The dynamics of a four-step feedback procedure to control chaos

In this paper we make a description of the dynamics of a four-step procedure to control the dynamics of the logistic map. Some massive calculations are made for computing the topological entropy with prescribed accuracy. This provides us the parameter regions where the model has a complicated dynamical behavior. Our computations also show the dynamic Parrondo’s paradox “simple+simple=complex”, which should be taking into account to avoid undesirable dynamics.


Introduction
In [18], a four-step feedback procedure to control a chaotic map proposed by [24] is studied. In particular, the authors consider the logistic map f µ (x) = µx(1 − x) and the scheme    x n+1 = (1 − λ 3 )y n + λ 3 f µ (y n ) = g λ 3 ,µ (y n ), y n = (1 − λ 2 )z n + λ 2 f µ (z n ) = g λ 2 ,µ (z n ), z n = (1 − λ 1 )x n + λ 1 f µ (x n ) = g λ 1 ,µ (x n ), which can be rewritten as They analyze a switching strategy to control chaos which can lead to the Parrondo's paradox "chaos1 + chaos2 = order" or "undesirable1 + undesirable2= desirable" dynamical behavior. Roughly speaking (see e.g. [7]), we consider two discrete dynamical systems given by continuous maps f i : X → X, i = 1, 2, on a metric space X, usually a subset of R n , n ∈ N. The Parrondo's paradox appears when the dynamical behavior of both maps is simple (resp. chaotic) and that of f 1 • f 2 is chaotic (resp. simple). Of course, this paradox can involve more maps (see e.g. [5]). Mathematical examples includes interval dynamics (see [8] and [5]), dynamics of complex maps (see [2] and [3]), local stability problems ( [14] and [16]) among others. In addition, we can find applications of this paradox to Physics (see [12] and [19]), Biology (see [23], [25] and [26]) and Social Sciences (see [13] and [20]). It is stated in [18] that "the range of stability of the dynamical system can be increased by using the switching strategy". This is shown by simulations where µ takes two different values, but identical values of λ i , i = 1, 2, 3. We think that the control parameter should vary instead of µ, because the system that we want to control depends on this parameter.
However, for a fixed value of µ they analyze just the case λ 1 = λ 2 = λ 3 = λ. In this case, the model (1) reads as Hence, no paradox is possible here and the dynamics it given by the third iterate g 3 λ,µ , which has a similar behavior to that of the map g λ,µ . Probably, they made the assumption on λ because the proposed model depends on four parameters and then, as we will show, two problems appear in order to study the model. On one hand, four parameters implies that the computations increase. For instance, if we fix λ i ∈ (0, 1), i = 1, 2, 3, and µ ∈ [3,4] with step size 0.01, the number of cases under study is 10 8 , while for λ 1 = λ 2 = λ 3 = λ is just the middle. On the other hand, presenting the results is quite complicated because usually bifurcation diagrams are showed in dimension two and graphs of functions, e.g. the estimation of Lyapunov exponents, only can be done in dimension three.
The aim of this paper is to analyze properly the system, including the general cases. We will make use of the topological entropy, which allows us to compute the parameter regions where the dynamics is complicated with prescribed accuracy. As a necessary complement, we will also estimate the Lyapunov exponents. We cannot make anything to reduce the number of parameters, so we will fix some of them, changing the remaining ones in order to see the evolution of chaotic parametric regions. For that, we previously make an study of the map F λ 1 ,λ 2 ,λ 3 ,µ , because its shape has influence in the techniques we have to use to make our analysis.
The paper is organized as follows. Next section will be devoted to study the map F λ 1 ,λ 2 ,λ 3 ,µ and establish some basic facts which we use to analyze the model. This will be done in Section 3, where we distinguish two particular cases, namely λ 1 = λ 2 = λ 3 and λ 1 = λ 2 , and the general one. The paper finishes with some conclusions.

Model facts
Let f µ (x) = µx(1 − x) be he logistic family where µ > 0 and x a real number. It is well-known that if µ ≤ 4, then the interval I = [0, 1] is invariant by f µ , that is, f µ (I) ⊆ I. It is easy to see that for any x ∈ R \ [0, 1] its orbit converges to −∞. If µ > 4, the interval I is not invariant because there exists a (maximal) interval I 0 = (a, b), containing 1/2, such that f µ (I 0 ) ⊂ (−∞, 0), and so any orbit with initial condition in I 0 converges to −∞ as well. In fact, in this case almost every orbit in I converges to −∞, although the map f µ is chaotic since {[0, a], [b, 1]} is a two-horseshoe and therefore f µ has positive topological entropy (see [1,Chapter 4]). The chaotic set, that is, the set where f µ has a complicated dynamical behavior lies in a zero measure subset of I. Then, we will restrict our analisys to the case µ ∈ (0, 4]. So, fix g λ,µ (x) = (1−λ)x+λf µ (x), λ ∈ [0, 1]. Note that g λ,µ leaves invariant the interval I because it is a convex combination of two points x and f µ (x) lying in this interval. However, for practical reasons we may consider the map restricted to the interval [0, c], where c is the smallest positive real number such that g λ,µ (c) = 0. Now, we consider the map under study This map also leaves invariant the interval I, but again for practical reasons, we consider it as defined in [0, c], where where c is the smallest positive real number such that F λ 1 ,λ 2 ,λ 3 ,µ (c) = 0. This will simplify somehow the description below.
We need to compute the local extrema of F λ 1 ,λ 2 ,λ 3 ,µ , given by the equation which reads as which gives us the equations It is straightforward to check that the solution of equation (3) is Then, equation (4) reads as with solutions Finally, equation (5) reads as with solutions when they exist as real number. It is easy to see that when the seven extrema exist, then are not real, then x ± 2 are also non real. Finally, it is easy to see by equations (6) and (7) that . We summarize the possible cases as follows. Figure 1 depicts examples of some of these cases.
(F2) As S(F λ 1 ,λ 2 ,λ 3 ,µ ) < 0, the Milnor attractors (see [22]) of any orbit, except perhaps the fixed point 0, must attract the orbit of some extrema. Then, the number of such attractors is bounded by 3 for cases (C1) and (C2), 2 in the cases (C3), (C4) and (C6) and one otherwise. There are many possible trajectories that are not attracted by the Milnor attractors, but they lie in a set of zero Lebesgue measure and therefore, we cannot observe them in numerical simulations.
(F3) If x is a maximum or minimum of F λ 1 ,λ 2 ,λ 3 ,µ , then its Lyapunov exponent is given by When the orbit of x is attracted by a periodic orbit it is negative. Then, positive Lyapunov exponents is necessary to have a complicated dynamical behavior.
The above definition of topological entropy is not suitable for computing it. So, we use the algorithms to compute it with prescribed accuracy that can be found in [4] for the cases (C5), (C7) and (C8), theone from [9] for the cases (C3), (C4) and (C6) and the one from [10] for the remaining cases.
Taking into account all the above facts, we can compute the topological entropy of F λ 1 ,λ 2 ,λ 3 ,µ . The parameter regions where the topological entropy is positive are capable to exhibit chaos, although it is not always physically observable.
The parameter values with positive topological entropy but negative Lyapunov exponents are those for which chaos is not observed physically. This situation is common along the paper, although for most of the cases we only depict the topological entropy. Table 1 shows the values of µ and λ 0 for which h(g λ,µ ) > 0 when λ > λ 0 . µ λ 0 4 0.85664895 3.9 0.88618857 3.8 0.91783816 3.7 0.95183216 3.6 0.98844104 Table 1. Values of λ 0 for which h(g λ,µ ) > 0 when λ > λ 0 .
Note that 8 and 2λµ and 2λµ .
It is easy to see that ∆(λ) > 0 and ∆(λ) + 2 ∆(λ) > 0 when So, it is possible to have three different bifurcation diagrams when x ± i , i ∈ {1, 2, 3} exist. For instance, Figure 3 shows and example of this fact. 3.2 The case λ 1 = λ 2 = λ Next, we will assume that λ 1 = λ 2 = λ and let F λ,λ 3 ,µ = F λ 1 ,λ 2 ,λ 3 ,µ . Now, we fix the parameter µ and check how the dynamics of F λ,λ 3 ,µ changes when λ and λ 3 vary. We fix µ = 4 and show in Figure 4 the computation of the topological entropy and the estimation of the Lyapunov exponents. Figure 4 gives the region where the map F λ,λ 3 ,µ can be chaotic, provided by positive topological entropy. The colored region of Lyapunov exponents shows when this chaos can be physically observable. We will not show more figures with estimations of Lyapunov exponents. Note that for µ = 4, the map g λ,4 has positive topological entropy when λ 0 > 0.85664895. The zoomed region of Figure 4 (g)-(h) shows that F λ,λ 3 ,4 may have positive topological entropy for these parameter values. But note that λ 3 > λ 0 , and therefore g λ 3 ,4 is also chaotic. It is remarkable that we can obtain positive values of entropy with parameter λ 3 close to zero, and hence g λ 3 ,µ close to the identity.
The zoomed region of Figure 4 (e)-(f) shows the existence of parameter values for which the map F λ,λ 3 ,4 has positive entropy when λ and λ 3 are both smaller than λ 0 . In particular, when λ is close enough to 0.84 and λ 3 to 0.62. This is an example of "simple + simple = complex" paradox. This is not only for µ = 4. Figure 5 shows this zoomed region where Parrondo's paradox is possible for several values of µ. Table 2 shows the values of µ, λ and λ 3 for which h(F λ,λ 3 ,µ ) > 0. The topological entropy remains positive in a suitable neighborhood of these values.  Table 2. Values of µ, λ and λ 3 for which h(F λ,λ 3 ,µ ) > 0.

The general case
Now, we assume that the parameters λ 1 , λ 2 and λ 3 need not be the same. Note that by [17], we have that Hence, all the parameters λ 1 , λ 2 and λ 3 play the same role. This allows us to fix λ 3 = λ and vary the parameters λ 1 and λ 2 within (0, 1). In the previous section we check that when µ decreases, so it does the region where the dynamic is complex. So, we will focuss on the variation of λ. For that, we consider a fixed value of µ = 3.9. Figure 7 shows the computation of the topological entropy for some values of λ. Our computations suggest that when λ = λ 3 decreases, so it does the area of the parameter region where the topological entropy is positive. Note that when λ < λ 0 = 0.88618857 the map g λ,3.9 has zero topological entropy and Figure 7 (c), (d) and (e) shows that we can obtain positive topological entropy for all the values λ i < λ 0 , i = 1, 2, 3. This is again a new example of the Parrondo's paradox "simple + simple =complex". Note that the values of λ = λ 3 0.87 and 0.65 come from Table 2. Figure 8 shows bifurcation diagrams for these values. Finally, note that the graphs of Figure 7 are not symmetric with respect to the diagonal of the square. So, there can exist values of λ i , i = 1, 2, 3 such that F λ 1 ,λ 2 ,λ 3 ,µ is chaotic while F λ 2 ,λ 1 ,λ 3 ,µ is not. It suffices to take λ 1 = 0.85, λ 2 = 0.78 and λ 3 = 0.87, which gives us h(F λ 1 ,λ 2 ,λ 3 ,3.9 ) = 0.12022146651591863 while h(F λ 2 ,λ 1 ,λ 3 ,3.9 ) = 0.

Conclusion
Although the map g λ,µ is useful in controlling the chaotic dynamics of the logistic family f µ , it must be carefully handled if we want to apply it with different values of λ. Here, the "simple + simple = complex" Parrondo's paradox can produce undesirable dynamics. In general, if λ cannot be fixed, we should take these values in an open open interval with suitable length, as the mentioned paradox is only possible for different enough λ's. We want to mention that this study can be adapted for other chaotic maps different from the logistic family.