J\geq1\), then every solution \(\{ x_{n} \} \) of (1) satisfies \(\lim_{n\rightarrow \infty}x_{n}=\infty\).
Proof
Theorem 10
If \(A+B+C+D+E+F+G+H+I > 1\) or \(J>0\) and \(A+B+C+D+E+F+G+H+I\geq1\), then the box \([ 1,\infty ) ^{2}\) is a part of the basin of attraction of the point at infinity of (1).
Proof
Theorem 11
If \(( H+I1 ) ^{2}<4J ( E+F+G ) \), then every solution \(\{ x_{n} \} \) of (1) satisfies \(\lim_{n\rightarrow\infty}x_{n}=\infty\).
Proof
In view of the results for (5) in [1], see Lemma 2, there is no prime periodtwo solution when \(( H+I1 ) ^{2}\leq4J ( E+F+G ) \) or \(E\geq G\) or \(H+I>1\). So if there is no equilibrium point and there is no prime periodtwo solution of (5), then every solution \(\{ y_{n} \} \) of (5) satisfies \(\lim_{n\rightarrow\infty}y_{n}=\infty\), which implies \(\lim_{n\rightarrow\infty}x_{n}=\infty\). □
Remark 2
Theorems 8, 9, 10, and 11 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 periodtwo 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 11 that the escape region of (5) is a subset of the escape region of (1). In the subsequent sections we will consider global dynamics of (5) in the complement of the parametric region described by Theorems 8, 9, 10, and 11.
2 Local stability analysis of equilibrium solutions
2.1 The case \(J>0\)
(i) If \(\Delta<0\), then \(g^{\prime} ( x ) >0\) for all x and the function \(g ( x ) \) is monotonically increasing, which implies that there is no nonnegative root of \(g ( x ) =0\).

If \(H+I>1\), then (6) has no positive solutions.

If \(H+I=1\), then \(x_{2}=0\) and \(g(x)\geq g ( x_{2} ) =J>0\) so there is no positive solution of \(g ( x ) =0\).

If \(H+I<1\), then \(x_{2}>0\), \(g^{\prime} ( x ) <0\) for \(x\in ( x_{1},x_{2} ) \) so \(g ( x_{1} ) >g ( 0 ) =J>g ( x_{2} ) \) and:
(a) If \(g ( x_{2} ) \in ( 0,J ) \) there is no positive solution of \(g ( x ) =0\).
2.2 The case \(J=0\)
(i) If \(\Delta_{0}<0\), then the zero equilibrium is the only equilibrium.
 (a)
If \(H+I>1\), then (12) has no positive solutions.
 (b)
If \(H+I=1\), then \(\overline{x}_{2}=0\), and there is no positive equilibrium solutions.
 (c)
If \(H+I<1\), then the only positive equilibrium solution is \(\overline{x}_{2}\).
Proposition 1
 (a)
locally asymptotically stable if \(H+I<1\),
 (b)
nonhyperbolic and locally stable if \(H+I=1\),
 (c)
unstable if \(H+I>1\),
 (d)
a saddle point if \(H>I1\),
 (e)
a repeller if \(1I< H< I1\).
Proposition 2
 (a)
locally asymptotically stable if \(p+q<1\),
 (b)
nonhyperbolic and locally stable if \(p+q=1\),
 (c)
unstable if \(p+q>1\),
 (d)
a saddle point if \(p>q1\),
 (e)
a repeller if \(1q< p< q1\).
The next theorems will describe the local stability for the positive equilibrium(s) in more detail.
Theorem 12
 (a)
If \(g ( x_{2} ) <0\), then there are two positive equilibrium solutions \(\overline{x}_{1}\in ( 0,x_{2} ) \) and \(\overline{x}_{2}\in ( x_{2},+\infty ) \) of (1). Furthermore, \(\overline{x}_{1}\) is locally asymptotically stable and \(\overline{x}_{2}\) is unstable.
 (b)
If \(g ( x_{2} ) =0\), then there is one positive equilibrium solution \(\overline{x}=x_{2}\), which is nonhyperbolic and locally stable.
Proof

\(\overline{x}_{2}\) is a saddle point if \(qp<1\),

\(\overline{x}_{2}\) is a repeller if \(qp>1\).
Theorem 13
Proof

\(\overline{x}_{+}\) is saddle point if \(qp<1\);

\(\overline{x}_{+}\) is repeller if \(qp>1\).
Example 1
3 Local stability of periodtwo solutions
Theorem 14
Let \(D\leq A\), \(A+C\leq B+D\) (or \(C+D\leq A+B\)), \(G\leq E \), \(J\in ( 0,1 ) \), and \(H+I<1\). If \(\Delta\leq0\) or \(\Delta>0\) and \(g ( x_{2} ) \in ( 0,J ) \), then every solution \(\{ x_{n} \} \) of (1) satisfies \(\lim_{n\rightarrow\infty}x_{n}=\infty\). Here \(g ( x ) \) and Δ are defined by (7) and (9), respectively, and \(x_{2}\) is the greater root of equation \(g^{\prime} ( x ) =0\).
Proof
Since \(H+I<1\), \(D\leq A\), \(A+C\leq B+D\) (or \(C+D\leq A+B\)), and \(G\leq E\), we see that (18) implies that the periodtwo solutions do not exist. If \(\Delta\leq0\) or \(\Delta>0\) and \(g ( x_{2} ) \in ( 0,J ) \), then there is no positive solution of \(g ( x ) =0\), so there are no positive equilibriums of (1). As a consequence of Theorem 2 every bounded solution of (1) approaches either an equilibrium solution or a periodtwo solution and every unbounded solution is asymptotic to the point at infinity in a monotonic way. Hence, every solution \(\{ x_{n} \} \) of (1) satisfies \(\lim_{n\rightarrow\infty} x_{n}=\infty\). □
Remark 3
Lemma 1
The eigenvalues of the Jacobian matrix of \(J_{T^{2}}\) at a periodtwo solution are nonnegative numbers.
In order to solve system of equations (20) and (21), we consider two different cases.
Remark 4
Equation (31) shows that (1) can have at most three periodtwo solutions. Lemma 3 gives an upper bound of the number of periodtwo solutions of equation (1) in some special cases. Equation (31) can be solved but its solutions are very complicated and would depend on 10 parameters. In the remaining part of the paper we will work under the assumptions that (1) has between zero and three periodtwo solutions and we will present the global dynamics in all possible cases. In particular, Theorem 20 describes a global dynamics in the case when (1) has one or three periodtwo solutions, while Theorem 21 gives a global dynamics in the case when (1) has zero or two periodtwo solutions. The existence of at least one periodtwo solution is guaranteed by Theorem 19.
4 Global behavior
In this section we present the global dynamics of (1) in different parametric regions.
4.1 The case that there exists a minimal periodtwo solution on the coordinate axes (\(\Phi=0\) and \(\Psi>0\))
4.1.1 The case: \(A=D=E=G=H=J=0\), \(I=1\)
4.1.2 The case: \(A=E=H=J=0\) and \(I<1\)

It is clear that if \(B\Psi^{2}+F\Psi+I>1\), then \(( a1 ) ( d1 ) >0=bc\) and \(ad>1\) or \(\mathcal{S}<1+\mathcal{D}\) and \(\mathcal{D}>1\), which implies \(( \Phi,\Psi ) = ( 0,\frac{G+ \sqrt{G^{2}+4D ( 1I ) }}{2D} ) \) is a repeller. For example, if \(B=D\) and \(F>G\) or \(B>D\) and \(F=G\), then \(qp1<0\), which implies that the positive equilibrium \(\overline{x}_{+}\) is a saddle point.

In the case when \(B\Psi^{2}+F\Psi+I=1\), then \(( a1 ) ( d1 ) =0=bc\) or \(\mathcal{S}=1+\mathcal{D}\), which implies that the periodtwo solution \(( \Phi,\Psi ) = ( 0,\frac{G+\sqrt{ G^{2}+4D ( 1I ) }}{2D} ) \) is a nonhyperbolic point. A special case when this holds is \(B=D\) and \(G=F\). In this case every point on the curve (35) is a periodtwo solution of (34), except the equilibrium point, Since \(qp1=0\), the positive equilibrium \(\overline{x}_{+}\) is a nonhyperbolic point. Also, the points \(( \Phi,\Psi ) = \{ ( 0,\frac{F+\sqrt{F^{2}+4B ( 1I ) }}{2B} ) , ( \frac{F+\sqrt{F^{2}+4B ( 1I ) }}{ 2B},0 ) \} \) are the endpoints of the curve (35) in the first quadrant.

If \(B\Psi^{2}+F\Psi+I<1\), then \(( a1 ) ( d1 ) <0=bc\) or \(\mathcal{S}>1+\mathcal{D}\), which implies that the periodtwo solution \(( \Phi ,\Psi ) = ( 0,\frac{G+\sqrt{G^{2}+4D ( 1I ) }}{2D} ) \) is a saddle point. That is true for example if \(B=D\) and \(F< G\) or \(B>D\) and \(F=G\), then \(qp1>0\), which implies that the positive equilibrium \(\overline{x}_{+}\) is a repeller.
One can prove that if \(B=D\), \(F\neq G\) or \(F=G\), \(B\neq D\) the map T has no other minimal periodtwo solutions except \(( \Phi,\Psi ) = ( 0,\frac{G+\sqrt{G^{2}+4D ( 1I ) }}{2D} )\). This leads to the following result.
Theorem 15
Let \(A=E=H=J=0\) and \(I<1\), \(D>0\), then (34) has locally stable zero equilibrium and the positive equilibrium \(\overline{x}_{+}\). The following statements hold:
(b) If \(B=D\), \(F>G\) or \(B>D\), \(F=G\), then the positive equilibrium \(\overline{x}_{+}\) is a saddle point and there exists the periodtwo solution which is a repeller. Global behavior of (34) is described by Theorem 3 where we set \(E_{1}(0,0)\) and \(E_{2}(\overline{x}_{+},\overline {x}_{+})\). (See Figure 1(b).)
(c) If \(B=D\) and \(F=G\), then every point on the curve (35), which is passing through the equilibrium \(E=(\bar{x}_{+},\bar{x}_{+})\), is a periodtwo solution of (34) except point E. Periodtwo solutions and the positive equilibrium \(\overline{x}_{+}\) are nonhyperbolic points, while the zero equilibrium is locally stable. The curve (35) divides the first quadrant into two regions. The region below (35) in the first quadrant is a basin of attraction of the zero equilibrium and the region above the (35) in the first quadrant is a basin of attraction of point at infinity. (See Figure 1(c).)
Proof
(a) The existence of four curves \(\mathcal{W} ^{s}(P_{1})\), \(\mathcal{W}^{s}(P_{2})\), \(\mathcal{W}^{u}(P_{1})\), and \(\mathcal{W}^{u}(P_{2})\) with the described properties is guaranteed by Theorems 1 and 4 of [18] applied to the map \(T^{2}\) given by (23). The global result follows from Theorem 3.
(b) All conditions of Theorem 3 are satisfied, which implies the proof.

if \(\det\mathcal{V}>0\Leftrightarrow2B>C\), then \(\mathcal{U}\) is either an ellipse or a circle if \(C=0\),

if \(\det\mathcal{V}<0\Leftrightarrow2B<C\), then \(\mathcal{U}\) is a hyperbole,

if \(\det\mathcal{V}=0 \Leftrightarrow2B=C\), then \(\mathcal {U}\) is union of two parallel lines.
4.1.3 The case: \(A=D=E=H=J=0\) and \(I<1\)

If \(B ( 1I ) +G ( FG ) >0\), then \(\mathcal{S} \mathcal{D}1<0\), \(\mathcal{D}>2I>1\), and the periodtwo solution \(( \Phi ,\Psi ) = ( 0,\frac{1I}{G} ) \) is a repeller. For example, if \(F>G\), \(B\geq0\) or \(F=G\), \(e B>0\), then \(qp1<0\) and the positive equilibrium \(\overline{x}_{+}\) is a saddle point.

If \(B ( 1I ) +G ( FG ) =0\), then \(\mathcal{S} \mathcal{D}1=0\) and the periodtwo solution \(( \Phi,\Psi ) = ( 0,\frac{1I}{G} ) \) is a nonhyperbolic point. For example if \(F=G\) and \(B=0\), then the positive equilibrium \(\overline{x}_{+}\) is a nonhyperbolic point and every point on the curve \(Cxy+F ( x+y ) +I1=0\) is a periodtwo solution of (36), except \(( \overline{x} _{+},\overline{x}_{+} ) \). The points \(( \Phi,\Psi ) = \{ ( 0,\frac{1I}{G} ) , ( \frac{1I}{G},0 ) \} \) are the endpoints of the curve \(Cxy+F ( x+y ) +I1=0\) in the first quadrant.

If \(B ( 1I ) +G ( FG ) <0\), then \(\mathcal{S} \mathcal{D}1>0\) and the periodtwo solution \(( \Phi,\Psi ) = ( 0,\frac{1I}{G} ) \) is a saddle point and \(B ( 1I ) +G ( FG ) <0\). For example if \(F< G\) and \(B=0\), then \(qp1>0\) and the positive equilibrium \(\overline{x}_{+}\) is repeller.
One can prove that in this case the map T has no other minimal periodtwo solutions except \(( \Phi,\Psi ) = ( 0,\frac{G+\sqrt{G^{2}+4D ( 1I ) }}{2D} )\). This leads to the following theorem.
Theorem 16
Let \(A=D=E=H=J=0\) and \(I<1\), then (36) has locally stable zero equilibrium and the positive equilibrium \(\overline{x}_{+}\). The following statements hold:
(a) If \(B=0\) and \(F< G\), then the positive equilibrium \(\overline{x}_{+}\) is repeller and there exists the periodtwo solution which is a saddle point. In this case there exist four continuous curves \(\mathcal{W} ^{s}(P_{1})\), \(\mathcal{W}^{s}(P_{2})\), \(\mathcal{W}^{u}(P_{1})\), and \(\mathcal{W}^{u}(P_{2})\). The curves \(\mathcal{W}^{s}(P_{1})\) and \(\mathcal{W}^{s}(P_{2})\) are passing through the point \(E_{+}=(\bar{x}_{+},\bar{x}_{+})\) and they are graphs of decreasing functions. The curves \(\mathcal{W}^{u}(P_{1})\) and \(\mathcal{W} ^{u}(P_{2})\) are the graphs of increasing functions and are starting at \(E_{0} ( 0,0 ) \). Every solution \(\{ x_{n} \} \) which starts below \(\mathcal{W}^{s}(P_{1})\cup\mathcal{W}^{s}(P_{2})\) in the NorthEast ordering converges to \(E_{0}\) and every solution \(\{ x_{n} \} \) which starts above \(\mathcal{W}^{s}(P_{1})\cup\mathcal{W} ^{s}(P_{2})\) in the NorthEast ordering satisfies \(\lim_{n\rightarrow\infty}x_{n}=\infty\). (See Figure 1(a).)
(b) If \(F>G\), \(B\geq0\) or \(F=G\), \(B>0\), then the positive equilibrium \(\overline{x}_{+}\) is a saddle point and there exists the periodtwo solution which is repeller. Global behavior of (34) is described by Theorem 3 where we set \(E_{1}(0,0)\) and \(E_{2}(\overline{x}_{+},\overline {x}_{+})\). (See Figure 1(b).)
(c) If \(F=G\) and \(B=0\), then every point on the curve \(\mathcal{J}_{1}=Cxy+F ( x+y ) +I=1\), which is passing through the point \(E=(\bar{x} _{+},\bar{x}_{+})\), is a periodtwo solution of (36) except E. All periodtwo solutions and the positive equilibrium \(\overline{x}_{+}\) are nonhyperbolic points and the zero equilibrium is locally stable. The curve \(\mathcal{J}_{1}\) divides the first quadrant into two regions. The region below \(\mathcal{J}_{1}\) in the first quadrant is the basin of attraction of the zero equilibrium and the region above \(\mathcal{J}_{1}\) in the first quadrant is the basin of attraction of point at infinity. (See Figure 1(c).)
Proof
(a) The existence of four curves \(\mathcal{W} ^{s}(P_{1})\), \(\mathcal{W}^{s}(P_{2})\), \(\mathcal{W}^{u}(P_{1})\), and \(\mathcal{W}^{u}(P_{2})\) with the described properties is guaranteed by Theorems 1 and 4 of [18] applied to the map \(T^{2}\) given by (23). The global result follows from Theorem 3.
(b) All conditions of Theorem 3 are satisfied and the proof follows from Theorem 3.
The special cases studied in Theorem 16 leads to the following general result.
Theorem 17
Proof
4.1.4 The Case: \(J=0\), \(H+I\geq1\)
Next we consider the case where \(J=0\) and \(H+I\geq1\). In this case (1) has exactly one equilibrium which is the zero equilibrium, which in view of Proposition 1 is unstable.
The following result describes a global dynamics of (1) in this case.
Theorem 18
 (a)
If \(1H< I\leq1\) and (1) does not possess a periodtwo solution, then every solution \(\{x_{n}\}\) of (1) satisfies \(\lim_{n\rightarrow\infty }x_{n}=\infty\);
 (b)
If \(1H=I\leq1\), then every solution \(\{x_{n}\}\) of (1) is either a periodtwo solution or it satisfies \(\lim_{n\rightarrow\infty}x_{n}=\infty\);
 (c)
If \(1+H< I\) and (1) does not possess a periodtwo solution, then every solution \(\{x_{n}\}\) of (1) satisfies \(\lim_{n\rightarrow\infty }x_{n}=\infty\).
Proof
(a) In view of Theorem 3 there exist the global stable manifold \(W^{s}(0,0)\) and the global unstable manifold \(W^{u}(0,0)\), where \(W^{s}(0,0)\) is a graph of continuous decreasing curve and \(W^{u}(0,0)\) is a graph of continuous increasing curve and both manifolds are invariant sets. The only decreasing curve in the first quadrant \(Q_{1}\) passing through \((0,0)\) is the union of the coordinate axes, but this set is clearly not an invariant set, which means that \(W^{s}(0,0)\) is not a part of \(Q_{1}\). On the other hand \(W^{u}(0,0)\) exists and all solutions are asymptotic to \(W^{u}(0,0)\). Thus if (1) does not possess a periodtwo solution, then every solution \(\{x_{n}\}\) of (1) satisfies \(\lim_{n\rightarrow\infty} x_{n}=\infty\).
(c) In view of \(I>1+H\) from (1) we have \(x_{n+1}>Ix_{n1}\), which implies \(x_{n+1}>I^{k}x_{1}\) or \(x_{n+1}>I^{k}x_{0}\) for some k such that \(k\rightarrow\infty\) as \(n\rightarrow\infty\). Consequently every solution \(\{x_{n}\}\) of (1) satisfies \(\lim_{n\rightarrow\infty} x_{n}=\infty\). □
4.2 The case of two equilibrium points and a finite number of hyperbolic minimal periodtwo solutions
It is easy to see that \(f(y,x)=x\) and \(f(x,y)=y\) if and only if \(\tilde {F}(x,y)=\tilde{G}(x,y)=0\). We can view \(\tilde{F}\) and \(\tilde{G}\) as polynomials in y with coefficients in \(\mathbb{R}[x]\) and as polynomials in x with coefficients in \(\mathbb{R}[y]\). Let \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})\) and \(\operatorname{Res}_{y}(\tilde{F},\tilde{G})\) be resultants (see [14, 15]) of the polynomials \(\tilde{F}(x,y)\) and \(\tilde{G}(x,y)\) with respect to the variables x and y, respectively.
The following lemma gives some properties of \(\tilde{F}(x,y)\), \(\tilde {G}(x,y)\), \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})\), and \(\operatorname{Res}_{y}(\tilde{F},\tilde{G})\).
Lemma 2
 (i)
\(\tilde{F}(x,y)=\tilde{G}(y,x)\) and \(\operatorname{deg}_{x} (\tilde{F})=\operatorname{deg}_{y} (\tilde{G})\leq3\), and \(\operatorname{deg}_{y} (\tilde{F})=\operatorname{deg}_{x} (\tilde{G})\leq3\), where the indices indicate in which variable we consider that polynomial.
 (ii)
If \(s(y)=\operatorname{Res}_{x}(\tilde{F},\tilde{G})\) and \(r(x)=\operatorname{Res}_{y}(\tilde {F},\tilde{G})\), then \(r(x)=(1)^{\operatorname{deg}_{x}(\tilde{F})\cdot \operatorname{deg}_{x}(\tilde {G})} s(x)\).
Proof
 (i)
The proof follows from the fact \(\tilde{F}(x,y)=f(y,x)x\) and \(\tilde{G}(x,y)=f(x,y)y\).
 (ii)
Let \(\tilde{F}(x,y):=\sum_{i=0}^{3} a_{i}(y)x^{i}\) and \(\tilde{G}(x,y):=\sum_{i=0}^{3} b_{i}(y)x^{i}\). Since \(\tilde{F}(x,y)=\tilde {G}(y,x)\), we obtain \(\tilde{F}(x,y):=\sum_{i=0}^{3} b_{i}(x)y^{i} \) and \(\tilde{G}(x,y):=\sum_{i=0}^{3} a_{i}(x)y^{i}\). From this and the definition of the Sylvester matrix we get \(r(x)=(1)^{\operatorname{deg}_{x}(\tilde{F})\cdot \operatorname{deg}_{x}(\tilde{G})} s(x)\).
Let \(P_{i}\), \(i=2,3,4,5\) be the polynomials in the Appendix. The following lemma gives us information as regards the number of minimal periodtwo solutions.
Lemma 3
 (i)
If \(A>0\) and \(D>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})=P_{1}(y) P_{2}(y)\). If \(P_{2}\not\equiv0\), then \(\sharp PS\leq3\).
 (ii)
If \(A=0\) and \(D>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde {G})=D^{3\operatorname{deg}_{x} (\tilde{G})}P_{1}(y)P_{2}(y)\). If \(P_{2}\not\equiv0\), then \(\sharp PS\leq3\).
 (iii)
If \(D=0\) and \(A>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde {G})=(1)^{\operatorname{deg}_{x} (\tilde{F})}A^{3\operatorname{deg}_{x} (\tilde{F})}P_{1}(y)P_{2}(y)\). If \(P_{2}\not\equiv0\), then \(\sharp PS\leq3\).
 (iv)
If \(A=D=0\), \(C+B>0\), \(C+G>0\), and \(B+E>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})=P_{1}(y)P_{3}(y)\). If \(P_{3}\not\equiv0\), then \(\sharp PS\leq2\).
 (v)
If \(A=D=0\), \(B=E=0\), and \(C>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde {G})=P_{1}(y)P_{4}(y)\). If \(P_{4}\not\equiv0\), then \(\sharp PS\leq1\).
 (vi)
If \(A=D=0\), \(C=G=0\), and \(B>0\), then \(\operatorname{Res}_{x}(\tilde{F},\tilde {G})=P_{1}(y)P_{5}(y)\). If \(P_{5}\not\equiv0\), then \(\sharp PS\leq1\).
Proof
Suppose that \(\{\Phi,\Psi\}\) is a minimal periodtwo solution (\(\Phi\neq\Psi\)). Then \(\tilde{F}(\Phi,\Psi)=\tilde{G}(\Psi ,\Phi)=0\), Lemma 2, and Theorem 7 imply \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})(\Psi) =\operatorname{Res}_{y}(\tilde{F},\tilde{G})(\Phi)=0\), and we see that \(P(\Phi)=P(\Psi)=0\). This implies that there exist at most \(\lfloor \operatorname{deg}(P)/2\rfloor\) isolated minimal periodtwo solutions. By using the package Mathematica, one can see that the statements (i)(vi) are true. □
The following lemma gives the necessary conditions under which an isolated periodtwo solution is nonhyperbolic.
Lemma 4
 (i)
If (1) has a nonhyperbolic periodtwo solution, then \(\operatorname{Dis}(P)=0\).
 (ii)If (1) has a nonhyperbolic equilibrium point, thenwhere \(\operatorname{Dis}(P_{1})\) is given in the Appendix.$$\operatorname{Dis}(P_{1}\cdot P)=\operatorname{Dis}(P_{1}) \operatorname{Dis}(P)\operatorname{Res}(P_{1},P)^{2}=0, $$
Proof
Let \(\mathcal{B}(E_{})\) be the basin of attraction of \(E_{}(\bar{x}_{},\bar{x}_{})\) and \(\mathcal{B}(\infty,\infty)\) be the basin of attraction of \((\infty,\infty)\). The following lemma is true.
Lemma 5
 (i)
If \(Q_{1}(E_{+})=\{(x,y): x\geq\bar{x}_{+} \textit{ and } y\geq\bar{x}_{+}\}\), then \(\operatorname{int}( Q_{1}(E_{+}))\subset\mathcal{B}(\infty,\infty)\).
 (ii)
If \(Q_{3}(E_{+})=\{(x,y): 0\leq x\leq\bar{x}_{+} \textit{ and } 0\leq y\leq\bar {x}_{+}\}\), then \(\operatorname{int}(Q_{3}(E_{+}))\subset\mathcal{B}(\bar{x}_{},\bar{x}_{})\).
Proof
Assume that \((x_{0},y_{0})\in \operatorname{int}(Q_{3}(E_{+}))\). By Theorem 6 [18] there exists \((\tilde{x}_{0},\tilde{x}_{1})\in \operatorname{int}(Q_{3}(E_{+}))\) such that \((x_{0},y_{0})\preceq_{ne}(\tilde{x}_{0},\tilde{x}_{1})\) and \((\bar {x}_{},\bar{x}_{})\preceq_{ne}(\tilde{x}_{0},\tilde{x}_{1})\), and \(T(\tilde {x}_{0},\tilde{x}_{1})\preceq_{ne}(\tilde{x}_{0},\tilde{x}_{1})\). By monotonicity of T we obtain \(T^{i+1}(\tilde{x}_{0},\tilde {x}_{1})\preceq_{ne}T^{i}(\tilde{x}_{0},\tilde{x}_{1})\prec_{ne} E_{+}\), which implies \(T^{n}(\tilde{x}_{0},\tilde{x}_{1})\to(\bar{x}_{},\bar{x}_{})\) as \(n\to\infty\). Since \((0,0)\preceq_{ne}T(0,0)\) we have \(T^{i}(0,0)\preceq _{ne}T^{i+1}(0,0)\prec_{ne} (\bar{x}_{},\bar{x}_{})\), which implies \(T^{n}(0,0)\to(\bar{x}_{},\bar{x}_{})\) as \(n\to\infty\). Similarly, one can prove \(\operatorname{int}( Q_{1}(E_{+}))\subset\mathcal{B}(\infty,\infty)\). □
In view of Theorem 2 it is easy to see that \(\{ T^{n}(x_{0},y_{0})\}\) is either asymptotic to \((\infty,\infty)\) or converges to a periodtwo solution, for all \((x_{0},y_{0})\in\mathcal{R}=[0,\infty)^{2}\). In view of Lemma 3 we can suppose that \(\operatorname{Res}_{x}(\tilde{F},\tilde{G})=P_{1}(y) P(y)\), \(P\in\mathbb{R} [y]\). If \(P\not\equiv0\) and \(\operatorname{\operatorname{Dis}}(P_{1}\cdot P)=\operatorname{Dis}(P_{1}) \operatorname{Dis}(P)\operatorname{Res}(P_{1},P)^{2}\neq0\), then by Theorem 12 and Lemma 4 we see that \(E_{+}(\bar{x}_{+},\bar {x}_{+})\) is a repeller or a saddle point and all minimal periodtwo solutions are hyperbolic. By Lemma 5 we see that \(\operatorname{int}(Q_{3}(E_{+}))\subset\mathcal {B}(\bar{x}_{},\bar{x}_{})\) and \(\operatorname{int}( Q_{1}(E_{+}))\subset\mathcal{B}(\infty,\infty)\). Let \(\mathcal{S}_{1}\) denote the boundary of \(\mathcal{B}(\bar{x}_{},\bar{x}_{})\) considered as a subset of \(Q_{2}(E)\) and \(\mathcal{S}_{2}\) denote the boundary of \(\mathcal{B}(\bar{x}_{},\bar{x}_{})\) considered as a subset of \(Q_{4}(E_{+})\). It is easy to see that \(E_{+}\in\mathcal{S}_{1}\), \(E_{+}\in\mathcal{S}_{2}\) and \(T(\mathcal{R})\subset \operatorname{int}(\mathcal{R})\).
The proof of the following lemma for a cooperative map is the same as the proof of Claims 1 and 2 [21] for a competitive map, so we skip it.
Lemma 6
 (a)
If \((x_{0},y_{0})\in\mathcal{B}(\bar{x}_{},\bar{x}_{})\), then \((x_{1},y_{1})\in\mathcal{B}(\bar{x}_{},\bar{x}_{})\) for all \((x_{1},y_{1})\preceq_{ne} (x_{0},y_{0})\).
 (b)
If \((x_{0},y_{0})\in\mathcal{S}_{1}\cup\mathcal{S}_{2}\), then \((x_{1},y_{1})\in \operatorname{int}(\mathcal{B}(\bar{x}_{},\bar{x}_{}))\) for all \((x_{1},y_{1})\ll_{ne} (x_{0},y_{0})\).
 (c)
\(\mathcal{S}_{1} \cap \operatorname{int}(Q_{2}(E_{+}))\neq\emptyset\) and \(\mathcal {S}_{2} \cap \operatorname{int}(Q_{4}(E_{+}))\neq\emptyset\).
 (d)
\(T(\mathcal{S}_{1}\cup\mathcal{S}_{2})\subseteq\mathcal {S}_{1}\cup\mathcal{S}_{2}\).
 (e)
\((x_{0},y_{0}), (x_{1},y_{1})\in\mathcal{S}_{1}\cup\mathcal {S}_{2}\Rightarrow(x_{0},y_{0})\ll_{se} (x_{1},y_{1})\textit{ or } (x_{1},y_{1})\ll _{se} (x_{0},y_{0})\).
 (f)
\(\mathcal{S}_{1}\cup\mathcal{S}_{2}\) is the graph of continuous strictly decreasing function.
Theorem 19
Suppose that \(P\not\equiv0\) and \(\operatorname{Dis}(P_{1}\cdot P)\neq0\). If \(E_{+} \) is a repeller, then there exists at least one minimal periodtwo solution \(\{(\phi,\psi),(\psi,\phi)\}\), which is a saddle point, such that \((\phi,\psi)\ll_{se}E_{+}\ll_{se} (\psi,\phi)\).
Proof
In view of Lemma 6 we see that \((\mathcal{S}_{1}\cup \mathcal{S}_{2},\ll_{se})\) is a totally ordered set which is invariant under T. If \((x_{0},y_{0})\in(\mathcal{S}_{1}\cup\mathcal{S}_{2})\setminus \{E_{+}\}\), then \(\{T^{(2n)}(x_{0},y_{0})\}\) is eventually componentwise monotone. Since \(\mathcal{S}_{1}\cup\mathcal{S}_{2}\) is the graph of a continuous strictly decreasing function, there exists a minimal periodtwo solution \(\{(\Phi,\Psi),(\Psi,\Phi)\} \in(\mathcal{S}_{1}\cup\mathcal {S}_{2})\setminus\{E_{+}\}\) such that \(T^{(2n)}(x_{0},y_{0})\to(\Phi,\Psi)\) as \(n\to\infty\). Since \(\mathcal {S}_{1}\cup\mathcal{S}_{2}=\partial\mathcal{B}(\bar{x}_{},\bar{x}_{})\) is a closed set, we see that \(\{(\Phi,\Psi),(\Psi,\Phi)\}\) belongs to \((\mathcal {S}_{1}\cup\mathcal{S}_{2})\setminus\{E_{+}\}\). By Lemma 4 all periodtwo solutions are hyperbolic. Since \(\{(\Phi,\Psi),(\Psi,\Phi )\}\in\partial\mathcal{B}(\bar{x}_{},\bar{x}_{})\), it is not locally asymptotically stable. Thus \(\{(\Phi,\Psi),(\Psi,\Phi)\}\) must be a saddle point. □
Corollary 1
Suppose that \(P\not\equiv0\) and \(\operatorname{Dis}(P_{1}\cdot P)\neq0\). If \(E_{+}(\bar {x}_{+},\bar{x}_{+})\) is a repeller, then \(\operatorname{int}(Q_{2}(E_{+})) \cup \operatorname{int}(Q_{4}(E_{+}))\) contains one or three distinct minimal periodtwo solutions. If T has one minimal periodtwo solution \(\{ (\Phi_{1},\Psi_{1}),(\Psi_{1},\Phi_{1})\}\), then it is a saddle point and \((\Phi _{1},\Psi_{1})\ll_{se} E_{+} \ll_{se}(\Psi_{1},\Phi_{1})\). If T has three minimal periodtwo solutions \(\{(\Phi_{i},\Psi_{i}), (\Psi_{i},\Phi_{i})\} _{i=1}^{3}\), then they are ordered in the SouthEast ordering. If \((\Phi _{3},\Psi_{3})\ll_{se}(\Phi_{2},\Psi_{2})\ll_{se}(\Phi_{1},\Psi_{1})\ll_{se} E_{+} \ll _{se}(\Psi_{1},\Phi_{1})\ll_{se}(\Psi_{2},\Phi_{2})\ll_{se}(\Psi_{3},\Phi_{3})\), then odd indexed periodtwo points are saddles and even indexed periodtwo points are repellers.
Proof
By Lemma 4 all equilibrium points and minimal periodtwo solutions are hyperbolic. In view of Theorem 7 we see that \(\tilde{F}\) and \(\tilde{G}\) have no common component. From Lemma 3 the number of minimal periodtwo solutions is at most three. In view of Theorem 19, T has at least one minimal periodtwo solution, which is a saddle point. Assume that T has two minimal periodtwo solutions \(\{(\Phi_{1},\Psi_{1}),(\Psi_{1},\Phi_{1})\}\) and \(\{(\Phi_{2},\Psi_{2}),(\Psi _{2},\Phi_{2})\}\). Assume that \(\{(\Phi_{1},\Psi_{1}),(\Psi_{1},\Phi_{1})\}\) is a saddle point and \((\Phi_{1},\Psi_{1})\ll_{se} E_{+} \ll_{se}(\Psi_{1},\Phi_{1})\). Further, suppose that \((\Phi_{2},\Psi_{2})\ll_{se}(\Phi_{1},\Psi_{1})\ll_{se} E_{+} \ll_{se}(\Psi_{1},\Phi_{1})\ll_{se}(\Psi_{2},\Phi_{2})\). The map \(T^{2}\) satisfies all conditions of Theorem 3, which yields the existence of the global stable manifolds \(\mathcal{W}^{s}(\{(\Phi_{1},\Psi_{1}),(\Psi_{1},\Phi_{1})\} )\), the union of two curves \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\) and \(\mathcal{W}^{s}((\Psi_{1},\Phi_{1}))\) that have a common endpoint \(E_{+}\). Then \(\mathcal{W}^{s}(\Phi_{1},\Psi_{1})\) has the second endpoint at \((\Phi _{2},\Psi_{2})\) and \(\mathcal{W}^{s}(\Psi_{1},\Phi_{1})\) has the second endpoint at \((\Psi_{2},\Phi_{2})\). Furthermore, the minimal periodtwo solution \(\{(\Phi_{2},\Psi_{2}),(\Psi _{2},\Phi_{2})\}\) is a repeller. Since the global stable manifold is unique, the set \((\mathcal{S}_{1}\cap Q_{2}(\Phi_{2},\Psi_{2}))\cup(\mathcal {S}_{2}\cap Q_{4}(\Psi_{2},\Phi_{2}))\) is invariant under T. Similarly, as in Theorem 19, one can prove that \(\operatorname{int}(Q_{2}(\Phi_{2},\Psi_{2}))\cup \operatorname{int}(Q_{4}(\Psi_{2},\Phi_{2}))\) contains exactly one minimal periodtwo solution, which is a saddle point. Hence, if T has two minimal periodtwo solutions, then there exists a third minimal periodtwo solution. This proves the lemma. If \((\Phi_{1},\Psi_{1})\ll_{se}(\Phi_{2},\Psi _{2})\ll_{se} E_{+} \ll_{se}(\Psi_{2},\Phi_{2})\ll_{se}(\Psi_{1},\Phi_{1})\) the proof is similar and will be omitted. □
Corollary 2
Assume that \(P\not\equiv0\) and \(\operatorname{Dis}(P_{1}\cdot P)\neq0\). If \(E_{+}(\bar {x}_{+},\bar{x}_{+})\) is a saddle point, then \(\operatorname{int}(Q_{2}(E_{+})) \cup \operatorname{int}(Q_{4}(E_{+}))\) contains either zero or two minimal periodtwo solutions \(\{(\Phi_{i},\Psi_{i}),(\Psi_{i},\Phi_{i})\}\), \(i=1,2\), which are ordered to the SouthEast ordering. If there exist two minimal periodtwo solutions such that \((\Phi_{2},\Psi_{2})\ll_{se}(\Phi _{1},\Psi_{1})\ll_{se} E_{+} \ll_{se}(\Psi_{1},\Phi_{1})\ll_{se}(\Psi_{2},\Phi_{2})\), then an even indexed periodtwo point is a saddle and an odd indexed periodtwo point is a repeller.
Proof
The proof is similar to the proof of Corollary 1 and it will be omitted. □
Theorem 20
Suppose that \(P\not\equiv0\) and \(\operatorname{Dis}(P_{1}\cdot P)\neq0\). If \(E_{+}(\bar {x}_{+},\bar{x}_{+})\) is a repeller, then \(\operatorname{int}(Q_{2}(E_{+}))\cup \operatorname{int}(Q_{4}(E_{+}))\) contains one or three minimal periodtwo solutions \(\{(\Phi_{i},\Psi_{i}), (\Psi_{i},\Phi_{i})\}_{i=1}^{2n+1}\), where \(n=0\) or \(n=1\), such that \((\Phi_{i+1},\Psi_{i+1})\ll_{se} (\Phi_{i},\Psi_{i})\ll_{se} E_{+}\) and \(E_{+}\ll_{se}(\Psi_{i},\Phi_{i})\ll_{se} (\Psi_{i+1},\Phi_{i+1})\), and \((\Psi_{i},\Phi_{i})=T(\Phi_{i},\Psi_{i})\). Furthermore, the odd indexed periodtwo points are saddles and the even indexed periodtwo points are repellers and the following hold:
Proof
Theorem 21
Suppose that \(P\not\equiv0\) and \(\operatorname{Dis}(P_{1}\cdot P)\neq0\). If \(E_{+}(\bar {x}_{+},\bar{x}_{+})\) is a saddle point, then either T has no minimal periodtwo solution or \(\operatorname{int}(Q_{2}(E_{+}))\cup \operatorname{int}(Q_{4}(E_{+}))\) contains two distinct minimal periodtwo solutions and the following hold:
(b) If \(\operatorname{int}(Q_{2}(E_{+}))\cup \operatorname{int}(Q_{4}(E_{+}))\) contains two minimal periodtwo solutions \(\{(\Phi_{1},\Psi_{1}), (\Psi_{1},\Phi_{1})\}\) and \(\{(\Phi _{2},\Psi_{2}),(\Psi_{2},\Phi_{2})\}\) such that \((\Psi_{2},\Phi_{2})\ll_{se}(\Phi _{1},\Psi_{1})\ll_{se} E_{+} \ll_{se}(\Psi_{1},\Phi_{1})\ll_{se}(\Psi_{2},\Phi_{2})\) then \(\{(\Phi_{1},\Psi_{1}),(\Psi_{1},\Phi_{1})\}\) is a repeller and \(\{(\Phi _{2},\Psi_{2}),(\Psi_{2},\Phi_{2})\}\) is a saddle point.
Proof
The proof is similar to the proof of Theorem 20 and it will be omitted. □
All figures are generated by the software package Dynamics 3 [22].
Declarations
Acknowledgements
The authors are grateful to two anonymous referees for a number of helpful and constructive suggestions, which improved the presentation of results.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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