Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation.

1. Introduction and Preliminaries

In this paper, we investigate the local and global dynamics of the following difference equation:

[mathematical expression not reproducible]. (1)

where the parameters B, F, b, c are positive real numbers and initial conditions [x.sub.-1] and [x.sub.0] are arbitrary positive real numbers. Equation (1) is the special case of a general second order quadratic fractional equation of the form

[mathematical expression not reproducible], (2)

with nonnegative parameters and initial conditions such that A + B + C > 0, a + b + c + d + e + f > 0 and a[x.sup.2.sub.n] + b[x.sub.n][x.sub.n-1] +c[x.sup.2.sub.n-1] + d[x.sub.n] + e[x.sub.n-1] + f > 0, n = 0, 1, .... Several global asymptotic results for some special cases of Equation (2) were obtained in [1-11]. Also, Equation (1) is a special case of the equation

[mathematical expression not reproducible], (3)

with positive parameters and nonnegative initial conditions [x.sub.-1], [x.sub.0]. Local and global dynamics of Equation (3) was investigated in .

The special case of Equation (3) when B = C = 0, i.e.,

[mathematical expression not reproducible]. (4)

was studied in . The authors performed the local stability analysis of the unique equilibrium point and gave the necessary and sufficient conditions for the equilibrium to be locally asymptotically stable, a repeller or nonhyperbolic equilibrium. Also, it was shown that Equation (4) exhibits the Naimark-Sacker bifurcation.

The special case of Equation (3) (when B = F = 0 and C = 1) is the following equation:

[mathematical expression not reproducible], (5)

where the parameters b, c, and f are nonnegative numbers with condition b + c > 0, f [not equal to] 0 and the initial conditions [x.sub.-1], [x.sub.0] arbitrary nonnegative numbers such that [x.sub.-1] +[x.sub.0] > 0. Equation (5) is a perturbed Sigmoid Beverton-Holt difference equation and it was studied in . The special case of Equation (5) for b = 0 is the well-known Thomson equation

[mathematical expression not reproducible], (6)

where the parameters c and f are positive numbers and the initial conditions [x.sub.-1], [x.sub.0] are arbitrary nonnegative numbers, is used in the modelling of fish population .

The dynamics of (6) is very interesting and follows from the dynamics of related equation

[x.sub.n+1] = [[x.sup.2.sub.n]/c[x.sup.2.sub.n] + f], n = 0, 1, .... (7)

Indeed (6) is delayed version of (7) and so it exhibits the existence of period-two solutions.

Two interesting special cases of Equation (2) are the following difference equations:

[mathematical expression not reproducible], (8)

studied in , and

[mathematical expression not reproducible], (9)

studied in . In both equations, (8) and (9), the associated map changes its monotonicity with respect to its variable.

In this paper, in some cases when the associated map changes its monotonicity with respect to the first variable in an invariant interval, we will use Theorems 1 and 2 below in order to obtain the convergence results. However, if F = [F.sub.g] = [(B/b).sup.3]c, we would not be able to use this method, so we will use the semicycle analysis; see  to show that each of the following four sequences [mathematical expression not reproducible] converges to the unique equilibrium point.

Also, we will show that Equation (1) exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

Note that the problem of determining invariant intervals in the case when the associated map changes its monotonicity with respect to its variable has been considered in [17, 18].

In this paper, we will use the following well-known results, Theorem 2.22, in , and Theorem 1.4.7 in .

Theorem 1. Let [a, b] be a compact interval of real numbers and assume that f : [a, b] x [a, b] [right arrow] [a, b] is a continuous function satisfying the following properties:

(a) f(x, y) is nondecreasing in x [member of] [a, b] for each y [member of] [a, b], and f(x, y) is nonincreasing in y [member of] [a, b] for each x [member of] [a, b];

(b) If (m, M) [member of] [a, b] x [a, b] is a solution of the system

f (m, M) = m and f (M, m) = M, (10)

then m = M.

Then

[x.sub.n+1] = f ([x.sub.n], [x.sub.n-1]), n = 0, 1, ... (11)

has a unique equilibrium [bar.x] [member of] [a, b] and every solution of Equation (11) converges to [bar.x].

Theorem 2. Let [a, b] be an interval of real numbers and assumethat f: [a, b] x [a, b] [right arrow] [a, b] is a continuous function satisfying the following properties:

(a) f(x, y) is nonincreasing in both variables

(b) If (m, M) [member of] [a, b] x [a, b] is a solution of the system

f (m, m) = M and f (M, M) = m, (12)

then m = M.

Then, (11) has a unique equilibrium [bar.x] [member of] [a, b] and every solution of Equation (11) converges to [bar.x].

Remark 3. As is shown in  the unique equilibrium in Theorems 1 and 2 is globally asymptotically stable.

The rest of this paper is organized as follows. The second section presents the local stability of the unique positive equilibrium solution and the nonexistence of the minimal period-two solution. The third section gives global dynamics in certain regions of the parametric space. The results and techniques depend on monotonic character of the transition function f(x, y) which is either decreasing in both arguments or increasing in first and decreasing in second argument. In simpler situations Theorems 1 and 2 are sufficient to prove global stability of the unique equilibrium. In more complicated situations we use the semicycle analysis, which is extensively used in [15, 19] for many linear fractional equations, to prove that every solution has four convergent subsequences, which leads to the conclusion that every solution converges to period-four solution. In some parts of parametric space we prove that there is no minimal period-four solution and so every solution converges to the equilibrium, while in other parts of parametric space we prove that the period-four solution exists. The semicycle analysis presented here uses innovative techniques based on analysis of systems of polynomial equations which coefficients depend on four parameters. Finally in the region of parameters complementary to the one where the period-four solution exists we prove that the Naimark-Sacker bifurcation takes place which produces locally stable periodic solution. All numerical simulations indicate that the equilibrium solution is globally asymptotically stable whenever it is locally asymptotically stable and that the dynamics is chaotic whenever the equilibrium is repeller. An interesting feature of Equation (1) is that it gives an example of second order difference equation with period-four solution for which period-two solution does not exist. The global dynamics of Equation (11) when the transition function f(x, y) is either increasing in both arguments or decreasing in the first and increasing in the second argument is fairly simple as every solution {[x.sub.n]} breaks into two eventually monotonic subsequences {[x.sub.2n]} and {[x.sub.2n+1]}; see [21-23]. The global dynamics of Equation (11) when the transition function f(x, y) is either decreasing in both arguments or increasing in the first and decreasing in the second argument could be quite complicated ranging from global asymptotic stability of the equilibrium, see [19, 21, 22, 24-26] to conservative and nonconservative chaos, see [3, 19, 26]. Interesting applications can be found in .

2. Linearized Stability

In this section, we present the local stability of the unique positive equilibrium of Equation (1) and the nonexistence of the minimal period-two solution of Equation (1).

In view of the above restriction on the initial conditions of Equation (1), the equilibrium points of Equation (1) are the positive solutions of the equation

[bar.x] = [B[[bar.x].sup.2] + F/(b + c) [[bar.x].sup.2], (13)

or equivalently

(b + c) [[bar.x].sup.3] - B[[bar.x].sup.2] - F = 0. (14)

Equation (1) has the unique positive solution [bar.x] given as

[mathematical expression not reproducible], (15)

where

[DELTA] = 2[B.sup.3] + 27F [(b + c).sup.2]. (16)

Now, we investigate the stability of the positive equilibrium of Equation (1). Set

F (u, v) = [Buv + F/buv + [cv.sup.2]] = [Buv + F/v(bu + cv)], (17)

and observe that

[mathematical expression not reproducible]. (18)

The linearized equation associated with Equation (1) about the equilibrium point [bar.x] is

[z.sub.n+1] = s[z.sub.n] + t[z.sub.n-1] (19)

where

s = [f'.sub.u] ([bar.x], [bar.x]) and t = [f'.sub.v] ([bar.x], [bar.x]). (20)

Theorem 4. Let [F.sub.0] = [(B/c).sup.3]b. The unique equilibrium point [bar.x] of Equation (1) given by (15) is

(i) locally asymptotically stable if F < [F.sub.0],

(ii) a repeller if F > [F.sub.0],

(iii) a nonhyperbolic point of elliptic type if F = [F.sub.0].

Proof. In view of

[mathematical expression not reproducible], (21)

we have that

s + t = [2F/[[bar.x].sup.3] (b + c)] < 0 (22)

and

[mathematical expression not reproducible], (23)

and so [absolute value of s] < [absolute value of (1 - t)].

Also, we have

[mathematical expression not reproducible]. (24)

Since [absolute value of s] < [absolute value of (1 - t)], the equilibrium point [bar.x] will be nonhyperbolic if t = -1 and [absolute value of s] < 2. From t = -1 we obtain

[mathematical expression not reproducible], (25)

and by using (14), we have

[mathematical expression not reproducible]. (26)

Now,

[mathematical expression not reproducible] (27)

and the characteristic equation of (19) is of the form

[[lambda].sup.2] + [b - c/b + c][lambda] + 1 = 0, (28)

from which

[mathematical expression not reproducible], (29)

and [absolute value of [[lambda].sub.1,2]] = 1,

that is, [bar.x] is nonhyperbolic equilibrium point. Let us denote

g (x) = (b + c) [x.sup.3] - B[x.sup.2] - F. (30)

Then,

[mathematical expression not reproducible]. (31)

The condition [absolute value of s] < [absolute value of (1 - t)] = 1 - t is always satisfied. Hence, it holds: the equilibrium solution [bar.x] is locally asymptotically stable if

[mathematical expression not reproducible], (32)

i.e., F < [F.sub.0] and a repeller if t > -1, which is equivalent with [mathematical expression not reproducible], i.e., F > [F.sub.0]. See Figure 1.

Lemma 5. Equation (1) has no minimal period-two solution.

Proof. Otherwise Equation (1) has a minimal period-two solution ... x, y, x, y, ... which satisfies

[mathematical expression not reproducible]. (33)

Then,

[mathematical expression not reproducible], (34)

which yields

(x - y) (bxy + c ([x.sup.2] + xy + [y.sup.2])) = 0, (35)

which implies x = y. So, there is no a minimal period-two solution.

3. Global Results

In this section, we prove several global attractivity results in the parts of parametric space.

We notice that the function f(u, v) is always decreasing with respect to the second variable and can be either decreasing or increasing with respect to the first variable, depending on the sign of the nominator of [f'.sub.u]. Therefore,

[mathematical expression not reproducible], (36)

and the function f(u, v) is nonincreasing in both variables if v [less than or equal to] [square root of (bF/Bc)], and nondecreasing with respect to the first variable and nonincreasing with respect to the second variable if v [greater than or equal to] [square root of (bF/Bc)]. Since

[mathematical expression not reproducible], (37)

if we denote [F.sub.g] = [(B/b).sup.3]c, we can have three possible cases:

[mathematical expression not reproducible]. (38)

As we have been seen, the nature of the local stability of the equilibrium point depends on the parameter [F.sub.0], so we distinguish the following scenarios:

(1) [F.sub.g] [less than or equal to] [F.sub.0],

(2) [F.sub.g] > [F.sub.0].

Case 1 ([F.sub.g] [less than or equal to] [F.sub.0]). Notice first that [F.sub.g] < [F.sub.0] implies c < b and that [F.sub.g] = [F.sub.0] implies c = b. Now, we observe three subcases.

(a) F < [F.sub.g] [less than or equal to] [F.sub.0]. If F < [F.sub.g] [less than or equal to] [F.sub.0], the function f(u, v) is nondecreasing with respect to the first variable and nonincreasing with respect to the second variable on the invariant interval of Equation (1) which is given by

[L, U] = [[square root of (bF]/Bc)], B/b], (39)

i.e., it holds

[mathematical expression not reproducible]. (40)

Indeed, since

[mathematical expression not reproducible] (41)

we have that

[mathematical expression not reproducible], (42)

and

[mathematical expression not reproducible] (43)

which is true for c [less than or equal to] b and F < [F.sub.g].

Also, since F < ([B.sup.3]/[b.sup.3])c = F, we obtain

[mathematical expression not reproducible]. (44)

This means that the equilibrium point [bar.x] belongs to the invariant interval [L, U].

Theorem 6. If F < [F.sub.g] [less than or equal to] [F.sub.0], then the equilibrium point [bar.x] is globally asymptotically stable.

Proof. The system of algebraic equations

f (m, M) = m, f (M, m) = M, (45)

is reduced to the system

F + BMm = m (c[M.sup.2] + bmM), F + BMm = M ([cm.sup.2] + Mbm), (46)

which yields

Mm(b - c)(M - m) = 0. (47)

Since c [not equal to] b, then it implies that m = M = [bar.x]. Now, by using Theorems 1 and 4, the conclusion follows.

For some numerical values of parameters we give a visual evidence for Theorem 6 which indicates that in the case when F < [F.sub.g] < [F.sub.0], the corresponding orbit converges very quickly (see Figure 2(a)), and in the case when F < [F.sub.g] = [F.sub.0], the corresponding orbit converges significantly slower (see Figure 2(b)).

(b) [F.sub.g] < F < [F.sub.0]

Lemma 7. If F < [F.sub.d] = 4[B.sup.3]/[(b + c).sup.2], then the system of algebraic equations

f (m, m) = M and f (M, M) = m, (48)

has the unique solution (m, M) = ([bar.x], [bar.x])

Proof. From (48) we have that

B[m.sup.2] + F = M[m.sup.2] (b + c), B[M.sup.2] + F = m[M.sup.2] (b + c), (49)

that is,

[mathematical expression not reproducible]. (50)

from which m = M = [bar.x] or B(m + M - (b + c)mM) = 0.

If B(m + M-(b + c)mM) = 0,thenm = BM/(-B + M(b + c)). Since M = (B[m.sup.2] + F)/[m.sup.2](b + c) > B/(b + c) (see the first equation of system (49)), then m >0. After substituting m in the second equation of system (49), we get

[mathematical expression not reproducible], (51)

from which we have that

[M.sub.1,2] = 1/2[B.sup.2] (F (b + c) [+ or -] [square root of ([F.sup.2] [(b + c).sup.2] - 4[B.sup.3]F)])] > 0. (52)

Straightforward calculation show that [m.sub.1] = [M.sub.2] and [m.sub.2] = [M.sub.1]. Notice that the solution ([m.sub.2], [M.sub.2]) is exactly the same as the solution ([M.sub.1], [m.sub.1]), and that system (48) has a unique solution m = M = [bar.x] if F [less than or equal to] 4[B.sup.3]/[(b + c).sup.2] = [F.sub.d].

Theorem 8. If [F.sub.g] < F [less than or equal to] [F.sub.d] < [F.sub.0], where [F.sub.d] = 4[B.sup.3]/[(b + c).sup.2], then the equilibrium [bar.x] is globally asymptotically stable.

Proof. If [F.sub.g] < F < [B.sup.3]/bc, then

[mathematical expression not reproducible], (53)

which means that the interval [L, U] = [B/b, [square root of (bF/Bc)]] is an invariant interval.

Indeed, since the function f is nonincreasing in both variables on the invariant interval, then

[mathematical expression not reproducible], (54)

and we obtain that

[mathematical expression not reproducible], (55)

and

[mathematical expression not reproducible]. (56)

Hence,

[mathematical expression not reproducible]. (57)

The following calculation will show that [F.sub.d] < [B.sup.3]/bc. Indeed,

[mathematical expression not reproducible], (58)

which is true.

Also, since g([square root of (bF/Bc)])g(B/b) < 0, it means that the equilibrium point [bar.x] belongs to the invariant interval [L, U].

Now, by using Lemma 7, Theorems 2 and 4, we get the conclusion that the equilibrium [bar.x] is globally asymptotically stable.

For some numerical values of parameters we give a visual evidence for Theorem 8. See Figure 3.

Lemma 9. Assume that F [not equal to] [F.sub.g].

(i) If [x.sub.n-1] < [square root of (bF/Bc)], then [x.sub.n+1] > B/b.

(ii) If [x.sub.n-1] > [square root of (bF/Bc)], then [x.sub.n+1] < B/b.

(iii) If [x.sub.n-1] = [square root of (bF/Bc)], then [x.sub.n+1] = B/b.

Proof. Since the map associated with the right-hand side of Equation (1) is always decreasing in the second variable, we have that

[mathematical expression not reproducible]. (59)

Note that under assumption of Lemma 9, the following inequality holds:

[mathematical expression not reproducible]. (60)

(c) F = [F.sub.g] < [F.sub.0]. By substituting parameter F = [F.sub.g] = [(B/b).sup.3]c = [E.sup.3]c, where [bar.x] = E = B/b, in Equation (1), we obtain

[mathematical expression not reproducible]. (61)

Lemma 10. (i) Assume that F = [F.sub.g] < [F.sub.0], i.e., c < b. Then Equation (61) does not possess a minimal period-four solution.

(ii) Assume that F = [F.sub.g] = [F.sub.0], i.e., c = b. Then Equation (61) has the minimal period-four solutions of the form

[mathematical expression not reproducible] (62)

where E = [bar.x] = (B/b)(= B/e) and t > 0 is an arbitrary number depending on initial conditions [x.sub.0] and [x.sub.-1].

Proof. Suppose that Equation (61) has a minimal period-four solution ... x, y, z, t, x, y, z, t, ... ; then it holds

[mathematical expression not reproducible], (63)

where E = [bar.x]. By eliminating x and y we obtain

z (z - E)W(t, z) = 0, t (t - E)U(t, z) = 0, (64)

where the functions W(t, z) and U(t, z) can be written in the polynomial form as

[mathematical expression not reproducible], (65)

[mathematical expression not reproducible], (66)

where

[mathematical expression not reproducible]. (67)

Since z [not equal to] 0 and t [not equal to] 0, from system (64), we obtain the following four cases:

(1) The system

z - E = 0, t - E = 0, (68)

implies z = t = E, and by using (63), weget x = y = E.

(2) The system

z - E = 0, U (t, z) = 0, (69)

implies z = E and

U (t, E) = [E.sup.3] (b - c) (b + c) ([ct.sup.2] + tE (b + c) + c[E.sup.2]) x [(cE + bt).sup.2] > 0 (70)

if c < b. If b = c, then U(t, z) = 0 is satisfied for every t >0, and by using system (63), it follows that the periodic solution of the minimal period four is of the form (62).

(3) The system

W(t, z) = 0, T - E = 0, (71)

implies t = E and

[mathematical expression not reproducible], (72)

so the conclusion is the same as in the previous case.

(4) The system

W (t, z) = 0, U (t, z) = 0, (73)

demands more detailed analysis.

(a) Assume that b > c. Then we can write b = c + [xi], [xi] > 0.

Consider the polynomials W(t, z) and U(t, z) as polynomials in one variable t:

[mathematical expression not reproducible], (74)

where

[mathematical expression not reproducible]. (75)

If z [member of] [[c.sup.2]E/[(c + [xi]).sup.2], E], then [b.sub.i] [greater than or equal to] 0, for i = 0, 1, 4 and [b.sub.2] > 0, [b.sub.3] > 0, so we have that

U(t, z) > 0 for z [member of] [[c.sup.2]E/[(c + [xi]).sup.2], E]. (76)

Since W(t, z) and U(t, z) are polynomials of the fifth and fourth degrees, respectively, the resultant of these polynomials is the determinant of the ninth degree:

[mathematical expression not reproducible], (77)

from which we obtain

[mathematical expression not reproducible], (78)

where

[mathematical expression not reproducible]. (79)

If the equation [res.sub.t](W, U) = 0 has solutions for variable z, then they are the common roots of both equations in system (73) for a fixed value of t. One of these positive roots is [z.sub.1] = E, but for z = E and t >0 system (73) has no solutions since U(t, E) > 0, see (76). Therefore, in this case, Equation (61) has no minimal period-four solution.

The positive solution of the equation [[??].sub.1] (z) = 0 is

[mathematical expression not reproducible]. (80)

We will show later that [z.sub.2] can not be a component of any positive solutions of system (73).

The positive solution of the equation [[??].sub.2] (z) = 0 is

[mathematical expression not reproducible], (81)

and [z.sub.3] [member of] ([c.sup.2]E/[(c + [xi]).sup.2], E). Namely,

[mathematical expression not reproducible], (82)

which is true, and

[mathematical expression not reproducible], (83)

which is also true. So, system (73) has no solutions since U(t, E) > 0, see (76).

Now, we prove that the eventually positive roots of the equation [PHI](z, E, c + [xi], c) = 0 for variable z can belong only to the interval ([c.sup.2]E/[(c + [xi]).sup.2], E) and that every z [member of] ([c.sup.2]E/[(c + [xi]).sup.2], E) can not be a component of any positive solution of system (73). First, we prove that [PHI](z, E, c + [xi], c) > 0 for z > E. Let z = E + [gamma], [gamma] > 0; then we obtain

[mathematical expression not reproducible], (84)

i.e., [PHI](E + [gamma], E, c + [xi], c) > 0. It means that the function [PHI](z, E, c + [xi], c) eventually has the positive roots in the interval (0, E]. Since we already considered the case when z = E, now we investigate the existence of the positive roots of the equation [PHI](z, E, b, c) = 0 for 0 < z < E. As we have seen, U(t, z) > 0 for [c.sup.2]E/[(c + [xi]).sup.2] < z < E and t >0, so system (73) has no solution in the case when the equation [PHI](z, E, c + [xi], c) = 0 has the positive roots in the interval ([c.sup.2]E/[(c + [xi]).sup.2], E). This implies that Equation (61) has no minimal period-four solution whenever any root of equation [res.sub.t](W, U) = 0 lies in the interval ([c.sup.2]E/[(c + [xi]).sup.2], E).

Now, we prove that the equation [PHI](z, E, c + [xi], c) = 0 has no root for variable z if z [member of] (0, [c.sup.2]E/[(c + [xi]).sup.2]) and [xi] > 0. It is easy to see the following:

[[PHI].sub.0] = [PHI] (0, E, c + [xi], c) = 16[c.sup.3][E.sup.4] [(c + [xi]).sup.8] > 0. (85)

For [[PHI].sub.1] = [PHI]([c.sup.2]E/[(c + [xi]).sup.2], E, c + [xi], c) we have

[mathematical expression not reproducible], (86)

and

[mathematical expression not reproducible]. (87)

Now, it is sufficient to prove d[PHI](z, E, b, c)/dz < 0 in (0, [c.sup.2]E/[(c + [xi]).sup.2]), [xi] > 0. Since

[mathematical expression not reproducible], (88)

where

[mathematical expression not reproducible], (89)

then for z = [c.sup.2]E/k[(c + [xi]).sup.2], [xi] > 0, k > 1 we obtain

[mathematical expression not reproducible], (90)

with

[mathematical expression not reproducible]. (91)

Similarly, now we will consider W(t, z) and U(t, z) as polynomials in the variable t (with the coefficients [[alpha].sub.i], [[beta].sub.j], i [member of] {0, 1, ... , 5}, j [member of] {0, 1, ... , 4}). The resultant of these polynomials is

[mathematical expression not reproducible], (92)

where

[mathematical expression not reproducible]. (93)

If the equation [res.sub.z](W, U) = 0 has solutions for variable t, then they are the common roots of both equations in system (73) for a fixed value of z. One of these positive roots is [t.sub.1] = E, and for t = E and z > 0 system (73) has no solutions since

[mathematical expression not reproducible] (94)

for [xi] > 0. It means that Equation (61) has no minimal period-four solution.

Similarly, for the second root [t.sub.2] = cE/(c + [xi]), [xi] > 0 we obtain [??] = U(cE/(c + [xi]), z, c + [xi], c),

[mathematical expression not reproducible], (95)

where [xi] > 0. It means that Equation (61) has no minimal period-four solution in this case. As we have already seen, the equation [PHI](t, E, c + [xi], c) = 0 has eventually positive roots only in the interval t [member of] ([c.sup.2]E/[(c + [xi]).sup.2], E), [xi] > 0. Then,

[mathematical expression not reproducible]. (96)

This implies W(t, z) > 0 for [c.sup.2]E/[(c + [xi]).sup.2] < t < E and z > 0, so system (73) has no solution in the case when the equation [PHI](t, E, c + [xi], c) = 0 has the positive roots in the interval ([c.sup.2]E/[(c + [xi]).sup.2], E), which further means that Equation (61) has no minimal period-four solution.

Also, the positive solution of the equation [[LAMBDA].sub.2](t) = 0 is

[mathematical expression not reproducible]. (97)

Note that [t.sub.3] = [z.sub.2]. Now, we prove that ([z.sub.2], [z.sub.2]) can not be solution of system (73). Indeed, suppose the opposite, i.e.,

[mathematical expression not reproducible], (98)

which is a contradiction with the assumption that c > 0 and [xi] > 0.

Consequently system (73) does not have positive solutions when b > c.

(b) Assume that b = c. Then, system (73)

W(t, z, c, c) = 0, U(t, z, c, c) = 0, (99)

is of the form

[mathematical expression not reproducible], (100)

and combining those equations, we have the following four cases:

(i)

t - E = 0, z - E = 0, (101)

and the solution in this case is t = z = E,

(ii)

[mathematical expression not reproducible], (102)

and substituting t by E we obtain

[E.sup.2] (z - E)[(z + E).sup.3] = 0 (103)

from which we get that the solution is t = z = E,

(iii)

[mathematical expression not reproducible], (104)

i.e.,

[E.sup.2] (t - E)[(t + E).sup.3] = 0, (105)

and the solution is t = z = E,

(iv)

[mathematical expression not reproducible]. (106)

By subtracting we get

-2tE (z - E)(z + E) (zE + [E.sup.2] + tz + [z.sup.2]) = 0, (107)

i.e.,

z = E. (108)

Hence, the solution is t = z = E.

This means that (E, E) is a solution of system (73) and that Equation (61) does not possess a minimal period-four solution.

Thus, if F = [F.sub.g] < [F.sub.0], then Equation (61) does not possess a minimal period-four solution. Consequently if F = [F.sub.g] = [F.sub.0], then Equation (61) has the minimal period-four solutions of the form (62).

Theorem 11. Assume that F = [F.sub.g] = [(B/b).sup.3]c < [F.sub.0]. Then, the unique equilibrium point [bar.x] = B/b of Equation (61) is globally asymptotically stable. Also, every solution of Equation (61) oscillates about the equilibrium point [bar.x] with semicycles of length two.

Proof. Notice that

[mathematical expression not reproducible], (109)

i.e., [x.sub.n+1] and [x.sub.n-1] are from the different sides of the equilibrium point (see also Lemma 9, when [square root of (bF/Bc)] = B/b). Also, that means [x.sub.n+1] and [x.sub.n+5] are always from the same side of the equilibrium point [bar.x] = B/b. Since

[mathematical expression not reproducible], (110)

where [mathematical expression not reproducible] is a linear function in variable [x.sub.n], it can be seen that H = 0 [??] [x.sub.n+4] = [x.sub.n] = B/b because Equation (61) has no period-two solutions nor period-four solutions (and it holds that [x.sub.n] = B/b [??] [x.sub.n+2] = B/b [??] [x.sub.n+4] = B/b, see Lemma 9). Also,

[mathematical expression not reproducible], (111)

which means that every sequence [mathematical expression not reproducible] is monotone and bounded. That implies that each of the sequences is convergent. Since, by Lemmas 5 and 10, Equation (61) has neither minimal period-two nor period-four solutions, it holds

[mathematical expression not reproducible] (112)

which implies that equilibrium [bar.x] is an attractor and by using Theorem 4, which completes the proof of the theorem.

For some numerical values of parameters we give a visual evidence for Theorem 11. See Figure 4.

Remark 12. One can see from Theorems 6, 8, and 11 that the equilibrium point [bar.x] is globally asymptotically stable for all values of parameter F such that 0< F [less than or equal to] [F.sub.d], where [F.sub.g] < [F.sub.0], i.e., c < b (see Figure 5(a)) and for all values of parameter F such that 0 < F < [F.sub.g] = [F.sub.0] = [F.sub.d], i.e., c = b (see Figure 5(b)).

(d) F = [F.sub.g] = [F.sub.0]. Since [F.sub.g] = [F.sub.0] implies c = b, Equation (61) is of the form

[mathematical expression not reproducible]. (113)

In this case, by using Lemma 10, we see that Equation (61) has minimal period-four solutions of the form (62). Based on our many numerical simulations and the proof of Theorem 11, we believe that the following conjectures are true.

Conjecture 13. If F = [F.sub.g] = [F.sub.0] (that is b = c), then every solution of Equation (61) converges to some period-four solution of the form (62) or to the equilibrium point [bar.x].

For some numerical values of parameters we give a visual evidence for this case. See Figures 6 and 7.

Case 2 (F < [F.sub.0] < [F.sub.g]). We give a visual evidence for some numerical values of parameters which indicates very interesting behaviour and verifies our suspicion that the equilibrium point [bar.x] is globally asymptotically stable in this case also. See Figures 8 and 9.

Conjecture 14. If F < [F.sub.0] < [F.sub.g] (that is, b < c), then the equilibrium point [bar.x] of Equation (1) is globally asymptotically stable.

4. Naimark-Sacker Bifurcation for b [not equal to] c

In this section, we consider bifurcation of a fixed point of map associated with Equation (1) in the case where the eigenvalues are complex conjugates and of unit module. We use the following standard version of the Naimark-Sacker result, see [28, 29]

Theorem 15 (Naimark-Sacker or Poincare-Andronov-Hopf Bifurcation for maps). Let

F : R x [R.sup.2] [right arrow] [R.sup.2]; ([lambda], x) [right arrow] F ([lambda], x) (114)

be a [C.sup.4] map depending on real parameter [lambda] satisfying the following conditions:

(i) F([lambda], 0) = 0 for [lambda] near some fixed [[lambda].sub.0];

(ii) DF([lambda], 0) has two nonreal eigenvalues [mu](([lambda]) and [bar.[mu]([lambda])] for [lambda] near [[lambda].sub.0] with [absolute value of [mu]([[lambda].sub.0])] = 1;

(iii) (d/d[lambda])[absolute value of [mu]([lambda])] = d([[lambda].sub.0]) [not equal to] 0 at [lambda] = [[lambda].sub.0];

(iv) [[mu].sup.k]([[lambda].sub.0]) [not equal to] 1 for k = 1, 2, 3, 4.

Then there is a smooth [lambda]-dependent change of coordinate bringing F into the form

F([lambda], x) = F ([lambda], x) + O([[parallel]x[parallel].sup.5]) (115)

and there are smooth function a([lambda]), b([lambda]), and [omega]([lambda]) so that in polar coordinates the function F([lambda], x) is given by

[mathematical expression not reproducible]. (116)

If a([[lambda].sub.0]) > 0, then there is a neighborhood U of the origin and a [delta] > 0 such that for [absolute value of [lambda] - [[lambda].sub.0]] < [delta] and [x.sub.0] [member of] U, then [omega]-limit set of [x.sub.0] is the origin if [lambda] < [[lambda].sub.0] and belongs to a closed invariant [C.sup.1] curve [GAMMA]([lambda]) encircling the origin if [lambda] > [[lambda].sub.0]. Furthermore, [GAMMA]([[lambda].sub.0]) = 0.

If a([[lambda].sub.0]) < 0, then there is a neighborhood U of the origin and a [delta] > 0 such that for [absolute value of ([lambda] - [[lambda].sub.0])] < [delta] and [x.sub.0] [member of] U, then [alpha]-limit set of [x.sub.0] is the origin if [lambda] > [[lambda].sub.0] and belongs to a closed invariant [C.sup.1] curve [GAMMA]([lambda]) encircling the origin if [lambda] < [[lambda].sub.0]. Furthermore, [GAMMA]([[lambda].sub.0]) = 0.

Consider a general map F([lambda], x) that has a fixed point at the origin with complex eigenvalues [mu](X) = [alpha](X) + i[beta]([lambda]) and [bar.[mu]([lambda])] = [alpha]([lambda]) - i[beta]([lambda]) satisfying ([alpha][([lambda]).sup.2] + ([beta][([lambda])).sup.2] = 1 and [beta]([lambda]) [not equal to] 0. By putting the linear part of such a map into Jordan Canonical form, we may assume F to have the following form near the origin:

[mathematical expression not reproducible]. (117)

Then the coefficient a([[lambda].sub.0]) of the cubic term in Equation (116) in polar coordinate is equal to

[mathematical expression not reproducible], (118)

where

[mathematical expression not reproducible], (119)

[mathematical expression not reproducible], (120)

[mathematical expression not reproducible], (121)

and

[mathematical expression not reproducible]. (122)

Theorem 16. Assume that b, c, B > 0, [F.sub.0] = [(B/c).sup.3]b, and [bar.x] = B/c.

(i) If [k.sub.2]c < b < c or c < b < [k.sub.3]c, where [k.sub.2] and [k.sub.3] are positive solutions of the equation 3[k.sub.3] - 9[k.sub.2] - 3k + 1 = 0, then there is a neighborhood U of the equilibrium point [bar.x] and [rho] > 0 such that for [absolute value of F - [F.sub.0]] < [rho] and [x.sub.0], [x.sub.-1] [member of] U then [omega]-limit set of solution of Equation (1), with initial condition [x.sub.0], [x.sub.-1] is the equilibrium point [bar.x] if F < [F.sub.0] and belongs to a closed invariant [C.sup.1] curve [GAMMA]([F.sub.0]) encircling the equilibrium point [bar.x] if F > [F.sub.0]. Furthermore, [GAMMA]([F.sub.0]) = 0.

(ii) If 0 < b < [k.sub.2]c or [k.sub.3]c < b < +[infinity], then there is a neighborhood U of the equilibrium point [bar.x] and a [rho] > 0 such that for [absolute value of F - [F.sub.0]] < [rho] and [x.sub.0], [x.sub.-1] [member of] U then [alpha]-limit set of [x.sub.0], [x.sub.-1] is the equilibrium point [bar.x] if F > [F.sub.0] and belongs to a closed invariant [C.sup.1] curve [GAMMA]([F.sub.0]) encircling the equilibrium point [bar.x] if F < [F.sub.0]. Furthermore, [GAMMA]([F.sub.0]) = 0.

Proof. See Figures 10 and 11 for visual illustration. In order to apply Theorem 15 we make a change of variable

[mathematical expression not reproducible]. (123)

Set

[mathematical expression not reproducible], (124)

then

[mathematical expression not reproducible]. (125)

Let us define the function

[mathematical expression not reproducible]. (126)

Then K(u, v) has the unique fixed point (0, 0). The Jacobian matrix of K(u, v) is given by

[mathematical expression not reproducible]. (127)

and its value at the zero equilibrium is

[mathematical expression not reproducible], (128)

i.e.,

[mathematical expression not reproducible]. (129)

The eigenvalues [mu](F), [bar.[mu](F)], using (128), are

[mathematical expression not reproducible] (130)

because

[mathematical expression not reproducible]. (131)

Then

[mathematical expression not reproducible], (132)

where

[mathematical expression not reproducible]. (133)

Denote [F.sub.0] = [(B/c).sup.3]b. For F = [F.sub.0] we have [mathematical expression not reproducible]. The eigenvalues of [J.sub.0] are [mu]([F.sub.0]) and [bar.[mu]([F.sub.0])] where

[mathematical expression not reproducible]. (134)

The eigenvectors corresponding to [mu]([F.sub.0]) and [bar.[mu]([F.sub.0])] are v([F.sub.0]) and [bar.v([F.sub.0])] where

[mathematical expression not reproducible]. (135)

Further,

[mathematical expression not reproducible], (136)

and [[mu].sup.k]([F.sub.0]) [not equal to] 1 for k = 1, 2, 3, 4 for c > 0, b > 0, and b [not equal to] c. For F = [F.sub.0] and [bar.x] = B/c

[mathematical expression not reproducible], (137)

and

[mathematical expression not reproducible]. (138)

Hence, for F = [F.sub.0] system (125) is equivalent to

[mathematical expression not reproducible]. (139)

Let [mathematical expression not reproducible], where

[mathematical expression not reproducible]. (140)

Then system (125) is equivalent to its normal form

[mathematical expression not reproducible] (141)

where

[mathematical expression not reproducible]. (142)

Let

[mathematical expression not reproducible]. (143)

By the straightforward calculation we obtain that

[mathematical expression not reproducible], (144)

where

[mathematical expression not reproducible], (145)

and

[mathematical expression not reproducible]. (146)

Further,

[mathematical expression not reproducible]. (147)

Now, by using (118), (119), (120), (121), and (122) we obtain

[mathematical expression not reproducible] (148)

and finally,

[mathematical expression not reproducible]. (149)

If we substitute b with kc we obtain

[mathematical expression not reproducible]. (150)

So,

[mathematical expression not reproducible]. (151)

Solutions, determined numerically, are [k.sub.1] [approximately equal to] -0.48445, [k.sub.2] [approximately equal to] 0.21014, and [k.sub.3] [approximately equal to] 3.2743. Since b > 0 and c > 0 it must be k > 0. Now,

[mathematical expression not reproducible]. (152)

Further,

[mathematical expression not reproducible] (153)

and [mathematical expression not reproducible], i.e., [mathematical expression not reproducible]. By differentiating the equilibrium equation

(b + c)[x.sup.3] - B[x.sup.2] - F = 0 (154)

with respect to F and solving for x'(F) we obtain x'(F) = 1/(3(b + c)[x.sup.2] - 2Bx), i.e., [mathematical expression not reproducible].

[mathematical expression not reproducible]. (155)

Now,

[mathematical expression not reproducible]. (156)

By substituting x'(F) in the above expression and considering the fact that [absolute value of [mu]([F.sub.0])] = 1, we obtain

[mathematical expression not reproducible], (157)

i.e.,

[mathematical expression not reproducible] (158)

and that completes the proof of the theorem.

https://doi.org/10.1155/2018/7052935

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

The article is a joint work of all four authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Acknowledgments

This work is supported by the Fundamental Research Funds of Bosnia and Herzegovina FM001/2-3995-V/17.

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M. R. S. Kulenovic (iD), (1) S. Moranjkic, (2) M. Nurkanovic, (2) and Z. Nurkanovic (iD) (2)

(1) Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA

(2) Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

Correspondence should be addressed to M. R. S. Kulenovic; mkulenovic@mail.uri.edu

Received 18 July 2018; Accepted 17 October 2018; Published 8 November 2018

Guest Editor: Abdul Qadeer Khan

Caption: Figure 1: If F < [F.sub.0], then [bar.x] < [[bar.x].sub.nh], i.e., [bar.x] is LAS, and if F > [F.sub.0], then [bar.x] > [[bar.x].sub.nh], i.e., [bar.x] is repeller.

Caption: Figure 2: The orbit for (a) b = 2, c = 1, B = 4, F = 7, [F.sub.g] = 8, [F.sub.0] = 128, and ([x.sub.0], [x.sub.-1]) = (0.3, 2.8) and (b) b = 0.5, c = 0.5, B = 2, F = 30, [F.sub.g] = [F.sub.0] = 32, and ([x.sub.0], [x.sub.-1]) = (1.1, 1) generated by Dynamica 4 .

Caption: Figure 3: The orbit and the phase portrait for b = 2, c = 1, B = 4, F = 28, [F.sub.g] = 8, [F.sub.0] = 128, and ([x.sub.0], [x.sub.-1]) = (0.3,2.8) generated by Dynamica 4 .

Caption: Figure 4: The orbit and the phase portrait for b = 2, c = 1, B = 4, F = 8, [F.sub.g] = 8, [F.sub.0] = 128, and ([x.sub.0], [x.sub.-1]) = (0.3, 2.8) generated by Dynamica 4 .

Caption: Figure 5: Visual representation of local and global asymptotic stability of Equation (1) when (a) [F.sub.g] < [F.sub.d] < [F.sub.0], i.e., c < b, and (b) [F.sub.0] = [F.sub.g], i.e., c = b.

Caption: Figure 6: The orbit and phase portraits for b = 1, c = 1, B = 2, and F = [F.sub.g] = [F.sub.0] = 8 generated by Dynamica 4 .

Caption: Figure 7: Bifurcation diagram in (F, x) plane for b = 1, c = 1, and B = 2, generated by Dynamica 4 .

Caption: Figure 8: The orbit and phase portrait for c = 2, b = 1, B = 4, F = 7, [F.sub.0] = 8, [F.sub.g] = 128, and ([x.sub.0], [x.sub.-1]) = (0.3, 2.8) generated by Dynamica 4 .

Caption: Figure 9: The orbit and phase portrait for c = 2, b = 1, B = 4, F = 3, [F.sub.0] = 8, [F.sub.g] = 128, and ([x.sub.0], [x.sub.-1]) = (0.3, 2.8) generated by Dynamica 4 .

Caption: Figure 10: Bifurcation diagrams in (F, x) plane for b = 0.5, c =1, and B = 2, generated by Dynamica 4 .

Caption: Figure 11: Phase portraits when c = 1, b = 0.5, B = 2, [x.sub.-1] = [x.sub.0] = 2.1 (green), and [x.sub.-1] = [x.sub.0] = 6.8 (blue), generated by Dynamica 4 .
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