A NOTE ON GLOBAL STABILITY IN THE PERIODIC LOGISTIC MAP

. In this paper, the dynamics of the celebrated p − periodic one-dimensional logistic map is explored. A result on the global stability of the origin is provided and, under certain conditions on the parameters, the local stability condition of the p − periodic orbit is shown to imply its global stability.


RAFAEL LU ÍS AND SANDRA MENDONC ¸A
where x n ∈ [0, 1], as before.Here, the dynamics of the equation is given by the composition of maps f 0 , f 1 , f 2 , . .., where If the sequence of parameters r n is periodic, i.e., if there exists an integer p > 1 (the period) such that r n+p = r n , for all n ∈ N, then f n+p = f n , for all n ∈ N.This is equivalent to say that Equation ( 2) is non-autonomous p−periodic or simply p−periodic.In this case, the dynamics of Equation ( 2) is determined by the following composition map A fixed point of Φ i p generates a periodic cycle in the sequence defined by Equation (2).It follows from [2] that this fixed point generates a periodic cycle of minimal period p of the equation (2).Let us represent this p−periodic cycle by the following ordered set of points C p = {x 0 , x 1 , . . ., x p−1 } .
Since Φ i p (x i ) = x i , the dynamics of Equation ( 2) is completely determined by the composition operator Φ 0 p = Φ p .Throughout this paper we study Equation (2) when the sequence of maps f n is p−periodic.
The local dynamics of the periodic logistic map has been investigated in the last decades.The composition of two quadratic maps was already considered by M. Kot and W. M. Schaffer [10] in 1984, in the framework of seasonal populations.In this work, the authors study the effects of seasonality on the dynamics of a bivoltine population with discrete, nonoverlapping generations and conclude that large seasonality is inevitably destabilizing but that mild seasonality may have a pronounced stabilizing effect.
In 1992, Li [12] studied a periodically fluctuating environment in a populationmodelling process that generates non-autonomous periodic difference equations.In this work, the existence and uniqueness of periodic solutions are studied, a sufficient condition for existence and a necessary condition for uniqueness are obtained, and the stability of the periodic solutions is also investigated.Results are illustrated with several examples including the periodic logistic.Later, Grinfeld et. al. in 1996 [8] explored the logistic equation modified by a periodic time dependence.AlSharawi et.al. [2], in 2006, proved that the p-periodic logistic equation has cycles (periodic solutions) of minimal periods 1, p, 2p, 3p, . ... The authors have extended Singer's theorem to periodic difference equations, and used it to show that the p-periodic logistic equation has at most p stable cycles.The paper also includes computational methods to explore the stable cycles in the cases p = 2 and p = 3 and the determination of bifurcation curves.
Our main focus in this paper will be the global stability of C p , defined in (5).We will follow the techniques (partially) employed by Eduardo Liz in [13].In this groundbreaking paper, Liz showed global stability of the positive periodic cycle of the non-autonomous 2−periodic Ricker equation (and 3−periodic as well) given by x n+1 = x n e pn−xn , p n > 0 for all n = 0, 1, 2, . . . .Particularly, we will explore the conditions under which the local stability of a fixed point of Φ p implies its global stability, which is the same to say that Equation (2) has a globally asymptotically stable p−periodic cycle.It should be mentioned that in [11] the authors used different tools and studied the boundedness and the periodicity of non-autonomous periodic logistic map (3) when the sequence of parameters is periodic such that 0 < r n ≤ 2 for all n, and that sufficient conditions are given to support the existence of asymptotically stable and unstable p−periodic orbits, in this case.
In this paper we show that the global stability condition may be extended for a larger set of parameters, as long as the parameters are such that p−1 i=0 r i > 1 and that the absolute value of the derivative along the periodic orbit is less or equal than 1.Under these conditions and an assumption on the critical points of the composition operator Φ p , we establish global stability of the nontrivial fixed point of Φ p in Theorem 4.2.(or global stability of the p−periodic cycle of the p−periodic logistic equation ( 2)), which is the main goal of this work.
2. General results in stability.Let us consider the difference equation given by where, for each n, f n : X → X, for some topological space X, is a C 3 map.Here, the orbit of a point x 0 is generated by the composition of the sequence of maps Let f n+p = f n for all n = 0, 1, 2, . . .and define the composition operator Φ as follows Notice that we represent the n−th composition of Φ i p by Φ i,n p and when i = 0 we write Φ 0 p = Φ p = f p−1 • . . .• f 1 • f 0 .Let x i mod r to be a fixed point of Φ i s (guaranteed by Brouwer's fixed point theorem [15]).In other words we have that Φ i s (x i mod r ) = x i mod r .Notice that a nontrivial fixed point of Φ i s generates a periodic cycle in Equation (6).In order to study the behaviour of the composition map Φ i p , we follow the definition of stability present in Elaydi [7], page 19.However, this definition of stability may not be the most practical tool to show the stability of a fixed point of Φ i p .There exists a simple but powerful criterion that guarantees the local stability of fixed points that we recall here in the following two results.Theorem 2.1 (Local stability criteria, Elaydi [7], page 25).Let x i be a hyperbolic fixed point of a map Φ i p , i.e, a fixed point such that |(Φ i p ) (x i )| = 1, where Φ i p is continuously differentiable at x i .The following statements hold true: The stability criteria for nonhyperbolic fixed points are more complex and may be found in Elaydi's book [7], pages 28-30.
The precedent result gives conditions that guarantee local stability.In 1955 W. Coppel stated the following result concerning global dynamics: Theorem 2.2 (Global dynamics, Coppel [3]).Let I = [a, b] ⊆ R and f : I → I be a continuous map.If the equation f (f (x)) = x has no roots except the roots of the equation f (x) = x , then every orbit under the map f converges to a fixed point.
Coppel's theorem ensures global stability only when the map f has a unique fixed point.However, solving analytically the equation f (f (x)) = x can be a hard work, if not impossible, especially in periodic equations.
The following result is a consequence of Singer's results [18]: Theorem 2.3 (Global stability, Singer's theorem [18]).If a unimodal map with negative Schwarzian derivative has a unique fixed point x * which is locally asymptotically stable, then x * is globally stable.
Theorem 2.3 is a powerful result in global stability.However, when dealing with composition of maps, only in particular cases the composition map Φ i p is unimodal.In 2008, El-Morshedy and Jiménez López presented the following result concerning global stability, a consequence of Theorem 2.3: Assume that there are numbers c and d with a has at most one turning point, and (whenever it makes sense) f (x) ≤ f (c) for every x ≤ c, and f (x) ≥ f (d) for every x ≥ d.In addition, if f is decreasing at x * , assume additionally that Sf (x) < 0 for all x ∈ (c, d) except at most one critical point of f , and −1 ≤ f (x * ) < 0. Then x * is globally stable.

Trivial solution.
In this section we establish the global stability of the trivial fixed point of Equation (2), i.e, equation when {r n , n ∈ N} is a p−periodic sequence of numbers of the interval (0, 4].Notice that x * = 0 is a fixed point of Equation (8).Define Φ 0 (x) = x, and for p ≥ 1 let A condition that guarantees the local stability of x * = 0 is given by p−1 i=0 r i ≤ 1.In order to see this, notice that Proof.If 1 − Φ i (x), i = 0, 1, 2 . . . is either 0 or 1, then we are done.Suppose Φ i (x) = 1 and Φ i (x) = 0, i = 0, 1, 2 . ... Since 1 − Φ i (x) ∈ (0, 1) for all x ∈ (0, 1), the relation Multiplying both sides of this last inequality by x ∈ (0, 1] we obtain Hence, the origin is the unique fixed point of Φ p in the unit interval. Moreover, Φ p (Φ p (x)) < Φ p (x) < x, for all x ∈ (0, 1].Thus, the unique solution of the equation Φ p (Φ p (x)) = x in the unit interval is x * = 0. Thus, applying Coppel's Theorem it follows that every orbit under the map Φ p , in the unit interval, converges to x * = 0.
4. Nontrivial solution.In this section we prove the global stability of the p−periodic solution of the periodic logistic equation (8).There are two cases to consider: (i) the composition map Φ p has a unique critical point (unimodal map) and (ii) the composition map Φ p has more than one critical point.
The critical points of Φ p are the solutions of the equation Φ p (x) = 0, or equivalently, of the equation 2 is the unique critical point of each individual map f i , 0 ≤ i ≤ p − 1, it follows that the critical points of Φ p are the solutions of the following p equations , the p equations are equivalent to It is not possible to write the solutions of the last p − 1 equations explicitly for all cases.However, it is not hard to see that the number of solutions is an odd number not greater than p j=1 2 j−1 = 2 p − 1.For instance, when p = 3 we obtain the equations Since Φ 0 (x) = x and Φ 1 (x) = f 0 (x), we obtain the following six potencial solutions: 1.If r 0 > 2 and r 1 ≤ 2, then Φ 3 has three critical points, 1 2 and x ± .
2. If r 0 ≤ 2 and r 1 > 2, then, since , the composition operator has either one or three critical points, either 1  2 or 1 2 and x±,− .3. If r 0 > 2 and r 1 > 2, then Φ 3 has either five or seven critical points, i.e., either Notice that the condition in the product is needed in order to find a positive fixed point of Φ p , i.e., a p−periodic cycle of the difference equation ( 8).
We shall now prove the following result: Theorem 4.1.Under the conditions on the parameters r i , i = 0, ..., p − 1, given by ( 11), the p−periodic logistic equation ( 8) has a globally asymptotically stable p−periodic cycle.
Proof.The individual maps f i are increasing on 0, 1 2 and their positive fixed points are x * i = ri−1 ri < 1 2 , i = 0, 1, ....Moreover, 1 2 is the unique and common critical point of all individual maps.Notice that conditions (10) and ( 11) together imply that 1  2 is also the unique critical point of the composition map Φ p .Let us take the interval [m, M ], where m = min 0≤i≤p−1 {x * i } and M = max 0≤j≤p−1 {x * j }.Clearly, Φ p (m) ≥ m and Φ p (M ) ≤ M .Thus [m, M ] is invariant under Φ p .Moreover, it is compact and a convex subset of R and Φ p is continuous.Hence, from Brouwer's fixed-point theorem [9] the composition map Φ p has a nontrivial fixed point in the interval [m, M ].
Under conditions (11), it is clear that the origin is an unstable fixed point.So there is a neighborhood of the origin where Φ p (x) > x.A simple induction shows that Φ p 1 2 ≤ 1 2 .Moreover, Φ p (0) = Φ p (1) = 0 and Φ p (x) is a polynomial of degree 2 p with a negative leading coefficient.The composition function Φ p (x) is thus a concave down function in [0, 1] with a unique critical point 1  2 .More precisely, Φ p (x) is increasing in 0, 1  2 since it is a composition of increasing functions on 0, 1 2 and decreasing in 1  2 , 1 since it is the composition of decreasing functions on 1 2 , 1 .In other words, the composition map Φ p is a unimodal map in [0, 1].Moreover, Φ p (x) has a unique positive fixed point x * in (0, 1) in which x * ≤ 1 2 .Thus, x * is global attractor.

4.2.
Non-unimodal map.In this sub-section we study the p−periodic logistic equation when the composition operator Φ p (x) is not unimodal and it has a unique positive fixed point but has no periodic points.This scenario happens when p−1 i=0 ri > 1 and some of the parameters r i , i = 0, 1, . . ., p − 1 are greater than 2. We will follow the techniques employed by Liz in [13] for a p−periodic Ricker map.
Assumption H1: Let us assume that Φ p has more than one critical point, the sequence of parameters satisfy p−1 i=0 r i > 1, i = 0, 1, . . ., p − 1, and Φ p has a unique positive real fixed point x * such that The discussion that follows depends on the number of critical points of the map Φ p and on the period of the composition map.Moreover, the analysis for p = 2 and p = 3 gives us the guidelines for the proof of the general case.
and attracting (the map Ψ is increasing where Ψ(x) > x for all x < x * and Ψ(x) < x for all x > x * ).Hence, the global stability follows by monotonicity of Ψ in the interior of I k .

4.2.2.
When p = 3.The map Φ 3 has either three, five or seven critical points as follows: (i) c 2 = 1 2 , c 1 and c 3 (solutions of the equation f 0 (x) = 1 2 ) such that c 1 < c 2 < c 3 , (ii) c 3 = 1 2 , c 2 and c 4 (solutions of the equation f 0 (x) = 1 2 ), and c 1 and c 5 (solutions of the equation Φ 2 (x) = 1  2 ) such that c i < c i+1 , i = 0, 1, . . ., 4 (iii) c 4 = 1 2 , c 2 and c 6 (solutions of the equation f 0 (x) = 1  2 ) and c 1 , c 3 , c 5 , and c 7 as solutions of the equation Φ 2 (x) = 1  2 such that c i < c i+1 , i = 0, 1, . . ., 6.The exact values of these critical points were presented in the beginning of this section.Case 1.The map Φ 3 has 3 critical points.Similar to p = 2. Case 2. The map Φ 3 has 5 critical points.We have to split our analysis according the map Φ 3 is decreasing or increasing at x * : 1.The map Φ 3 is decreasing at x * .There are 3 cases: We now summarize these ideas in the following result: Theorem 4.2.Under Assumptions H1 and H2, every locally asymptotically stable fixed point of Φ p is a globally asymptotically stable fixed point.
Since a fixed point of Φ p generates a periodic point of period p in the periodic logistic equation (2), from Theorem 4.2 the following corollary follows: Corollary 1.Under Assumptions H1 and H2, a locally asymptotically stable p−periodic cycle of the p−periodic logistic equation x n+1 = r n x n (1−x n ), r n+p = r n for all n, is globally asymptotically stable.
5. space for Φ 2 .Following the techniques employed in [2], one can find the region of local stability of the fixed points of the composition map Φ 2 = f 1 • f 0 (and of Φ 3 as well) by calculating the solution of the equation |Φ 2 (x * )| = 1 which is equivalent to the following set of equations: The implicit solution, in the parameter space r 0 Or 1 , of System ( 13) is −r 6 0 r 3 1 (r 0 r 1 − 1) 2 r 2 1 r 2 0 − 4r 1 r 2 0 − 4r 2 1 r 0 + 18r 1 r 0 − 27 = 0 (15) while the implicit solution of System ( 14) is Drawing implicitly the curves where the two previous equations are satisfied, we are able to find the region where the stability of the fixed points of Φ 2 occurs.This region is represented in Figure 1, in the parameter space r 0 Or 1 .The solutions of Equation ( 15) are represented by the dashed curves while the solutions of Equation ( 16) are represented by the solid curves.
If the parameters r 0 and r 1 belong to the region O, then the origin is a fixed point globally asymptotically stable.Once the parameters cross the dashed curve, from Region O to Region S, a bifurcation occurs, known as saddle-node bifurcation.The fixed point x * = 0 becomes unstable and a new globally asymptotically stable fixed point of Φ 2 is born.This fixed point is, in fact, a 2−periodic cycle of the 2−periodic equation (2).Now if the parameters r 0 and r 1 cross the dashed pitchfork, a saddlenode bifurcation occurs.In this case, the 2−periodic cycle becomes unstable and a new locally asymptotically stable 2−periodic cycle is born.
At the solid curve, a new type of bifurcation occurs known as a period-doubling bifurcation.Hence, when the parameters cross the solid curve, the globally 2−periodic cycle of equation ( 2) becomes unstable and a new locally asymptotically stable 4−periodic cycle is born.
Before ending this section, it should be mentioned that similar conclusions can be drawn for the triples (r 0 , r 1 , r 2 ) in the case of p = 3.

1 2 , 1 .
x ± and x±,− or 1 2 , x ± and x±,± .4.Unimodal map.In this subsection we study the p−periodic solution of the periodic logistic equation when the composition map Φ p is unimodal.So let us assume the following condition on the parameters p−1 i=0 r i > 1 and r i ≤ 2 for all i = 0, 1, 2, . . ., p − 1.

4. 2 . 1 .
When p = 2.In this case, the map Φ 2 has three critical points, c 2 = 1 2 (local minimum) and the solutions of the equation f 0 (x) = 1 2 (local maxima).Let c 1 and c 3 to be the left and the right solution of this equation, respectively.Notice that c 1 < 1 2 < c 3 .There are two cases: Case 1.The map Φ 2 is decreasing at x * .If the fixed point x * belongs to (c 3 , 1), we choose a = 0, c = c 3 and b = d = 1, and all the conditions of Theorem 2.4 are satisfied.If the fixed point x * belongs to (c 1 , c 2 ), we choose a = 0, c = c 1 and d = c 2 and b = c 3 .It is clear that the interval I = [a, b] is invariant and attracts all the orbits.Moreover, the map Φ 2 | I satisfies all the conditions of Theorem 2.4.Hence, x * is a globally asymptotically stable fixed point under Assumption H1.Case 2. The map Φ 2 is increasing at x

1 Figure 1 .
Figure 1.Regions of stability, in the parameter space r 0 Or 1 , of the fixed points of f 1• f 0 , with f i (x) = r i x(1 − x), i = 0, 1.