Global dynamics of quadratic second order ... - Semantic Scholar
Applied Mathematics and Computation 227 (2014) 50–65
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Global dynamics of quadratic second order difference equation in the first quadrant J. Bekteševic´ a, M.R.S. Kulenovic´ b,⇑, E. Pilav c a
Division of Mathematics, Faculty of Mechanical Engineering, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA c Department of Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina b
a r t i c l e
i n f o
a b s t r a c t We investigate the global behavior of a quadratic second order difference equation
Keywords: Attractivity Basins Difference equation Invariant Period-two solutions Stable manifold Unstable manifold
xnþ1 ¼ Ax2n þ Bxn xn1 þ Cx2n1 þ Dxn þ Exn1 þ F;
n ¼ 0; 1; . . .
with non-negative parameters and initial conditions. We find the global behavior for all ranges of parameters and determine the basins of attraction of all equilibrium points. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Consider the following difference equation
xnþ1 ¼ Ax2n þ Bxn xn1 þ Cx2n1 þ Dxn þ Exn1 þ F;
n ¼ 0; 1; . . .
ð1Þ
with non-negative parameters and initial conditions. We assume that A þ B þ C > 0 in order to avoid trivial linear case from our consideration and also A þ B þ C > 0 and A þ B þ D > 0; B þ C þ E > 0 in order to avoid the well-known cases of first order quadratic difference equations. Polynomial difference equations and corresponding maps have been studied in both real and complex domain and many results have been obtained, see [3–7,10]. Most known case of quadratic polynomial equations is Hénon equation
xnþ1 ¼ 1 þ bxn1 ax2n ;
n ¼ 0; 1; . . .
ð2Þ
which was considered by many authors, see [3–5,10] and references therein. Eq. (2) is a special case of Eq. (1). In this paper we restrict our attention to non-negative initial conditions and non-negative parameters which will make our results more special but also more precise and applicable. First papers on quadratic polynomial difference equations with non-negative initial conditions and non-negative parameters are [1,2], where the special case
xnþ1 ¼ Bxn xn1 þ Exn1 þ F;
n ¼ 0; 1; . . .
ð3Þ
of Eq. (1) was considered and results on global dynamics have been obtained. The obtained results in [1,2] provided the parts of the basins of attraction of different equilibrium points and periodic solutions. In this paper we give the precise description of all basins of attraction of all attractors of Eq. (1) as well as the point at infinity. In particular, we find the regions of the plane of initial conditions for which all solutions are bounded as well as the escape regions, that is the basins of attraction ⇑ Corresponding author. E-mail address: [email protected] (M.R.S. Kulenovic´). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.10.048
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of 1. Such sets have complicated structure in complex dynamics, see [3,6,7]. Our results are based on number of theorems which hold for monotone difference equations and corresponding monotone maps, which will be described in the next section. An interesting consequence of our results is the necessary and sufficient condition for the existence of periodtwo solutions. For example, if D ¼ E ¼ F ¼ 0 that is Eq. (1) contains only quadratic terms, the necessary and sufficient condition for the existence of the period-two solution is C > 3A þ B, in which case the period-two solution is a saddle point which stable manifold serves as the separatrix between two basins of attraction of a locally stable equilibrium point and the point at 1. The paper is organized as follows. The Section 2 presents the preliminary results from the theory of monotone maps. The Section 3 briefly gives the local stability analysis of all equilibrium points for all values of parameters. The Section 4 contains stability analysis of period-two solution which plays major role in our results. The Section 5 gives global results in all regions of parameters. The Section 6 provides two illustrative examples of global dynamics of some special cases of Eq. (1). 2. Preliminaries Consider the difference equation
xnþ1 ¼ f ðxn ; xn1 Þ;
n ¼ 0; 1; . . .
ð4Þ
where f is a continuous and increasing function in both variables. There are several global attractivity results for Eq. (4) which give the sufficient conditions for all solutions to approach a unique equilibrium. These results were used efficiently in monograph [14] to study the global behavior of solutions of second order linear fractional difference equation. Here we list some of these results that will be needed in this paper. The first result was obtained in [14] and it was extended to the case of higher order difference equations and systems in [12,15,16,18]. Theorem 1. Let ½a; b be an interval of real numbers and assume that
f : ½a; b ½a; b ! ½a; b is a continuous function satisfying the following properties: (a) f ðx; yÞ is non-decreasing in each of its arguments; (b) Eq. (4) has a unique equilibrium x 2 ½a; b. Then every solution of Eq. (4) converges to x. The following result has been obtained in [1]. Theorem 2. Let I # R and let f 2 C½I I; I be a function which increases in both variables. Then for every solution of Eq. (4) the 1 subsequences fx2n g1 n¼0 and fx2nþ1 gn¼1 of even and odd terms of the solution do exactly one of the following: (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 Eq. (1) approaches either an equilibrium solution or periodtwo solution or the finite point at the boundary and every unbounded solution is asymptotic to the point at infinity in a monotonic way. Thus the major problem in dynamics of Eq. (1) is the problem of determining the basins of attraction of three different types of attractors: the equilibrium solutions, period-two solution(s) and the point(s) at infinity. The following two results can be proved by using the techniques of proof of Theorem 11 in [8]. Theorem 3. Consider Eq. (4) where f is increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E1 ðx1 ; y1 Þ and E2 ðx2 ; y2 Þ are two consecutive equilibrium points in North-East ordering that satisfy
ðx1 ; y1 Þne ðx2 ; y2 Þ; that is x1 6 x2 ; y1 6 y2 . Assume that E1 is a saddle point or a non-hyperbolic point with second characteristic root in interval ð1; 1Þ, with the neighborhoods where f is strictly increasing and E2 is a local attractor. Then the basin of attraction BðE2 Þ of E2 is the region above the global stable manifolds W s ðE1 Þ. More precisely
BðE2 Þ ¼ fðx; yÞ : 9yl : yl < y;
ðx; yl Þ 2 W s ðE1 Þg:
The basin of attraction BðE1 Þ ¼ W s ðE1 Þ is exactly the global stable manifold of E1 . If there exists a period-two solution, then the end points of the global stable manifold are exactly the period two solution.
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Theorem 4. Consider Eq. (4) where f is increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E1 ðx1 ; y1 Þ and E2 ðx2 ; y2 Þ are two consecutive equilibrium points in North-East ordering that satisfy
ðx1 ; y1 Þne ðx2 ; y2 Þ and that E1 is a local attractor and E2 is a saddle point or a non-hyperbolic point with second characteristic root in interval ð1; 1Þ, with the neighborhoods where f is strictly increasing. Then the basin of attraction BðE1 Þ of E1 is the region below the global stable manifolds W s ðE2 Þ. More precisely
BðE1 Þ ¼ fðx; yÞ : 9yu : y < yu ;
ðx; yu Þ 2 W s ðE2 Þg:
The basin of attraction BðE2 Þ ¼ W s ðE2 Þ is exactly the global stable manifold of E2 . The global stable manifold extend to the boundary of the domain of Eq. (4). If there exists a period-two solution, then the end points of the global stable manifold are exactly the period two solution. Theorem 5. If D þ E > 1 then every solution fxn g of Eq. (1) satisfies limn!1 xn ¼ 1. Proof. If fxn g is a solution of Eq. (1) then fxn g satisfies the inequality
xnþ1 P Dxn þ Exn1 ;
n ¼ 0; 1; . . . ;
which in view of the result on difference inequalities, see Theorem 1.4.1 in [9,13], implies that xn P Ln ; n P 1 where fLn g is a solution of the initial value problem
Lnþ1 ¼ DLn þ ELn1 ;
L1 ¼ x1 ;
L0 ¼ x0 ;
n ¼ 0; 1; . . . :
Consequently,
1 xn P Ln ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEx1 þ kþ x0 Þknþ þ ðk x1 þ Ex0 Þkn ; 2 D þ 4E
n ¼ 1; 2; . . . ;
where
D
k ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þ 4E ; 2
which implies limn!1 xn ¼ 1.
h
Remark 1. Theorem 5 implies that interesting dynamics for Eq. (1) happens in the parametric region D þ E 6 1, which will be main objective of the rest of the paper. We will also describe more precisely global dynamics even in the parametric region D þ E > 1, where we describe the manifold that solutions will be following on their way to 1. 3. Local stability analysis of equilibrium solutions x of Eq. (1) satisfy the quadratic equation The equilibrium solutions
ðA þ B þ CÞx2 þ ðD þ E 1Þx þ F ¼ 0:
ð5Þ
This immediately leads to the following cases: (i) (ii) (iii) (iv)
there there there there
exists zero equilibrium x ¼ 0 if and only if F ¼ 0; exist no equilibrium solutions if and only if D ¼ ðD þ E 1Þ2 4FðA þ B þ CÞ < 0; exist one equilibrium solution if and only if D ¼ 0; exist two equilibrium solutions if and only if D > 0;
The positive equilibrium solutions, when exist are given as
x ¼
pffiffiffiffi 1DE D : 2ðA þ B þ CÞ
ð6Þ
The linearized equation at the zero equilibrium is
znþ1 ¼ Dzn þ Ezn1 :
ð7Þ
In view of local stability Theorem 1.1 from [14] we obtain the following result on local stability of the zero equilibrium: Proposition 1. The zero equilibrium of Eq. (1) is one of the following:
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(a) (b) (c) (d) (e)
53
locally asymptotically stable if D þ E < 1, non-hyperbolic and locally stable if D þ E ¼ 1, unstable if D þ E > 1, saddle point if D > jE 1j, repeller if 1 E < D < E 1.
x is The linearized equation at the positive equilibrium solutions
znþ1 ¼ pzn þ qzn1 ¼ ðð2A þ BÞx þ DÞzn þ ððB þ 2CÞx þ EÞzn1 ;
ð8Þ
In view of Theorem 1.1 from [14] we obtain the following result on local stability of the positive equilibrium: Proposition 2. The positive equilibrium solution of Eq. (1) is one of the following:
(a) (b) (c) (d) (e)
locally asymptotically stable if p þ q < 1, non-hyperbolic and locally stable if p þ q ¼ 1, unstable if p þ q > 1, saddle point if p > jq 1j, repeller if 1 q < p < q 1.
The precise description of local stability of positive equilibrium solutions is given with the following result. Theorem 6. If
D < 1 E and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then Eq. (1) has the equilibrium point x where
x ¼
pffiffiffiffi 1DE D 2ðA þ B þ CÞ
and the following holds: (i) x is locally asymptotically stable if
4ðA þ B þ CÞF < ðE þ D 1Þ2 : (ii) x a non-hyperbolic point if
4ðA þ B þ CÞF ¼ ðE þ D 1Þ2 :
Proof. It is easy to see that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2A BÞ ðD þ E 1Þ2 4FðA þ B þ CÞ 2ðE 1ÞA þ BðD E þ 1Þ þ 2CD @f p¼ ðx ; x Þ ¼ @u 2ðA þ B þ CÞ and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB 2CÞ ðD þ E 1Þ2 4FðA þ B þ CÞ þ 2ðEA CD þ CÞ þ BðD þ E þ 1Þ @f q¼ ðx ; x Þ ¼ @v 2ðA þ B þ CÞ and
1qp ¼
1qþp ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD þ E 1Þ2 4FðA þ B þ CÞ P 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC AÞ ðD þ E 1Þ2 4FðA þ B þ CÞ 2AðE 1Þ þ BðD E þ 1Þ þ 2CD AþBþC
q þ 1 ¼ ðB þ 2Cx Þ þ E þ 1 > 0: The proof of the statements of the theorem follows from Proposition 2. h
:
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Lemma 1. If
D þ E < 1 and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then
1 q þ p P 0: Proof. Since D þ E < 1 we have that E < 1 and
2AðE 1Þ þ BðD E þ 1Þ þ 2CD > 0; which implies 1 q þ p > 0 if C P A. Now, assume that C < A. One can show that
ð2Að1 EÞ þ Bð1 þ D EÞ þ 2CDÞ2 ðC AÞ2 D ¼ ðA þ B þ CÞðð1 þ D EÞðAðD þ 3E 3Þ þ BðD E þ 1Þ þCð3D þ E 1ÞÞ þ 4FðA CÞ2 ; where
D ¼ ðD þ E 1Þ2 4FðA þ B þ CÞ: Since D þ E < 1 we have that D þ 3E 3 < 0 and
BðD E þ 1Þ þ Cð3D þ E 1Þ ðB þ 2CÞðD E þ 1Þ C ¼ < 0; D þ 3E 3 D þ 3E 3 from which it follows that
BðD E þ 1Þ þ Cð3D þ E 1Þ < C < A; D þ 3E 3 which is equivalent to
AðD þ 3E 3Þ þ BðD E þ 1Þ þ Cð3D þ E 1Þ > 0: This implies
ð2Að1 EÞ þ Bð1 þ D EÞ þ 2CDÞ2 ðC AÞ2 D > 0; from which the proof follows. h Theorem 7. If
D < 1 E and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then Eq. (1) has the equilibrium point xþ where
xþ ¼
1DEþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD þ E 1Þ2 4FðA þ B þ CÞ 2ðA þ B þ CÞ
and the following holds: (i) xþ is a repeller if
A 0; WU > 0
and
q2 ðA B þ CÞ2 4ðA B þ CÞðAq2 þ Dq þ FÞ > 0, that is when
1þDE > 0 and A B þ C > 0 and q2 ðA B þ CÞ > 4ðAq2 þ Dq þ FÞ; CA
ð11Þ
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which is equivalent to
1þDE > 0; A B þ C > 0 and CA
ð1 þ D EÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ ðA CÞ2
> 4F:
ð12Þ
Lemma 2. For Eq. (1) the following holds: (i) If A < C then Eq. (1) has minimal period-two solution if and only if
E 1 < D and F < (ii) Eq. (a) (b) (c)
ðD E þ 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ 4ðA CÞ2
:
ð13Þ
(1) has no minimal period-two solution if any of the following conditions hold: A P C; E þ D > 1; D 6 0.
Proof. (i) It is sufficient to prove that (13) implies A B þ C > 0. Indeed, (13) implies that
AðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1Þ > 0; which is equivalent to
B
0 and AE þ A þ CD > AE þ A þ AD ¼ Að1 E þ DÞ > 0. This implies
B < 3A þ C þ
4DðA CÞ < A þ C; DEþ1
from which it follows that A B þ C > 0. (ii) (a) Assume that A P C. Then from (12) we obtain that 1 þ D E < 0 which implies E þ 3D 1 > 0. Since
AðD þ 3E 3Þ 2AðD E þ 1Þ A¼ > 0; 3D þ E 1 3D þ E 1 we have that
AðD þ 3E 3Þ > A P C; 3D þ E 1 which implies that
AðD þ 3E 3Þ Cð3D þ E 1Þ > 0; from which it follows that
ð1 þ D EÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ ðA CÞ2
4F < 0;
which is a contradiction to (12). This proves the statement (i). (b) Assume that E þ D > 1. If A P C the proof follows from (i). Now, suppose that A < C and that Eq. (1) has minimal period-two solution. Then from (12) we obtain that 1 þ D E > 0 and
AðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1Þ > 0; which is equivalent to
06B
0 which is a contradiction. h For example, the conditions for existence of period-two solutions are
B ¼ 2A; C A ¼ 1;
D ¼ E;
4ðA þ D þ F Þ < 1:
Set
un ¼ xn1
v n ¼ xn ;
and
n ¼ 0; 1; . . .
and write Eq. (1) in the equivalent form:
unþ1 ¼ v n
v nþ1 ¼ Av 2n þ Bv n un þ Cu2n þ Dv n þ Eun þ F: Let T be a function on ½0; 1Þ ½0; 1Þ defined by
T Then
u
v U W
¼
v 2
2
Av þ Bv u þ Cu þ Dv þ Eu þ F
:
is fixed point of T 2 , second iterate of T. Furthermore
T2
u
v
¼
g ðu; v Þ
hðu; v Þ
where
g ðu; v Þ ¼ Av 2 þ Bv u þ Cu2 þ Dv þ Eu þ F; hðu; v Þ ¼ A½g ðu; v Þ2 þ Bv g ðu; v Þ þ Dg ðu; v Þ þ C v 2 þ Ev þ F: The period-two solution is locally asymptotically stable if the eigenvalues of the Jacobian matrix J T 2 , evaluated at inside the unit disk. By definition
JT 2
U W
¼
@g ð @u @h ð @u
U; WÞ U; WÞ
@g ð @v @h ð @v
U; WÞ U; WÞ
! :
U W
lie
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By computing the partial derivatives of g and h and by using the fact that g ðU; WÞ ¼ U, we find the following identities:
@g ðU; WÞ ¼ BW þ 2C U þ E; @u @g ðU; WÞ ¼ 2AW þ BU þ D; @v and
@h @g ðU; WÞ ¼ ðU; WÞ½2Ag ðU; WÞ þ BW þ D @u @u ¼ ðBW þ 2C U þ EÞð2AU þ BW þ DÞ @h @g ðU; WÞ ¼ ðU; WÞ½2Ag ðU; WÞ þ BW þ D þ Bg ðU; WÞ þ 2C W þ E @v @v ¼ ð2AW þ BU þ DÞð2AU þ BW þ DÞ þ BU þ 2C W þ E: Set
@g @h ðU; WÞ þ ðU; WÞ @u @v ¼ ðBW þ 2AU þ DÞðBU þ 2AW þ DÞ þ ðB þ 2C ÞðU þ WÞ þ 2E
S¼
¼ 2ABðU þ WÞ2 þ ½DðB þ 2AÞ þ ðB þ 2C ÞðU þ WÞ þ ð2A BÞ2 UW þ D2 þ 2E ¼ 2ABq2 þ ½DðB þ 2AÞ þ ðB þ 2C Þq þ ð2A BÞ2
Aq2 þ Dq þ F þ D2 þ 2E ABþC
and
@g @h @g @h ðU; WÞ ðU; WÞ ðU; WÞ ðU; WÞ @u @v @v @u ¼ ðBW þ 2C U þ EÞðBU þ 2C W þ EÞ
D¼
¼ 2BC ðU þ WÞ2 þ EðB þ 2C ÞðU þ WÞ þ ð2C BÞ2 UW þ E2 ¼ 2BCq2 þ EðB þ 2C Þq þ ð2C BÞ2
Aq2 þ Dq þ F þ E2 : ABþC
It is easy to see that
S D 1 ¼ BðA C Þq2 þ 2ðAD þ C CEÞq þ 4ðA C Þ Aq2 þ Dq þ F þ ðD þ E 1ÞðD E þ 1Þ
2ðAD þ C CEÞ q 4 Aq2 þ Dq þ F þ ðD þ E 1Þq ¼ ðC AÞ Bq2 þ ð C AÞ
2AD ðA þ C ÞðE 1Þ 2 q ¼ ðC AÞ ð4A þ BÞq 3Dq 4F þ CA ¼
ðD þ E 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ þ 4FðA CÞ2 : AC
Theorem 8. If Eq. (1) has the minimal period-two solution then it is a saddle point. Proof. A minimal period-two solution is a saddle point if and only if
jSj > j1 þ Dj and S 2 4D > 0: Since S > 0 and D > 0 then (15) is equivalent to
S D 1 > 0; since
S 2 4D > ðD þ 1Þ2 4D ¼ ðD 1Þ2 P 0: In view of Lemma 2 we have that
SD1¼
ðD þ E 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ þ 4FðA CÞ2 > 0; AC
ð15Þ
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since A < C and
E 1 < D and F
þ ¼0 AþBþC AþBþC
p þ ðq 1Þ ¼ð2A þ BÞx þ D þ ðB þ 2CÞx þ E 1 ¼
and
p ðq 1Þ ¼ð2A þ BÞx þ D ððB þ 2CÞx þ EÞ þ 1 ¼
ð2A þ BÞð1 EÞ þ DðC AÞ ðC AÞð1 EÞ DðB þ 2CÞ AþBþC AþBþC
¼
ð3A þ B CÞð1 EÞ þ Dð3C þ B AÞ AþBþC
>
Dð3A þ B C Þ þ DðB A þ 3C Þ ¼ 2D > 0: AþBþC
Thus the results follows from Proposition2. h Now the global result follows from Theorem 4.
Theorem 10. Consider Eq. (1) under the conditions F ¼ 0 and D þ E < 1. Assume that Eq. (1) has the minimal period-two solution . . . ; U; W; U; W; . . ., that is assume that (13) is satisfied. Then the zero equilibrium is locally asymptotically stable and the positive equilibrium xþ is repeller. In this case there exist four continuous curves W s ðP 1 Þ; W s ðP 2 Þ; W u ðP 1 Þ; W u ðP2 Þ, where W s ðP1 Þ; W s ðP2 Þ are passing through the point Eþ ð xþ ; xþ Þ, and are graphs of decreasing functions. The curves W u ðP1 Þ; W u ðP 2 Þ are the graphs of increasing functions and are starting at E0 ¼ ð0; 0Þ. Every solution fxn g which starts below W s ðP1 Þ [ W s ðP 2 Þ in North-east ordering converges to E0 and every solution fxn g which starts above W s ðP 1 Þ [ W s ðP2 Þ in North-east ordering satisfies lim xn ¼ 1.
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Fig. 1. Visual illustration of Theorem 9. Dashed decreasing graph represents the stable manifold and dashed increasing graph represents the unstable manifold of Eþ .
Proof. For example, the condition (13) is satisfied if
B ¼ 2A;
C ¼ A þ 1;
D¼E
ðD þ EÞxn ¼ xn , which is a contradiction. If the subsequences fx2n g and fx2nþ1 g are eventually monotone, then without loss of generality we can assume that fx2n g is eventually non-decreasing and fx2nþ1 g is eventually non-increasing. In this case x2n ! 1 which would imply that x2nþ1 ! 1, which is a contradiction. Thus the remaining possibility is that fxn g is eventually increasing, which implies that xn ! 1 as n ! 1. Another way of proving the global behavior in the case when the equilibrium point E is non-hyperbolic was used in [11,17]. To complete this we will find the image of E þ tv , where t > 0 and v is the eigenvector that corresponds to the eigenvalue 1, under the map T. Since E þ tv ¼ ðt; tÞ, we have
TðE þ t v Þ ðE þ t v Þ ¼ Tððt; tÞÞ ðt; tÞ ¼ ð0; ðA þ B þ CÞt 2 þ ðD þ E 1ÞtÞ: By using the condition D þ E 1 ¼ 0, we have that
TðE þ t v Þ ðE þ t v Þ ¼ ð0; ðA þ B þ CÞt 2 Þ; which implies E þ t v ne TðE þ tv Þ for every t > 0. This shows that every point in ð x; 1Þ2 is a supersolution, see [17], and so every solution tends to 1. (c) In view of E > 1 þ D from Eq. (1) we have that xnþ1 > Exn1 which implies xnþ1 > Ek x1 or xnþ1 > Ek x0 for some k such that k ! 1 as n ! 1. Consequently every solution fxn g of Eq. (1) satisfies lim xn ¼ 1. h 5.2. Global dynamics of Eq. (1): case F > 0; D < 0 In this case Eq. (1) has no equilibrium solutions and in view of Theorem 2 every bounded solution of Eq. (1) converges to a period-two solution while every unbounded solution approaches 1.(See Fig. 3) Theorem 12. Consider Eq. (1) under the conditions F > 0; D < 0. Then every solution fxn g of Eq. (1) satisfies limn!1 xn ¼ 1 Proof. The proof follows from Lemma 2 which shows that the minimal period-two solution does not exist. h 5.3. Global dynamics of Eq. (1): case F > 0; D ¼ 0 In this case Eq. (1) has a unique equilibrium solution
x ¼
1DE ; 2ðA þ B þ CÞ
D þ E < 1;
and in view of Theorem 2 every bounded solution of Eq. (1) converges to this equilibrium or to a period-two solution while every unbounded solution approaches 1. The following result describes a global dynamics of Eq. (1) in this case. Theorem 13. Consider Eq. (1) under the conditions F > 0; D þ E < 1; D ¼ 0. The unique positive equilibrium is non-hyperbolic with characteristic roots 1 and R 2 ð1; 0Þ. There exists an invariant continuous curve W s ðEÞ, where Eð x; xÞ, which is the graph of a decreasing function, such that every solution fxn g of Eq. (1) for which ðx1 ; x0 Þ 2 W s ðEÞ is attracted to E as well as every solution fxn g of Eq. (1) for which ðx1 ; x0 Þne W s ðEÞ.
Fig. 3. Visual illustration of Theorems 11 and 12.
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Fig. 4. Visual illustration of Theorem 13. Dashed curve represents the global stable manifold of the equilibrium point E.
Every solution fxn g of Eq. (1) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ satisfies lim xn ¼ 1. In this case, the equilibrium x is semistable. (See Fig. 4) Proof. First we have
p þ q ¼ ð2A þ 2B þ 2CÞx þ D þ E ¼ 1 D E þ D þ E ¼ 1 and the characteristic equation at the positive equilibrium becomes k2 ð1 qÞk q ¼ 0 which yields k1 ¼ 1; k2 ¼ q 2 ð1; 0Þ. Theorem 4 implies the existence of the global stable manifold W s ðEÞ with the prescribed properties. In view of Lemma 2 the minimal period-two solution does not exist. In view of Theorem 4 the regions below and above W s ðEÞ in North-east ordering ne are invariant for the corresponding map T. Thus if we take any point ðx1 ; x0 Þne W s ðEÞ then by choosing the point ðx1 ; xW s Þ 2 W s ðEÞ we get that T n ððx1 ; x0 ÞÞne T n ððx1 ; xW s ÞÞ and since limn!1 T n ððx1 ; xW s ÞÞ ¼ E, we obtain that T n ððx1 ; x0 ÞÞ E, that is ðx1 ; x0 Þ 2 ½0; xÞ2 . Now we have
x1 ¼ f ðx0 ; x1 Þ < f ðx; xÞ ¼ x; x2 ¼ f ðx1 ; x0 Þ < f ðx; xÞ ¼ x; .. . xn ¼ f ðxn1 ; xn2 Þ < f ðx; xÞ ¼ x; where f ðu; v Þ ¼ Au2 þ Buv þ C v 2 þ Du þ Ev þ F. Hence, fx2n g and fx2nþ1 g are both eventually monotonic and bounded, thus x; x2nþ1 ! x because in this case there is no minimal period two solution. x2n ! Consider a solution fxn g of Eq. (1) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ. Choose the point ðx1 ; xW s Þ 2 W s ðEÞ we get that ðx1 ; xW s Þne ðx1 ; x0 Þ which implies T n ððx1 ; xW s ÞÞne T n ððx1 ; x0 ÞÞ and since x; 1Þ2 . Since the equilibrium point E is nonlimn!1 T n ððx1 ; xW ÞÞ ¼ E, we obtain that E T n ððx1 ; x0 ÞÞ, that is ððx1 ; x0 Þ 2 ½ hyperbolic the techniques for proving are more delicate and we will apply the technique used in [11,17]. To complete this we will find the image of E þ t v , where t > 0 and v is the eigenvector that corresponds to the eigenvalue 1, under the map T. Since E þ tv ¼ ð x þ t; x þ tÞ, we have
TðE þ tv Þ ðE þ t v Þ ¼ Tððx þ t; x þ tÞÞ ðx þ t; x þ tÞ ¼ ð0; ðA þ B þ CÞðx þ tÞ2 þ ðD þ EÞðx þ tÞ þ FÞ: By using the condition D ¼ 0, we have that
2 ! AþBþC ; TðE þ tv Þ ðE þ t v Þ ¼ ðA þ B þ CÞ 0; ðx þ tÞ þ 2ðD þ EÞ x; 1Þ2 is a supersolution, see [17], and so which implies E þ tv ne TðE þ tv Þ for every t > 0. This shows that every point in ð every solution tends to 1. h 5.4. Global dynamics of Eq. (1): case F > 0; D > 0 In this case Eq. (1) has zero or two positive equilibrium solutions and may have a minimal period-two solution.(See Fig. 5) Theorem 14. Consider Eq. (1) subject to the conditions F > 0; D > 0. Then the global dynamics of Eq. (1) is given as: (i) If D þ E P 1 then Eq. (1) does not have equilibrium solutions and every solution fxn g satisfies limn!1 xn ¼ 1;
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Fig. 5. Visual illustration of Theorem 14: cases (ii) and (iii). Dashed decreasing curve represents the global stable manifold of the equilibrium point Eþ in Fig. (a). Dashed decreasing curves represent the global stable manifolds of the periodic points P 1 and P 2 in Fig. (b). Dashed increasing curve represents the global unstable manifold of the equilibrium point Eþ in Fig. (a). Dashed increasing curves represent the global stable manifolds of the periodic points P 1 and P 2 in Fig. (b).
Fig. 6. Visual illustration of Theorem 14-case (iv). Dashed decreasing curve represents the global stable manifold of the equilibrium point Eþ .
(ii) If D þ E < 1 and the minimal period-two solution does not exist, that is (13) does not hold, then Eq. (1) has two equilibrium solutions x < xþ , where x is locally asymptotically stable and xþ is unstable. If xþ is a saddle equilibrium then there exist xþ ; xþ Þ, such that W s ðEþ Þ is a graph of two continuous curves W s ðEþ Þ and W u ðEþ Þ, both passing through the point Eþ ¼ ð u decreasing function and W ðEþ Þ is a graph of an increasing function. The first quadrant of initial condition Q 1 ¼ fðx1 ; x0 Þ : x1 P 0; x0 P 0g is the union of three disjoint basins of attraction, namely
Q 1 ¼ BðE Þ [ BðEþ Þ [ BðE1 Þ; where E and E1 denote the points ðx ; x Þ and ð1; 1Þ respectively, and BðEþ Þ ¼ W s ðEþ Þ,
BðE Þ ¼ fðx; yÞjðx; yÞne ðxEþ ; yEþ Þ for some ðxEþ ; yEþ Þ 2 W s ðEþ Þg; BðE1 Þ ¼ fðx; yÞjðxEþ ; yEþ Þne ðx; yÞ for some ðxEþ ; yEþ Þ 2 W s ðEþ Þg: In addition, for every ðx1 ; x0 Þ 2 Q 1 n W s ðEþ Þ every solution is asymptotic to W u ðEþ Þ. (iii) If D þ E < 1 and the minimal period-two solution does exist, that is (13) holds, then xþ is a repeller. In this case there exist xþ ; xþ Þ, four continuous curves W s ðP1 Þ; W s ðP 2 Þ; W u ðP1 Þ; W u ðP 2 Þ, where W s ðP 1 Þ; W s ðP 2 Þ are passing through the point Eþ ð and are graphs of decreasing functions. The curves W u ðP 1 Þ; W u ðP2 Þ are the graphs of increasing functions and are starting x ; x Þ. Every solution fxn g which starts below W s ðP1 Þ [ W s ðP 2 Þ in North-east ordering converges to E ð x ; x Þ and at E ð every solution fxn g which starts above W s ðP 1 Þ [ W s ðP2 Þ in North-east ordering satisfies lim xn ¼ 1. In other words, the basins of attraction are the same as in case (ii). (iv) Consider Eq. (1) subject to the conditions F > 0; D > 0; D þ E < 1; A < C and
F¼
ðD E þ 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ 4ðA CÞ2
:
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Then Eq. (1) has two equilibrium solutions x < xþ , where x is locally asymptotically stable and xþ is non-hyperbolic equilibrium point. In this case there exists continuous curves W s ðEþ Þ passing through the point Eþ ¼ ð xþ ; xþ Þ, such that W s ðEþ Þ is a graph of decreasing function. The first quadrant of initial condition Q 1 ¼ fðx1 ; x0 Þ : x1 P 0; x0 P 0g is the union of three disjoint basins of attraction, namely(See Fig. 6)
Q 1 ¼ BðE Þ [ BðEþ Þ [ BðE1 Þ; where E and E1 denote the points ðx ; x Þ and ð1; 1Þ respectively, and BðEþ Þ ¼ W s ðEþ Þ,
BðE Þ ¼ fðx; yÞjðx; yÞne ðxEþ ; yEþ Þ forsome ðxEþ ; yEþ Þ 2 W s ðEþ Þg; BðE1 Þ ¼ fðx; yÞjðxEþ ; yEþ Þne ðx; yÞ forsome ðxEþ ; yEþ Þ 2 W s ðEþ Þg:
Proof. (i) Proof of this part follows from Lemma 2 and is similar to the proof of Theorem 11. (ii) Since the minimal period-two solution does not exist, in view of Lemma 2 and Theorems 6 and 7 we have that x is locally asymptotically stable and xþ is a saddle or non-hyperbolic equilibrium point. Now the global result follows from Theorem 4. (iii) Since the minimal period-two solution does exist, in view of Lemma 2 and Theorems 6 and 7 we have that x is locally asymptotically stable and xþ is repeller. By Theorem 8 we obtain that the minimal period-two solution is a saddle point. The existence of four curves W s ðP 1 Þ; W s ðP2 Þ; W u ðP1 Þ; W u ðP2 Þ with the described properties is guaranteed by Theorems 1 and 4 of [17] applied to the map T 2 . (iv) The proof is similar to the proof of Part (iii) of Theorem 14 and will be omitted. h 6. Special cases In some special cases the global dynamics could be easily described as is clear from the following results. The first result describes the global dynamics of the Henon equation in the first quadrant, see [19]. Corollary 1. Consider the following equation
xnþ1 ¼ Ax2n þ Exn1 þ F;
n ¼ 0; 1; . . .
ð16Þ
where all parameters are positive. Then the global dynamics is as follows: (i) If either E P 1 or D1 ¼ ðB 1Þ2 4AF < 0, then every solution fxn g of Eq. (16) satisfies limn!1 xn ¼ 1; (ii) If E < 1 and D1 ¼ 0, then Eq. (16) has one non-hyperbolic equilibrium solution x and there exists an invariant continuous curve W s ðEÞ, where Eð x; xÞ, which is the graph of a decreasing function, such that every solution fxn g of Eq. (16) for which ðx1 ; x0 Þ 2 W s ðEÞ is attracted to E as well as every solution fxn g of Eq. (16) for which ðx1 ; x0 Þne W s ðEÞ. Every solution fxn g of Eq. (16) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ satisfies lim xn ¼ 1. (iii) If E < 1 and D1 > 0, then Eq. (16) has two equilibrium solutions 0 < x < xþ and the dynamics is described by Part (ii) of Theorem 14.
Proof. The proof of Part (i) follows from Part (i) of Theorem 14. The proof of Part (ii) follows from Theorem 13. The proof of Part (iii) follows from Theorem 14. Consider now the following special case
xnþ1 ¼ Cx2n1 þ Dxn þ F;
n ¼ 0; 1; . . .
ð17Þ
where all parameters are positive. The global dynamics is given with the following result. Corollary 2. Consider Eq. (17). Then the following result holds: < þ is a (i) If C < ð1þDÞð13DÞ then Eq. (17) has two equilibrium solutions 0 < x xþ , where x is locally asymptotically stable, x 4F repeller and the minimal period-two solution . . . ; U; W; . . . ; U < W is a saddle point. All non-equilibrium solutions fxn g converges to x , or to the period-two solution or are asymptotic to 1 with the basins of attraction described by Part (ii) of Theorem 14.
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(ii) If C ¼ ð1þDÞð13DÞ then Eq. (17) has two equilibrium solutions 0 < x < xþ , where x is locally asymptotically stable xþ and xþ 4F is a non-hyperbolic equilibrium solution. The dynamics is described by Part (ii) of Theorem 14. 2
(iii) If ð1þDÞð13DÞ < C < ð1DÞ then Eq. (17) has two equilibrium solutions 0 < x < xþ and no minimal period-two solutions. The 4F 4F dynamics is described by Part (ii) of Theorem 14. 2
(iv) If C ¼ ð1DÞ x and the global dynamics is the same as in Part (ii) of then Eq. (17) has one non-hyperbolic equilibrium solution 4F Corollary 1. 2
(v) If C > ð1DÞ then Eq. (17) neither has an equilibrium solution nor the minimal period-two solution and every solution fxn g of 4F Eq. (17) satisfies limn!1 xn ¼ 1;
Proof. Notice that the necessary and sufficient condition for existence of the minimal period-two solution becomes
ð1 þ DÞð1 3DÞ > 4CF; while the discriminant of the equilibrium solutions is D ¼ ð1 DÞ2 4CF. The proof of Part (i) follows from Part(i) of Theorem 14. The proof of Part (ii) follows from Theorem 13. The proof of Part (iii) follows from Part (ii) of Theorem 14. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part I, Int. J. Difference Equ. 3 (2008) 1–35. A.M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, Part II, Int. J. Difference Equ. 3 (2008) 195–225. E. Bedford, Complex Hénon maps with semi-parabolic fixed points, J. Difference Equ. Appl. 16 (2010) 425–426. E. Bedford, J. Smillie, Real polynomial diffeomorphisms with maximal entropy: tangencies, Ann. Math. 160 (2) (2004) 1–26. no. 1. E. Bedford, J. Smillie, Real polynomial diffeomorphisms with maximal entropy. II. Small Jacobian, Ergodic Theory Dyn. Syst. 26 (5) (2006) 1259–1283. A. Bonifant, J. Kiwi, J. Milnor, Cubic polynomial maps with periodic critical orbit. II. Escape regions, Conform. Geom. Dyn. 14 (2010) 68–112. A. Bonifant, J. Kiwi, J. Milnor, Errata for cubic polynomial maps with periodic critical orbit, Part II: escape regions, Conform. Geom. Dyn. 14 (2010) 190– 193. A. Brett, M.R.S. Kulenovic´, Basins of attraction of equilibrium points of monotone difference equations, Sarajevo J. Math. 5 (18) (2009) 211–233. E. Camouzis, G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Advances in Discrete Mathematics and Applications, vol.5, Chapman & Hall/CRC, Boca Raton, FL, 2008. J.-E. Fornaess, N. Sibony, Complex Hénon mappings in C2 and Fatou-Bieberbach domains, Duke Math. J. 65 (2) (1992) 345–380. M. Garic´-Demirovic´, M.R.S. Kulenovic´, M. Nurkanovic´, Global behavior of four competitive rational systems of difference equations in the plane, Discrete Dyn. Nat. Soc. (2009) 34. Art. ID 153058. E.A. Janowski, M.R.S. Kulenovic´, Attractivity and global stability for linearizable difference equations, Comput. Math. Appl. 57 (2009) 1592–1607. V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, Holland, 1993. M.R.S. Kulenovic´, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, London, 2001. M.R.S. Kulenovic´, O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, London, 2002. M.R.S. Kulenovic´, O. Merino, A global attractivity result for maps with invariant boxes, Discrete Cont. Dyn. Syst. Ser. B 6 (2006) 97–110. M.R.S. Kulenovic´, O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bifur. Chaos Appl. Sci. Eng. 20 (8) (2010) 2471– 2486. R.L. Nussbaum, Global stability, two conjectures and Maple, Nonlinear Anal. TMA 66 (2007) 1064–1090. C. Robinson, Stability, Symbolic Dynamics, and Chaos, CRC Press, Boca Raton, 1995.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Global dynamics of quadratic second order difference equation in the first quadrant J. Bekteševic´ a, M.R.S. Kulenovic´ b,⇑, E. Pilav c a
Division of Mathematics, Faculty of Mechanical Engineering, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA c Department of Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina b
a r t i c l e
i n f o
a b s t r a c t We investigate the global behavior of a quadratic second order difference equation
Keywords: Attractivity Basins Difference equation Invariant Period-two solutions Stable manifold Unstable manifold
xnþ1 ¼ Ax2n þ Bxn xn1 þ Cx2n1 þ Dxn þ Exn1 þ F;
n ¼ 0; 1; . . .
with non-negative parameters and initial conditions. We find the global behavior for all ranges of parameters and determine the basins of attraction of all equilibrium points. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Consider the following difference equation
xnþ1 ¼ Ax2n þ Bxn xn1 þ Cx2n1 þ Dxn þ Exn1 þ F;
n ¼ 0; 1; . . .
ð1Þ
with non-negative parameters and initial conditions. We assume that A þ B þ C > 0 in order to avoid trivial linear case from our consideration and also A þ B þ C > 0 and A þ B þ D > 0; B þ C þ E > 0 in order to avoid the well-known cases of first order quadratic difference equations. Polynomial difference equations and corresponding maps have been studied in both real and complex domain and many results have been obtained, see [3–7,10]. Most known case of quadratic polynomial equations is Hénon equation
xnþ1 ¼ 1 þ bxn1 ax2n ;
n ¼ 0; 1; . . .
ð2Þ
which was considered by many authors, see [3–5,10] and references therein. Eq. (2) is a special case of Eq. (1). In this paper we restrict our attention to non-negative initial conditions and non-negative parameters which will make our results more special but also more precise and applicable. First papers on quadratic polynomial difference equations with non-negative initial conditions and non-negative parameters are [1,2], where the special case
xnþ1 ¼ Bxn xn1 þ Exn1 þ F;
n ¼ 0; 1; . . .
ð3Þ
of Eq. (1) was considered and results on global dynamics have been obtained. The obtained results in [1,2] provided the parts of the basins of attraction of different equilibrium points and periodic solutions. In this paper we give the precise description of all basins of attraction of all attractors of Eq. (1) as well as the point at infinity. In particular, we find the regions of the plane of initial conditions for which all solutions are bounded as well as the escape regions, that is the basins of attraction ⇑ Corresponding author. E-mail address: [email protected] (M.R.S. Kulenovic´). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.10.048
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of 1. Such sets have complicated structure in complex dynamics, see [3,6,7]. Our results are based on number of theorems which hold for monotone difference equations and corresponding monotone maps, which will be described in the next section. An interesting consequence of our results is the necessary and sufficient condition for the existence of periodtwo solutions. For example, if D ¼ E ¼ F ¼ 0 that is Eq. (1) contains only quadratic terms, the necessary and sufficient condition for the existence of the period-two solution is C > 3A þ B, in which case the period-two solution is a saddle point which stable manifold serves as the separatrix between two basins of attraction of a locally stable equilibrium point and the point at 1. The paper is organized as follows. The Section 2 presents the preliminary results from the theory of monotone maps. The Section 3 briefly gives the local stability analysis of all equilibrium points for all values of parameters. The Section 4 contains stability analysis of period-two solution which plays major role in our results. The Section 5 gives global results in all regions of parameters. The Section 6 provides two illustrative examples of global dynamics of some special cases of Eq. (1). 2. Preliminaries Consider the difference equation
xnþ1 ¼ f ðxn ; xn1 Þ;
n ¼ 0; 1; . . .
ð4Þ
where f is a continuous and increasing function in both variables. There are several global attractivity results for Eq. (4) which give the sufficient conditions for all solutions to approach a unique equilibrium. These results were used efficiently in monograph [14] to study the global behavior of solutions of second order linear fractional difference equation. Here we list some of these results that will be needed in this paper. The first result was obtained in [14] and it was extended to the case of higher order difference equations and systems in [12,15,16,18]. Theorem 1. Let ½a; b be an interval of real numbers and assume that
f : ½a; b ½a; b ! ½a; b is a continuous function satisfying the following properties: (a) f ðx; yÞ is non-decreasing in each of its arguments; (b) Eq. (4) has a unique equilibrium x 2 ½a; b. Then every solution of Eq. (4) converges to x. The following result has been obtained in [1]. Theorem 2. Let I # R and let f 2 C½I I; I be a function which increases in both variables. Then for every solution of Eq. (4) the 1 subsequences fx2n g1 n¼0 and fx2nþ1 gn¼1 of even and odd terms of the solution do exactly one of the following: (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 Eq. (1) approaches either an equilibrium solution or periodtwo solution or the finite point at the boundary and every unbounded solution is asymptotic to the point at infinity in a monotonic way. Thus the major problem in dynamics of Eq. (1) is the problem of determining the basins of attraction of three different types of attractors: the equilibrium solutions, period-two solution(s) and the point(s) at infinity. The following two results can be proved by using the techniques of proof of Theorem 11 in [8]. Theorem 3. Consider Eq. (4) where f is increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E1 ðx1 ; y1 Þ and E2 ðx2 ; y2 Þ are two consecutive equilibrium points in North-East ordering that satisfy
ðx1 ; y1 Þne ðx2 ; y2 Þ; that is x1 6 x2 ; y1 6 y2 . Assume that E1 is a saddle point or a non-hyperbolic point with second characteristic root in interval ð1; 1Þ, with the neighborhoods where f is strictly increasing and E2 is a local attractor. Then the basin of attraction BðE2 Þ of E2 is the region above the global stable manifolds W s ðE1 Þ. More precisely
BðE2 Þ ¼ fðx; yÞ : 9yl : yl < y;
ðx; yl Þ 2 W s ðE1 Þg:
The basin of attraction BðE1 Þ ¼ W s ðE1 Þ is exactly the global stable manifold of E1 . If there exists a period-two solution, then the end points of the global stable manifold are exactly the period two solution.
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Theorem 4. Consider Eq. (4) where f is increasing function in its arguments and assume that there is no minimal period-two solution. Assume that E1 ðx1 ; y1 Þ and E2 ðx2 ; y2 Þ are two consecutive equilibrium points in North-East ordering that satisfy
ðx1 ; y1 Þne ðx2 ; y2 Þ and that E1 is a local attractor and E2 is a saddle point or a non-hyperbolic point with second characteristic root in interval ð1; 1Þ, with the neighborhoods where f is strictly increasing. Then the basin of attraction BðE1 Þ of E1 is the region below the global stable manifolds W s ðE2 Þ. More precisely
BðE1 Þ ¼ fðx; yÞ : 9yu : y < yu ;
ðx; yu Þ 2 W s ðE2 Þg:
The basin of attraction BðE2 Þ ¼ W s ðE2 Þ is exactly the global stable manifold of E2 . The global stable manifold extend to the boundary of the domain of Eq. (4). If there exists a period-two solution, then the end points of the global stable manifold are exactly the period two solution. Theorem 5. If D þ E > 1 then every solution fxn g of Eq. (1) satisfies limn!1 xn ¼ 1. Proof. If fxn g is a solution of Eq. (1) then fxn g satisfies the inequality
xnþ1 P Dxn þ Exn1 ;
n ¼ 0; 1; . . . ;
which in view of the result on difference inequalities, see Theorem 1.4.1 in [9,13], implies that xn P Ln ; n P 1 where fLn g is a solution of the initial value problem
Lnþ1 ¼ DLn þ ELn1 ;
L1 ¼ x1 ;
L0 ¼ x0 ;
n ¼ 0; 1; . . . :
Consequently,
1 xn P Ln ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEx1 þ kþ x0 Þknþ þ ðk x1 þ Ex0 Þkn ; 2 D þ 4E
n ¼ 1; 2; . . . ;
where
D
k ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2 þ 4E ; 2
which implies limn!1 xn ¼ 1.
h
Remark 1. Theorem 5 implies that interesting dynamics for Eq. (1) happens in the parametric region D þ E 6 1, which will be main objective of the rest of the paper. We will also describe more precisely global dynamics even in the parametric region D þ E > 1, where we describe the manifold that solutions will be following on their way to 1. 3. Local stability analysis of equilibrium solutions x of Eq. (1) satisfy the quadratic equation The equilibrium solutions
ðA þ B þ CÞx2 þ ðD þ E 1Þx þ F ¼ 0:
ð5Þ
This immediately leads to the following cases: (i) (ii) (iii) (iv)
there there there there
exists zero equilibrium x ¼ 0 if and only if F ¼ 0; exist no equilibrium solutions if and only if D ¼ ðD þ E 1Þ2 4FðA þ B þ CÞ < 0; exist one equilibrium solution if and only if D ¼ 0; exist two equilibrium solutions if and only if D > 0;
The positive equilibrium solutions, when exist are given as
x ¼
pffiffiffiffi 1DE D : 2ðA þ B þ CÞ
ð6Þ
The linearized equation at the zero equilibrium is
znþ1 ¼ Dzn þ Ezn1 :
ð7Þ
In view of local stability Theorem 1.1 from [14] we obtain the following result on local stability of the zero equilibrium: Proposition 1. The zero equilibrium of Eq. (1) is one of the following:
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(a) (b) (c) (d) (e)
53
locally asymptotically stable if D þ E < 1, non-hyperbolic and locally stable if D þ E ¼ 1, unstable if D þ E > 1, saddle point if D > jE 1j, repeller if 1 E < D < E 1.
x is The linearized equation at the positive equilibrium solutions
znþ1 ¼ pzn þ qzn1 ¼ ðð2A þ BÞx þ DÞzn þ ððB þ 2CÞx þ EÞzn1 ;
ð8Þ
In view of Theorem 1.1 from [14] we obtain the following result on local stability of the positive equilibrium: Proposition 2. The positive equilibrium solution of Eq. (1) is one of the following:
(a) (b) (c) (d) (e)
locally asymptotically stable if p þ q < 1, non-hyperbolic and locally stable if p þ q ¼ 1, unstable if p þ q > 1, saddle point if p > jq 1j, repeller if 1 q < p < q 1.
The precise description of local stability of positive equilibrium solutions is given with the following result. Theorem 6. If
D < 1 E and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then Eq. (1) has the equilibrium point x where
x ¼
pffiffiffiffi 1DE D 2ðA þ B þ CÞ
and the following holds: (i) x is locally asymptotically stable if
4ðA þ B þ CÞF < ðE þ D 1Þ2 : (ii) x a non-hyperbolic point if
4ðA þ B þ CÞF ¼ ðE þ D 1Þ2 :
Proof. It is easy to see that
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2A BÞ ðD þ E 1Þ2 4FðA þ B þ CÞ 2ðE 1ÞA þ BðD E þ 1Þ þ 2CD @f p¼ ðx ; x Þ ¼ @u 2ðA þ B þ CÞ and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB 2CÞ ðD þ E 1Þ2 4FðA þ B þ CÞ þ 2ðEA CD þ CÞ þ BðD þ E þ 1Þ @f q¼ ðx ; x Þ ¼ @v 2ðA þ B þ CÞ and
1qp ¼
1qþp ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD þ E 1Þ2 4FðA þ B þ CÞ P 0; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðC AÞ ðD þ E 1Þ2 4FðA þ B þ CÞ 2AðE 1Þ þ BðD E þ 1Þ þ 2CD AþBþC
q þ 1 ¼ ðB þ 2Cx Þ þ E þ 1 > 0: The proof of the statements of the theorem follows from Proposition 2. h
:
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Lemma 1. If
D þ E < 1 and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then
1 q þ p P 0: Proof. Since D þ E < 1 we have that E < 1 and
2AðE 1Þ þ BðD E þ 1Þ þ 2CD > 0; which implies 1 q þ p > 0 if C P A. Now, assume that C < A. One can show that
ð2Að1 EÞ þ Bð1 þ D EÞ þ 2CDÞ2 ðC AÞ2 D ¼ ðA þ B þ CÞðð1 þ D EÞðAðD þ 3E 3Þ þ BðD E þ 1Þ þCð3D þ E 1ÞÞ þ 4FðA CÞ2 ; where
D ¼ ðD þ E 1Þ2 4FðA þ B þ CÞ: Since D þ E < 1 we have that D þ 3E 3 < 0 and
BðD E þ 1Þ þ Cð3D þ E 1Þ ðB þ 2CÞðD E þ 1Þ C ¼ < 0; D þ 3E 3 D þ 3E 3 from which it follows that
BðD E þ 1Þ þ Cð3D þ E 1Þ < C < A; D þ 3E 3 which is equivalent to
AðD þ 3E 3Þ þ BðD E þ 1Þ þ Cð3D þ E 1Þ > 0: This implies
ð2Að1 EÞ þ Bð1 þ D EÞ þ 2CDÞ2 ðC AÞ2 D > 0; from which the proof follows. h Theorem 7. If
D < 1 E and D ¼ ðD þ E 1Þ2 4ðA þ B þ CÞF P 0; then Eq. (1) has the equilibrium point xþ where
xþ ¼
1DEþ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD þ E 1Þ2 4FðA þ B þ CÞ 2ðA þ B þ CÞ
and the following holds: (i) xþ is a repeller if
A 0; WU > 0
and
q2 ðA B þ CÞ2 4ðA B þ CÞðAq2 þ Dq þ FÞ > 0, that is when
1þDE > 0 and A B þ C > 0 and q2 ðA B þ CÞ > 4ðAq2 þ Dq þ FÞ; CA
ð11Þ
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which is equivalent to
1þDE > 0; A B þ C > 0 and CA
ð1 þ D EÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ ðA CÞ2
> 4F:
ð12Þ
Lemma 2. For Eq. (1) the following holds: (i) If A < C then Eq. (1) has minimal period-two solution if and only if
E 1 < D and F < (ii) Eq. (a) (b) (c)
ðD E þ 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ 4ðA CÞ2
:
ð13Þ
(1) has no minimal period-two solution if any of the following conditions hold: A P C; E þ D > 1; D 6 0.
Proof. (i) It is sufficient to prove that (13) implies A B þ C > 0. Indeed, (13) implies that
AðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1Þ > 0; which is equivalent to
B
0 and AE þ A þ CD > AE þ A þ AD ¼ Að1 E þ DÞ > 0. This implies
B < 3A þ C þ
4DðA CÞ < A þ C; DEþ1
from which it follows that A B þ C > 0. (ii) (a) Assume that A P C. Then from (12) we obtain that 1 þ D E < 0 which implies E þ 3D 1 > 0. Since
AðD þ 3E 3Þ 2AðD E þ 1Þ A¼ > 0; 3D þ E 1 3D þ E 1 we have that
AðD þ 3E 3Þ > A P C; 3D þ E 1 which implies that
AðD þ 3E 3Þ Cð3D þ E 1Þ > 0; from which it follows that
ð1 þ D EÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ ðA CÞ2
4F < 0;
which is a contradiction to (12). This proves the statement (i). (b) Assume that E þ D > 1. If A P C the proof follows from (i). Now, suppose that A < C and that Eq. (1) has minimal period-two solution. Then from (12) we obtain that 1 þ D E > 0 and
AðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1Þ > 0; which is equivalent to
06B
0 which is a contradiction. h For example, the conditions for existence of period-two solutions are
B ¼ 2A; C A ¼ 1;
D ¼ E;
4ðA þ D þ F Þ < 1:
Set
un ¼ xn1
v n ¼ xn ;
and
n ¼ 0; 1; . . .
and write Eq. (1) in the equivalent form:
unþ1 ¼ v n
v nþ1 ¼ Av 2n þ Bv n un þ Cu2n þ Dv n þ Eun þ F: Let T be a function on ½0; 1Þ ½0; 1Þ defined by
T Then
u
v U W
¼
v 2
2
Av þ Bv u þ Cu þ Dv þ Eu þ F
:
is fixed point of T 2 , second iterate of T. Furthermore
T2
u
v
¼
g ðu; v Þ
hðu; v Þ
where
g ðu; v Þ ¼ Av 2 þ Bv u þ Cu2 þ Dv þ Eu þ F; hðu; v Þ ¼ A½g ðu; v Þ2 þ Bv g ðu; v Þ þ Dg ðu; v Þ þ C v 2 þ Ev þ F: The period-two solution is locally asymptotically stable if the eigenvalues of the Jacobian matrix J T 2 , evaluated at inside the unit disk. By definition
JT 2
U W
¼
@g ð @u @h ð @u
U; WÞ U; WÞ
@g ð @v @h ð @v
U; WÞ U; WÞ
! :
U W
lie
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58
By computing the partial derivatives of g and h and by using the fact that g ðU; WÞ ¼ U, we find the following identities:
@g ðU; WÞ ¼ BW þ 2C U þ E; @u @g ðU; WÞ ¼ 2AW þ BU þ D; @v and
@h @g ðU; WÞ ¼ ðU; WÞ½2Ag ðU; WÞ þ BW þ D @u @u ¼ ðBW þ 2C U þ EÞð2AU þ BW þ DÞ @h @g ðU; WÞ ¼ ðU; WÞ½2Ag ðU; WÞ þ BW þ D þ Bg ðU; WÞ þ 2C W þ E @v @v ¼ ð2AW þ BU þ DÞð2AU þ BW þ DÞ þ BU þ 2C W þ E: Set
@g @h ðU; WÞ þ ðU; WÞ @u @v ¼ ðBW þ 2AU þ DÞðBU þ 2AW þ DÞ þ ðB þ 2C ÞðU þ WÞ þ 2E
S¼
¼ 2ABðU þ WÞ2 þ ½DðB þ 2AÞ þ ðB þ 2C ÞðU þ WÞ þ ð2A BÞ2 UW þ D2 þ 2E ¼ 2ABq2 þ ½DðB þ 2AÞ þ ðB þ 2C Þq þ ð2A BÞ2
Aq2 þ Dq þ F þ D2 þ 2E ABþC
and
@g @h @g @h ðU; WÞ ðU; WÞ ðU; WÞ ðU; WÞ @u @v @v @u ¼ ðBW þ 2C U þ EÞðBU þ 2C W þ EÞ
D¼
¼ 2BC ðU þ WÞ2 þ EðB þ 2C ÞðU þ WÞ þ ð2C BÞ2 UW þ E2 ¼ 2BCq2 þ EðB þ 2C Þq þ ð2C BÞ2
Aq2 þ Dq þ F þ E2 : ABþC
It is easy to see that
S D 1 ¼ BðA C Þq2 þ 2ðAD þ C CEÞq þ 4ðA C Þ Aq2 þ Dq þ F þ ðD þ E 1ÞðD E þ 1Þ
2ðAD þ C CEÞ q 4 Aq2 þ Dq þ F þ ðD þ E 1Þq ¼ ðC AÞ Bq2 þ ð C AÞ
2AD ðA þ C ÞðE 1Þ 2 q ¼ ðC AÞ ð4A þ BÞq 3Dq 4F þ CA ¼
ðD þ E 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ þ 4FðA CÞ2 : AC
Theorem 8. If Eq. (1) has the minimal period-two solution then it is a saddle point. Proof. A minimal period-two solution is a saddle point if and only if
jSj > j1 þ Dj and S 2 4D > 0: Since S > 0 and D > 0 then (15) is equivalent to
S D 1 > 0; since
S 2 4D > ðD þ 1Þ2 4D ¼ ðD 1Þ2 P 0: In view of Lemma 2 we have that
SD1¼
ðD þ E 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ þ 4FðA CÞ2 > 0; AC
ð15Þ
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59
since A < C and
E 1 < D and F
þ ¼0 AþBþC AþBþC
p þ ðq 1Þ ¼ð2A þ BÞx þ D þ ðB þ 2CÞx þ E 1 ¼
and
p ðq 1Þ ¼ð2A þ BÞx þ D ððB þ 2CÞx þ EÞ þ 1 ¼
ð2A þ BÞð1 EÞ þ DðC AÞ ðC AÞð1 EÞ DðB þ 2CÞ AþBþC AþBþC
¼
ð3A þ B CÞð1 EÞ þ Dð3C þ B AÞ AþBþC
>
Dð3A þ B C Þ þ DðB A þ 3C Þ ¼ 2D > 0: AþBþC
Thus the results follows from Proposition2. h Now the global result follows from Theorem 4.
Theorem 10. Consider Eq. (1) under the conditions F ¼ 0 and D þ E < 1. Assume that Eq. (1) has the minimal period-two solution . . . ; U; W; U; W; . . ., that is assume that (13) is satisfied. Then the zero equilibrium is locally asymptotically stable and the positive equilibrium xþ is repeller. In this case there exist four continuous curves W s ðP 1 Þ; W s ðP 2 Þ; W u ðP 1 Þ; W u ðP2 Þ, where W s ðP1 Þ; W s ðP2 Þ are passing through the point Eþ ð xþ ; xþ Þ, and are graphs of decreasing functions. The curves W u ðP1 Þ; W u ðP 2 Þ are the graphs of increasing functions and are starting at E0 ¼ ð0; 0Þ. Every solution fxn g which starts below W s ðP1 Þ [ W s ðP 2 Þ in North-east ordering converges to E0 and every solution fxn g which starts above W s ðP 1 Þ [ W s ðP2 Þ in North-east ordering satisfies lim xn ¼ 1.
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Fig. 1. Visual illustration of Theorem 9. Dashed decreasing graph represents the stable manifold and dashed increasing graph represents the unstable manifold of Eþ .
Proof. For example, the condition (13) is satisfied if
B ¼ 2A;
C ¼ A þ 1;
D¼E
ðD þ EÞxn ¼ xn , which is a contradiction. If the subsequences fx2n g and fx2nþ1 g are eventually monotone, then without loss of generality we can assume that fx2n g is eventually non-decreasing and fx2nþ1 g is eventually non-increasing. In this case x2n ! 1 which would imply that x2nþ1 ! 1, which is a contradiction. Thus the remaining possibility is that fxn g is eventually increasing, which implies that xn ! 1 as n ! 1. Another way of proving the global behavior in the case when the equilibrium point E is non-hyperbolic was used in [11,17]. To complete this we will find the image of E þ tv , where t > 0 and v is the eigenvector that corresponds to the eigenvalue 1, under the map T. Since E þ tv ¼ ðt; tÞ, we have
TðE þ t v Þ ðE þ t v Þ ¼ Tððt; tÞÞ ðt; tÞ ¼ ð0; ðA þ B þ CÞt 2 þ ðD þ E 1ÞtÞ: By using the condition D þ E 1 ¼ 0, we have that
TðE þ t v Þ ðE þ t v Þ ¼ ð0; ðA þ B þ CÞt 2 Þ; which implies E þ t v ne TðE þ tv Þ for every t > 0. This shows that every point in ð x; 1Þ2 is a supersolution, see [17], and so every solution tends to 1. (c) In view of E > 1 þ D from Eq. (1) we have that xnþ1 > Exn1 which implies xnþ1 > Ek x1 or xnþ1 > Ek x0 for some k such that k ! 1 as n ! 1. Consequently every solution fxn g of Eq. (1) satisfies lim xn ¼ 1. h 5.2. Global dynamics of Eq. (1): case F > 0; D < 0 In this case Eq. (1) has no equilibrium solutions and in view of Theorem 2 every bounded solution of Eq. (1) converges to a period-two solution while every unbounded solution approaches 1.(See Fig. 3) Theorem 12. Consider Eq. (1) under the conditions F > 0; D < 0. Then every solution fxn g of Eq. (1) satisfies limn!1 xn ¼ 1 Proof. The proof follows from Lemma 2 which shows that the minimal period-two solution does not exist. h 5.3. Global dynamics of Eq. (1): case F > 0; D ¼ 0 In this case Eq. (1) has a unique equilibrium solution
x ¼
1DE ; 2ðA þ B þ CÞ
D þ E < 1;
and in view of Theorem 2 every bounded solution of Eq. (1) converges to this equilibrium or to a period-two solution while every unbounded solution approaches 1. The following result describes a global dynamics of Eq. (1) in this case. Theorem 13. Consider Eq. (1) under the conditions F > 0; D þ E < 1; D ¼ 0. The unique positive equilibrium is non-hyperbolic with characteristic roots 1 and R 2 ð1; 0Þ. There exists an invariant continuous curve W s ðEÞ, where Eð x; xÞ, which is the graph of a decreasing function, such that every solution fxn g of Eq. (1) for which ðx1 ; x0 Þ 2 W s ðEÞ is attracted to E as well as every solution fxn g of Eq. (1) for which ðx1 ; x0 Þne W s ðEÞ.
Fig. 3. Visual illustration of Theorems 11 and 12.
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Fig. 4. Visual illustration of Theorem 13. Dashed curve represents the global stable manifold of the equilibrium point E.
Every solution fxn g of Eq. (1) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ satisfies lim xn ¼ 1. In this case, the equilibrium x is semistable. (See Fig. 4) Proof. First we have
p þ q ¼ ð2A þ 2B þ 2CÞx þ D þ E ¼ 1 D E þ D þ E ¼ 1 and the characteristic equation at the positive equilibrium becomes k2 ð1 qÞk q ¼ 0 which yields k1 ¼ 1; k2 ¼ q 2 ð1; 0Þ. Theorem 4 implies the existence of the global stable manifold W s ðEÞ with the prescribed properties. In view of Lemma 2 the minimal period-two solution does not exist. In view of Theorem 4 the regions below and above W s ðEÞ in North-east ordering ne are invariant for the corresponding map T. Thus if we take any point ðx1 ; x0 Þne W s ðEÞ then by choosing the point ðx1 ; xW s Þ 2 W s ðEÞ we get that T n ððx1 ; x0 ÞÞne T n ððx1 ; xW s ÞÞ and since limn!1 T n ððx1 ; xW s ÞÞ ¼ E, we obtain that T n ððx1 ; x0 ÞÞ E, that is ðx1 ; x0 Þ 2 ½0; xÞ2 . Now we have
x1 ¼ f ðx0 ; x1 Þ < f ðx; xÞ ¼ x; x2 ¼ f ðx1 ; x0 Þ < f ðx; xÞ ¼ x; .. . xn ¼ f ðxn1 ; xn2 Þ < f ðx; xÞ ¼ x; where f ðu; v Þ ¼ Au2 þ Buv þ C v 2 þ Du þ Ev þ F. Hence, fx2n g and fx2nþ1 g are both eventually monotonic and bounded, thus x; x2nþ1 ! x because in this case there is no minimal period two solution. x2n ! Consider a solution fxn g of Eq. (1) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ. Choose the point ðx1 ; xW s Þ 2 W s ðEÞ we get that ðx1 ; xW s Þne ðx1 ; x0 Þ which implies T n ððx1 ; xW s ÞÞne T n ððx1 ; x0 ÞÞ and since x; 1Þ2 . Since the equilibrium point E is nonlimn!1 T n ððx1 ; xW ÞÞ ¼ E, we obtain that E T n ððx1 ; x0 ÞÞ, that is ððx1 ; x0 Þ 2 ½ hyperbolic the techniques for proving are more delicate and we will apply the technique used in [11,17]. To complete this we will find the image of E þ t v , where t > 0 and v is the eigenvector that corresponds to the eigenvalue 1, under the map T. Since E þ tv ¼ ð x þ t; x þ tÞ, we have
TðE þ tv Þ ðE þ t v Þ ¼ Tððx þ t; x þ tÞÞ ðx þ t; x þ tÞ ¼ ð0; ðA þ B þ CÞðx þ tÞ2 þ ðD þ EÞðx þ tÞ þ FÞ: By using the condition D ¼ 0, we have that
2 ! AþBþC ; TðE þ tv Þ ðE þ t v Þ ¼ ðA þ B þ CÞ 0; ðx þ tÞ þ 2ðD þ EÞ x; 1Þ2 is a supersolution, see [17], and so which implies E þ tv ne TðE þ tv Þ for every t > 0. This shows that every point in ð every solution tends to 1. h 5.4. Global dynamics of Eq. (1): case F > 0; D > 0 In this case Eq. (1) has zero or two positive equilibrium solutions and may have a minimal period-two solution.(See Fig. 5) Theorem 14. Consider Eq. (1) subject to the conditions F > 0; D > 0. Then the global dynamics of Eq. (1) is given as: (i) If D þ E P 1 then Eq. (1) does not have equilibrium solutions and every solution fxn g satisfies limn!1 xn ¼ 1;
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Fig. 5. Visual illustration of Theorem 14: cases (ii) and (iii). Dashed decreasing curve represents the global stable manifold of the equilibrium point Eþ in Fig. (a). Dashed decreasing curves represent the global stable manifolds of the periodic points P 1 and P 2 in Fig. (b). Dashed increasing curve represents the global unstable manifold of the equilibrium point Eþ in Fig. (a). Dashed increasing curves represent the global stable manifolds of the periodic points P 1 and P 2 in Fig. (b).
Fig. 6. Visual illustration of Theorem 14-case (iv). Dashed decreasing curve represents the global stable manifold of the equilibrium point Eþ .
(ii) If D þ E < 1 and the minimal period-two solution does not exist, that is (13) does not hold, then Eq. (1) has two equilibrium solutions x < xþ , where x is locally asymptotically stable and xþ is unstable. If xþ is a saddle equilibrium then there exist xþ ; xþ Þ, such that W s ðEþ Þ is a graph of two continuous curves W s ðEþ Þ and W u ðEþ Þ, both passing through the point Eþ ¼ ð u decreasing function and W ðEþ Þ is a graph of an increasing function. The first quadrant of initial condition Q 1 ¼ fðx1 ; x0 Þ : x1 P 0; x0 P 0g is the union of three disjoint basins of attraction, namely
Q 1 ¼ BðE Þ [ BðEþ Þ [ BðE1 Þ; where E and E1 denote the points ðx ; x Þ and ð1; 1Þ respectively, and BðEþ Þ ¼ W s ðEþ Þ,
BðE Þ ¼ fðx; yÞjðx; yÞne ðxEþ ; yEþ Þ for some ðxEþ ; yEþ Þ 2 W s ðEþ Þg; BðE1 Þ ¼ fðx; yÞjðxEþ ; yEþ Þne ðx; yÞ for some ðxEþ ; yEþ Þ 2 W s ðEþ Þg: In addition, for every ðx1 ; x0 Þ 2 Q 1 n W s ðEþ Þ every solution is asymptotic to W u ðEþ Þ. (iii) If D þ E < 1 and the minimal period-two solution does exist, that is (13) holds, then xþ is a repeller. In this case there exist xþ ; xþ Þ, four continuous curves W s ðP1 Þ; W s ðP 2 Þ; W u ðP1 Þ; W u ðP 2 Þ, where W s ðP 1 Þ; W s ðP 2 Þ are passing through the point Eþ ð and are graphs of decreasing functions. The curves W u ðP 1 Þ; W u ðP2 Þ are the graphs of increasing functions and are starting x ; x Þ. Every solution fxn g which starts below W s ðP1 Þ [ W s ðP 2 Þ in North-east ordering converges to E ð x ; x Þ and at E ð every solution fxn g which starts above W s ðP 1 Þ [ W s ðP2 Þ in North-east ordering satisfies lim xn ¼ 1. In other words, the basins of attraction are the same as in case (ii). (iv) Consider Eq. (1) subject to the conditions F > 0; D > 0; D þ E < 1; A < C and
F¼
ðD E þ 1ÞðAðD þ 3E 3Þ þ BðD þ E 1Þ Cð3D þ E 1ÞÞ 4ðA CÞ2
:
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Then Eq. (1) has two equilibrium solutions x < xþ , where x is locally asymptotically stable and xþ is non-hyperbolic equilibrium point. In this case there exists continuous curves W s ðEþ Þ passing through the point Eþ ¼ ð xþ ; xþ Þ, such that W s ðEþ Þ is a graph of decreasing function. The first quadrant of initial condition Q 1 ¼ fðx1 ; x0 Þ : x1 P 0; x0 P 0g is the union of three disjoint basins of attraction, namely(See Fig. 6)
Q 1 ¼ BðE Þ [ BðEþ Þ [ BðE1 Þ; where E and E1 denote the points ðx ; x Þ and ð1; 1Þ respectively, and BðEþ Þ ¼ W s ðEþ Þ,
BðE Þ ¼ fðx; yÞjðx; yÞne ðxEþ ; yEþ Þ forsome ðxEþ ; yEþ Þ 2 W s ðEþ Þg; BðE1 Þ ¼ fðx; yÞjðxEþ ; yEþ Þne ðx; yÞ forsome ðxEþ ; yEþ Þ 2 W s ðEþ Þg:
Proof. (i) Proof of this part follows from Lemma 2 and is similar to the proof of Theorem 11. (ii) Since the minimal period-two solution does not exist, in view of Lemma 2 and Theorems 6 and 7 we have that x is locally asymptotically stable and xþ is a saddle or non-hyperbolic equilibrium point. Now the global result follows from Theorem 4. (iii) Since the minimal period-two solution does exist, in view of Lemma 2 and Theorems 6 and 7 we have that x is locally asymptotically stable and xþ is repeller. By Theorem 8 we obtain that the minimal period-two solution is a saddle point. The existence of four curves W s ðP 1 Þ; W s ðP2 Þ; W u ðP1 Þ; W u ðP2 Þ with the described properties is guaranteed by Theorems 1 and 4 of [17] applied to the map T 2 . (iv) The proof is similar to the proof of Part (iii) of Theorem 14 and will be omitted. h 6. Special cases In some special cases the global dynamics could be easily described as is clear from the following results. The first result describes the global dynamics of the Henon equation in the first quadrant, see [19]. Corollary 1. Consider the following equation
xnþ1 ¼ Ax2n þ Exn1 þ F;
n ¼ 0; 1; . . .
ð16Þ
where all parameters are positive. Then the global dynamics is as follows: (i) If either E P 1 or D1 ¼ ðB 1Þ2 4AF < 0, then every solution fxn g of Eq. (16) satisfies limn!1 xn ¼ 1; (ii) If E < 1 and D1 ¼ 0, then Eq. (16) has one non-hyperbolic equilibrium solution x and there exists an invariant continuous curve W s ðEÞ, where Eð x; xÞ, which is the graph of a decreasing function, such that every solution fxn g of Eq. (16) for which ðx1 ; x0 Þ 2 W s ðEÞ is attracted to E as well as every solution fxn g of Eq. (16) for which ðx1 ; x0 Þne W s ðEÞ. Every solution fxn g of Eq. (16) for which there exists ðxW ; yW Þ 2 W s ðEÞ such that ðxW ; yW Þne ðx1 ; x0 Þ; ðx1 ; x0 Þ R W s ðEÞ satisfies lim xn ¼ 1. (iii) If E < 1 and D1 > 0, then Eq. (16) has two equilibrium solutions 0 < x < xþ and the dynamics is described by Part (ii) of Theorem 14.
Proof. The proof of Part (i) follows from Part (i) of Theorem 14. The proof of Part (ii) follows from Theorem 13. The proof of Part (iii) follows from Theorem 14. Consider now the following special case
xnþ1 ¼ Cx2n1 þ Dxn þ F;
n ¼ 0; 1; . . .
ð17Þ
where all parameters are positive. The global dynamics is given with the following result. Corollary 2. Consider Eq. (17). Then the following result holds: < þ is a (i) If C < ð1þDÞð13DÞ then Eq. (17) has two equilibrium solutions 0 < x xþ , where x is locally asymptotically stable, x 4F repeller and the minimal period-two solution . . . ; U; W; . . . ; U < W is a saddle point. All non-equilibrium solutions fxn g converges to x , or to the period-two solution or are asymptotic to 1 with the basins of attraction described by Part (ii) of Theorem 14.
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(ii) If C ¼ ð1þDÞð13DÞ then Eq. (17) has two equilibrium solutions 0 < x < xþ , where x is locally asymptotically stable xþ and xþ 4F is a non-hyperbolic equilibrium solution. The dynamics is described by Part (ii) of Theorem 14. 2
(iii) If ð1þDÞð13DÞ < C < ð1DÞ then Eq. (17) has two equilibrium solutions 0 < x < xþ and no minimal period-two solutions. The 4F 4F dynamics is described by Part (ii) of Theorem 14. 2
(iv) If C ¼ ð1DÞ x and the global dynamics is the same as in Part (ii) of then Eq. (17) has one non-hyperbolic equilibrium solution 4F Corollary 1. 2
(v) If C > ð1DÞ then Eq. (17) neither has an equilibrium solution nor the minimal period-two solution and every solution fxn g of 4F Eq. (17) satisfies limn!1 xn ¼ 1;
Proof. Notice that the necessary and sufficient condition for existence of the minimal period-two solution becomes
ð1 þ DÞð1 3DÞ > 4CF; while the discriminant of the equilibrium solutions is D ¼ ð1 DÞ2 4CF. The proof of Part (i) follows from Part(i) of Theorem 14. The proof of Part (ii) follows from Theorem 13. The proof of Part (iii) follows from Part (ii) of Theorem 14. h References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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