Local dynamics and global attractivity of a certain second-order quadratic fractional difference equation

We investigate the local and global character of the equilibrium and the local stability of the period-two solution of the difference equation xn+1=βxnxn−1+γxn−12+δxnBxnxn−1+Cxn−12+Dxn where the parameters β, γ, δ, B, C, D are nonnegative numbers which satisfy B+C+D>0 and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that Bxnxn−1+Cxn−12+Dxn>0 for all n≥0. MSC:39A10, 39A11, 39A30.


Introduction and preliminaries
In this paper we study the global dynamics of the following rational difference equation: , n = , , , . . ., () where the parameters β, γ , δ, B, C, D are nonnegative numbers which satisfy B + C + D >  and the initial conditions x - and x  are arbitrary nonnegative numbers such that Bx n x n- + Cx  n- + Dx n >  for all n ≥ .Equation (), which has been studied in [-], is a special case of a general second-order quadratic fractional equation of the form with nonnegative parameters and initial conditions such that A + B + C > , a + b + c + d + e + f >  and ax  n + bx n x n- + cx  n- + dx n + ex n- + f > , n = , , . . . .Several global asymptotic results for some special cases of () were obtained in [-].
The change of variable x n = /u n transforms () into the difference equation , n = , , . . ., () where we assume that δ + β + γ >  and that the nonnegative initial conditions u - , u  are such that δu  n- + γ u n + βu n- >  for all n ≥ .Thus the results of this paper extend to ().The first systematic study of global dynamics of a special quadratic fractional case of () where A = C = D = a = c = d =  was performed in [, ].The dynamics of some related quadratic fractional difference equations was considered in the papers [-].In this paper we will perform the local stability analysis of the unique equilibrium and the period-two solution and we will give the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a saddle point, a repeller or a non-hyperbolic equilibrium.The local stability analysis indicates that some possible dynamics scenarios for () include period-doubling bifurcations and Naimark-Sacker bifurcation and global attractivity of the equilibrium, see [, ].This means that the techniques we used in [, , -] are applicable.We will also obtain the global asymptotic stability results for ().As we have seen in [] an efficient way of studying the dynamics of () is considering the dynamics of  special cases of () which are obtained when one or more coefficients are set to zero.Based on our results in [], it is difficult to prove global asymptotic stability results of the unique equilibrium even for linear fractional difference equations; there are still two remaining cases one needs to study to prove the general conjecture that the local stability of the unique equilibrium implies the global stability.
Some interesting special cases of (), which were thoroughly studied in [], are the following equations.
() The Beverton-Holt difference equation when γ = δ = C = : In the case of () Theorems  and  give the following special results.
Corollary  If the following condition holds: where L >  and U are lower and upper bounds of all solutions of (), then x is globally asymptotically stable.
Corollary  If the following condition holds: where m = min{x, x - , x  } >  and M = max{x, x - , x  } are lower and upper bounds of a specific solution of (), then the unique equilibrium x is globally asymptotically stable on the interval [m, M].
In this paper we present the local stability analysis for the unique equilibrium and the period-two solutions of () and then we apply Corollaries  and  to some special cases of () to obtain global asymptotic stability results for those equations.The obtained results will give the regions of the parametric space where the unique positive equilibrium of () is globally asymptotically stable.In an upcoming manuscript we will give more precisely the dynamics in some special cases of () such as the case where the right-hand side of () is decreasing in x n and increasing in x n- ; here the theory of monotone maps can be applied to give the global dynamics.The application of the monotone map theory requires precise information on the local stability of the equilibrium solutions and the period-two solutions which will be given in this paper.See [, ] for an application of the monotone maps techniques to some competitive systems of linear fractional difference equations.These results will give the parameter regions where a global period-doubling bifurcation takes place, see [].The special cases of () where the unique equilibrium changes its stability character from the local stability to repeller are cases where the Naimark-Sacker bifurcation occurs, see [, ], and these cases will be treated in an upcoming manuscript.Following the approach from [], we divide () into  special cases of types (k, m) where k (resp.m) denotes the number of positive parameters in the numerator (resp.denominator).We summarize information as regards the stability of both the equilibrium solution and the period-two solution as well as the monotonic character of the right-hand side of the special cases of types (, ), (, ), (, ), (, ) and (, ) of () in Tables -.We did not include the cases of the type (, ), which are well known from [] as well as  cases a non-hyp.eq.
LAS for cδ < 2 a repeller for cδ > 2 a non-hyp.eq. for cδ = 2 no period-two solution of types (, ), (, ), and (, ) for which global stability will be given in Section .Using the techniques established in [-] one can determine the rate of convergence for all regions of parameters for which we established convergence.Some special cases of () have very interesting dynamics such as  where, in the case β ≤ , every solution converges to  although  is out of range of this equation.Another interesting example is the equation , where, in the case γ ≤ , every solution converges to  or to the unique periodtwo solution.It is interesting to notice that  is out of the range of this equation.Another interesting example is the equation , which has the property that if β ≥  every solution approaches ∞.None of these dynamics scenarios were possible in the case of the linear fractional difference equation, which is also a special case of () and which was studied in great detail in [].

Local stability of the positive equilibrium
In this section we investigate the equilibrium points of Eq. () where ) and where the initial conditions x - and x  are arbitrary nonnegative real numbers such Bx n x n- + Cx  n- + Dx n >  for all n ≥ .In view of the above restriction on the initial conditions of (), the equilibrium points of () are positive solutions of the equation or equivalently no eq.point for a non-hyp.for δ = 2γ 2 possible Naimark-Sacker bifurcation a saddle point for 3γ < 1 the unique positive equilibrium of () is given by the unique positive equilibrium of () is given by Finally when δ >  and B + C >  the only equilibrium point of () is the positive solution of the quadratic equation ().

Period-two solution and stability
Partial derivatives no minimal period-two sol.
a non-hyp.eq. for

Period-two solution and stability
Partial derivatives no minimal period-two sol.
a non-hyp.eq. for no minimal period-two sol.
In summary, it is interesting to observe that when () has a positive equilibrium x, then x is unique and it satisfies () and ().This observation simplifies the investigation of the local stability of the positive equilibrium of ().
Next, we investigate the stability of the positive equilibrium of ().Set If x denotes an equilibrium point of (), then the linearized equation associated with () about the equilibrium point x is where p = f u (x, x) and q = f v (x, x).

Theorem  Assume that
Then the unique equilibrium point Proof It is easy to see that Then the proof follows from Theorem .. in [] and the fact that Then the unique equilibrium point Proof It is easy to see that Then the proof follows from Theorem .. in [] and the fact that As we previously mentioned if the only equilibrium point of () is the positive solution of the quadratic equation ().By using the identity () Now if we set A straightforward computation gives Lemma  Let p and q be partial derivatives given by () and ().Assume that (c) Then q +  >  if and only if Proof (a) The inequality qp >  is equivalent to we find that qp >  is always true.(b) There are three cases to consider.From (), we have pq +  >  if and only if and from which the proof follows.
(ii) Assume B(D + βγ ) + C(D + β + γ ) < .Then pq +  >  if and only if x < ρ  .It is easy to see that In view of the left-hand side of () we see that () is equivalent to from which the proof follows.
(iii) If B(D + βγ ) + C(D + β + γ ) = , then the proof follows from ().(c) The inequality q > - is equivalent to It is easy to see that from which it follows that () is equivalent to Since C > , in view of () we find that () and () are equivalent to Then the unique equilibrium point

asymptotically stable if and only if any of the following holds:
(a) (ii) a repeller if and only if the following holds: (iii) a saddle point if and only if the following holds: (iv) a non-hyperbolic equilibrium if and only if any of the following holds: Proof The proof follows from Theorem .. in [] and Lemma . http://www.advancesindifferenceequations.com/content/2014/1/68

Existence of period-two solutions
Assume that {φ, ψ} is a minimal period-two solution of ().Then which is equivalent to Assume that φψ = .Subtracting equations () and () we get Dividing () by φ and () by ψ and subtracting them we get If we set φ + ψ = x and φψ = y, where x, y > , then φ and ψ are positive and different solutions of the quadratic equation In addition to the conditions x, y >  it is necessary that x  -y > . http://www.advancesindifferenceequations.com/content/2014/1/68 From () and () we get the system Theorem  For () the following holds: (i) If γ =  then () has no a minimal period-two solution.
(ii) Assume that C =  and δ > .By using () we see that x and y satisfy the following equations: () Assume that γ (Dβ + γ ) -Bδ = .The solution of system () is given by In this case the equation has positive distinct solutions which are given by If γ (Dβ + γ ) -Bδ = , it is easy to see that system () has no solutions from which follows that () has no minimal period-two solution.
(iii) Assume that δ = , γ >  and C > .By using () we find that x and y satisfy the following equations: () Assume that y =  and B = C.The solution of system () is given by and x, y >  ∧ x  -y >  if and only if In this case the equation has positive distinct solutions which are given by . http://www.advancesindifferenceequations.com/content/2014/1/68 If y = , then from () we have x = γ C , which implies that { γ C , } is the minimal periodtwo solution.
If y =  and B = C, then the rest of the proof follows from Lemma .
(iv) The proof follows from the proof of Lemma .
Then for () the following holds: then () has one minimal period-two solution {φ + , ψ + } where φ + and ψ + are solutions of equation t x + t + y + = .(iii) In all other cases () has no minimal period-two solution.
Proof It is clear that (x ± , y ± ) are solutions of system ().Then minimal period-two solutions are solutions of the equation t  - One can show that the following identities hold: from which the proof follows.http://www.advancesindifferenceequations.com/content/2014/1/68 Assume now that B = C and C > .Solving the second equation of system () for y we get Substituting () in the first equation of system () we see that x satisfies the following equation: In a similar way one can show that y satisfies the following equation: The solutions of () are given by where Then the solutions of the system () are given by If x i , y i >  and x  i -y i > , then () has minimal period-two solutions given by Let g be a function given by Eliminating φ and ψ from () and () implies that if {φ, ψ} is a minimal period-two solution; then g(φ) =  and g(ψ) =  with φ = ψ, from which it follows that

Local stability of period-two solutions
Let {φ, ψ} be a minimal period-two solution of ().Set u n = x n- and v n = x n for n = , , . . .and write () in the equivalent form for n = , , . . . .Let T be the function defined by Then {φ, ψ} is a fixed point of T  , the second iterate of T, and where . http://www.advancesindifferenceequations.com/content/2014/1/68 By definition .
Theorem  If C = , γ >  and δ > , then () has the minimal period-two solution {φ, ψ} where φ and ψ are given by () and () if and only if In this case the minimal period-two solution {φ, ψ} is a saddle point.
Proof The existence of the minimal period-two solution follows from Theorem .Now, we prove that the minimal period-two solution is a saddle point.Since G(φ, ψ) = φ we have Using () and the fact C =  we see that the Jacobian matrix of T  at the point {φ, ψ} is given by The determinant of the Jacobian matrix () is given by The trace of the Jacobian matrix () is given by Substituting () and () into () and () we find that the determinant of the Jacobian matrix () is given by , http://www.advancesindifferenceequations.com/content/2014/1/68 and the trace of the Jacobian matrix () is given by The period-two solution {φ, ψ} is a saddle point if and only if One can see that Since we have we have which implies Hence, we prove that  + det J T  (φ, ψ)tr J T  (φ, ψ) < .From () we have . http://www.advancesindifferenceequations.com/content/2014/1/68 Let h(x) = x  -γ x +γ  .Since the discriminant of h is negative we have h(x) >  for x ∈ R, which implies Since we have Theorem  Assume δ = , γ >  and C > .Then () has the minimal period-two solution {φ, ψ} where if and only if The minimal period-two solution {φ, ψ} is (i) locally asymptotically stable if Proof The existence of the minimal period-two solution follows from Theorem .Now, we investigate the stability of {φ, ψ}.The Jacobian matrix of T  at the point {φ, ψ} is given by where The determinant of the Jacobian matrix () is given by we find that From () and () it follows that The rest of the proof follows from Lemma .
we find that from which the inequality follows.http://www.advancesindifferenceequations.com/content/2014/1/68() In view of the assumption of the lemma we have from which the proof follows.
Lemma  If C > , γ >  and Proof (i) Assume that D > β and β >  holds.Observe that Since B < C, we obtain Similarly Assume for the sake of contradiction that Dβγ ≥ .Then -D + β + γ >  and The solutions of the equation H  (B) =  for B are given by Note that the following three identities hold: from which it follows that In view of Lemma , (), and () we get (iii) Assume that D > β.As in case (ii) we have The solutions of the equation H  (B) =  for B are given by Note that the following two identities hold: In view of (), (), (), D > β, Dβγ < , which by Lemma  implies that B -> B + > B. This and the fact that the coefficient of B  is positive imply H  (B) > .

Boundedness of solutions of (1)
In view of Theorems  and  and Corollaries  and , any result on the existence of lower and upper bounds of the solutions of () yields some global asymptotic stability result for the unique equilibrium of ().See Section  for such results.As we show in Remark  global asymptotic results obtained by application of Theorems  and  and Corollaries  and  are not sharp, in the sense that they do not cover the whole parametric region of global asymptotic stability, but they are robust as they can be applied as soon as we have the lower and upper bounds of the solutions.The problem of boundedness of all solutions of () is more difficult than the corresponding problem for the linear fractional equation and in view of its importance requires a separate paper.An additional difficulty in studying boundedness is the presence of quadratic terms.http://www.advancesindifferenceequations.com/content/2014/1/68Here we will give some equations for which the boundedness of all solutions is clear and leave the problem of determining the boundedness of all solutions for a future study.The boundedness or existence of unbounded solutions for all nine special cases of () of the type (, ) follows immediately from the corresponding properties of the linear equation obtained by the substitution x n = e u n and it can range from boundedness of all solutions, periodicity of all solutions with the same period, to the unboundedness of all solutions.All three special cases of () of the type (, ), and all three special cases of () of the type (, ) as well as the special case of the type (, ) have all solutions uniformly bounded that is, there are constants L, U, L < U such that every solution satisfies L ≤ x n ≤ U, n = , , . . . .All six special cases of () of the type (, ), where the term in the nominator is also present in the denominator are also uniformly bounded as well as three special cases of () of the type (, ), with corresponding terms in the nominator and the denominator.Two special cases of (), where B = C =  and B = C = γ =  allow the existence of unbounded solutions.The remaining  special cases of () require detailed study and probably new methods in determining boundedness of solutions and complete classification of all special cases of ().
Proof In view of Corollary  we need to find the lower and upper bounds for all solutions of () for n ≥ .
(i) In this case the lower and upper bounds for all solutions of () for n ≥  are derived as Remark  Equation (), where D = δ =  and all other coefficients are positive, reduces to the well-known equation () which was studied in great detail in [, ] and for which we have shown that the unique equilibrium is globally asymptotically stable if and only if is locally asymptotically stable, that is, if and only if condition (i) of Theorem  holds.This result is certainly better than the global asymptotic result we derive from Corollaries  and .
Remark  Equation (), where either B =  or C = , and all other coefficients are positive, can be treated with Corollary  and the global asymptotic stability of the equilibrium (whenever it exists) follows from condition () in the interval [min{x, x - , x  }, max{x, x - , x  }] when min{x, x - , x  } > , that is, when x - x  > .Similarly, (), where exactly one of the coefficients β, γ or δ is zero, and all other coefficients are positive, can be treated with Corollary  and the global asymptotic stability of the equilibrium follows http://www.advancesindifferenceequations.com/content/2014/1/68 from condition () in the interval [min{x, x - , x  }, max{x, x - , x  }] when min{x, x - , x  } > , that is, when x - x  > .In this case max{x, x - , x  } can be replaced by U = max{β,γ ,δ} min{B,C,D} .
n x n- + γ x  n- + δx n Bx n x n- + Cx  n- = βx n x n- + γ x  n- Bx n x n- + Cx  http