Chaotic sub-dynamics in coupled logistic maps

Physica D 335 (2016) 45–53

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Physica D journal homepage: www.elsevier.com/locate/physd

Chaotic sub-dynamics in coupled logistic maps Marek Lampart a,b , Piotr Oprocha c,b,∗ a

Department of Applied Mathematics, VŠB - Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic

b

IT4Innovations, VŠB - Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic

c

AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland

article

info

Article history: Accepted 17 June 2016 Available online 18 July 2016 Communicated by I. Melbourne

abstract We study the dynamics of Laplacian-type coupling induced by logistic family fµ (x) = µx(1 − x), where

µ ∈ [0, 4], on a periodic lattice, that is the dynamics of maps of the form F (x, y) = ((1 − ε)fµ (x) + ε fµ (y), (1 − ε)fµ (y) + ε fµ (x))

Keywords: Coupled map lattices Logistic map Topological entropy Attractor

where ε > 0 determines strength of coupling. Our main objective is to analyze the structure of attractors in such systems and especially detect invariant regions with nontrivial dynamics outside the diagonal. In analytical way, we detect some regions of parameters for which a horseshoe is present; and using simulations global attractors and invariant sets are depicted. © 2016 Elsevier B.V. All rights reserved.

1. Introduction

for the particular choice of map g, which is taken from logistic family gµ (x) = 1 − µx2 for some µ (e.g. see [5,6]). Some other authors prefer to consider logistic family on [0, 1], i.e. fµ (x) = µx(1 − x) where in most considered cases µ = 4. Note that the dynamics of both interval maps x → 1 − µx2 and x → µx(1 − x) is ‘‘similar’’, in the sense that they share most of interesting dynamical features [7]. In [8] the authors consider so-called twoways communication (so-called Laplacian-type coupling)

Many complex systems coming from applications are studied from the qualitative, long term dynamics point of view. In the last years, this approach was successful in considerations on states evolution in the coupled map lattices (CML), which are a class of models of extended media in which the relations between temporal evolution and spatial translation play a crucial role [1,2]. It has been observed that CML systems show variation of space–time patterns which are ordinary to other spatially enlarged systems (see,e.g., [3] or [4] and references therein). If we consider one dimensional lattice, then there are various possibilities, and even for simply looking cases quite complex behavior can appear. If we browse the literature, then there are various concepts of communication between states. A simple, yet interesting, model, are one-way coupled logistic lattices which are  modeled as a lattice xi i∈Z ⊂ [−1, 1] with evolution modeled by a discrete dynamical system (n denotes values of lattice at nth iteration) xin+1 = (1 − ε)g (xin ) + ε g (xin−1 ),

(1.1)

∗ Corresponding author at: AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Kraków, Poland. E-mail addresses: [email protected] (M. Lampart), [email protected] (P. Oprocha). http://dx.doi.org/10.1016/j.physd.2016.06.010 0167-2789/© 2016 Elsevier B.V. All rights reserved.

xin+1 = (1 − ε)g (xin ) +

ε 2

g (xni−1 ) +

ε 2

g (xin+1 )

which is a restricted version of variable range coupling (e.g. see [9]): ′

xin+1

N ε  1 = (1 − ε)g ( ) + (g (xin−j ) + g (xin+j )) η(α) j=1 jα

xin

where xin ∈ [0, 1] is the state variable for the site i ∈ {1, 2, . . . , N } at time n, N > 0 is odd, ε ∈ [0, 1] is the coupling strength,

 ′ α ∈ [0, ∞) is the effective range, and η(α) = 2 Nj=1 j−α with N ′ = (N − 1)/2 (see e.g. [10] and references therein). Simply, we assume that α is very large (say α → ∞) so coupling introduced by terms with j ̸= 1 is not essential and can be ignored. Now, let j us assume that initial values on the lattice are periodic, say xi0 = x0 provided that i = j (mod 2). Then we can view Laplacian-type

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coupling as a two dimensional map, since then

ε

ε

g (xin−1 ) + g (xin+1 ) 2 2 = (1 − ε)g (xin ) + ε g (xin+1 )

xin+1 = (1 − ε)g (xin ) +

= (1 − ε)g (xin ) + ε g (xin−1 ).

(1.2)

In this paper we will consider the above mentioned situation. It is also clear that instead of 2 variables we can consider n variables, simply assuming n-periodicity instead of 2-periodicity on the lattice (also other type of coupling can be considered in that case). The model (1.2) was considered by many authors from different points of view in the last 30 years. Many interesting results were revealed, however most of them were obtained as a result of numerical simulations and much smaller insight was done using analytic methods. Spectral properties of the model (1.2) generalizations were discussed in [11] and conditions for the stability of spatially homogeneous chaotic solutions were presented. Some studies whether there is a synchronization were undertaken in [12] and later, e.g., in [13,14] or [15] (for more comments, see references therein). The existence of wavelike solution in the model (1.2) for which the spatiotemporal periodic pattern can be predicted was investigated in [16] (see also references therein). The bifurcation phenomena and loss of synchronization were examined in [17,5]. It was also observed that the model exhibits periodic structure, multiple attractors, entrained and phase reversed patterns as well as chaos, e.g., see [18,19]. The model (1.2) was also analyzed for negative coupling constant revealing existence of a forward invariant curve [20]. In [21] the authors proved a few basic facts on the model (1.2) including the existence of chaos in the sense of Li and Yorke for some range of parameters. It is very easy to see that the model exhibits chaotic behavior for zero coupling constant under the condition that fµ is chaotic, since in this case the model is acting just as a Cartesian product of chaotic fµ . There is also always an invariant diagonal, where dynamics is the same as fµ . In this direction a natural question arise, whether large offdiagonal regions of chaotic dynamics could be observed for nonzero coupling constant. One of our aims is to detect regions outside the diagonal, where chaotic dynamics can be supported. A natural approach is to split the region of parameters into several areas. For the first area all points are attracted to the main invariant subsystem, which is always embedded in the diagonal (see Theorem 3), and for the second case the model is showing chaotic motion outside the diagonal, i.e., there is a horseshoe located outside the diagonal (see Theorem 8). A sharp edge between these two opposite situations is yet to be determined. As we announced earlier, the study is focused on the two dimensional case (i.e. lattice with 2-periodic entries), since in higher dimension (i.e., larger period on the lattice) the problem could be dealt with similarly by analogous calculations and arguments. The paper is organized as follows. In Section 2 the model is presented, in Section 3 parameters which lead to the diagonal as an attractor are characterized, and finally in Section 4 nontrivial horseshoes (and invariant subsets) are detected. Appendix at the end of the paper contains description of all fixed points in the model. 2. The model Let us define the family of logistic maps fµ : [0, 1] → [0, 1], where 0 ≤ µ ≤ 4, by fµ (x) = µx(1 − x). The interval Iµ = [fµ2 (1/2), fµ (1/2)] is called the core of fµ , when µ ∈ (2, 4], see Fig. 1. For the choice of parameter µ ∈ [0, 2]

Fig. 1. Graph of fµ for µ = 3.8 and the graph restricted to the core Iµ (bounded by box).

the interval Iµ is still well defined but does not have such nice properties. Namely, I2 = {1/2}, I0 = {0} and for µ ∈ (0, 2) it is not invariant under fµ , hence not remarkable in these cases. The core Iµ is strongly invariant, that is fµ (Iµ ) = Iµ , and every point from (0, 1) is attracted to Iµ . The dynamics on the core can be very rich. For example, in [22] the authors show that for the family of tent maps the dynamics on the core is topologically exact for some range of parameters, which, generally speaking, means that most rich dynamical behavior is present in the core. In the case of logistic maps, the calculations are much harder and spectrum of possible dynamical behaviors is richer. However, it is known that for some parameters the dynamics on the core of logistic map is the same (in the sense of topological conjugacy) as on the core of tent map with slope corresponding to µ (e.g. see [7]). By coupled logistic map (with coupling constant ε ) the following map of the square F : [0, 1]2 → [0, 1]2 is meant, F (x, y) = (Fx (x, y), Fy (x, y))

(2.1)

where Fx (x, y) = (1 − ε)fµ (x) + ε fµ (y),

(2.2)

Fy (x, y) = (1 − ε)fµ (y) + ε fµ (x).

(2.3)

Simple calculations yield that Fix(F ) ⊇ {(0, 0), (pµ , pµ )} for µ > 1, where pµ = (µ − 1)/µ and they are the only fixed points of F on the diagonal. Note that the point pµ is fixed for fµ and is attracting or repulsive for µ ∈ (1, 3) or µ ∈ (3, 4], respectively. It may happen that Fix(F ) contains also points outside diagonal, however always #Fix(F ) ≤ 4. For the reader convenience we present formulas for all fixed points of F and their dependence on parameters µ, ε in the Appendix. The periodic structure of F was also studied by [23] (for more see references therein), where fixed points as well as two-cycles are listed with respect to the diagonal. Here, as usual, Fix(F ) and Per(F ) stand for the set of all fixed and periodic points of F , respectively; and the space R2 is endowed with the Euclidean norm ∥ · ∥. Remark 1. It is clear that ∆ = {(x, x) : x ∈ [0, 1]} is an invariant subset for F and that F |∆ can be identified with fµ by natural homeomorphism π : [0, 1] ∋ x → (x, x) ∈ ∆, that is F |∆ ◦ π = π ◦ fµ or equivalently F |∆ = π ◦ fµ ◦ π −1 . By Remark 1 it is clear that we have a lower bound for topological entropy of F given by htop (fµ ) ≤ htop (F ), F is chaotic in the sense of Li and Yorke, provided that fµ is chaotic in the sense of Li and Yorke, etc. The reader not familiar with theory of entropy is referred to books [24] or [25]. Many recent advances on chaos in the sense of Li and Yorke can be found in [26].

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the next result we show that for the (most of) parameters in R the dynamics of the map F is the same as the dynamics of fµ in the asymptotic sense. We will need the following fact from [28, Lemma 5]. Lemma 2. Let f : [0, 1] → [0, 1] be a continuous map such that Per(f ) = Fix(f ). Let 0 = d0 < d1 < · · · < dn = 1 Fig. 2. Bifurcation diagram for logistic family fµ for 0 ≤ µ ≤ 4. For µ > 3.6786 there is a strong 2-horseshoe for fµ2 .

be a division of [0, 1] such that di ̸∈ Fix(f ) for i ∈ {1, 2, . . . , n − 1}. Denote Ij = [dj−1 , dj ] for j ∈ {1, 2, . . . , n}. Then there exists δ > 0 such that if for some m ∈ N there are points x0 , . . . , xm−1 such that |f (xi ) − x(i+1) mod m | < δ for i ∈ {1, 2, . . . , m} (i.e., x0 , . . . , xm−1 form δ − m cycle) then there is j such that xi ∈ Ij for every i (i.e., all δ cycles for sufficiently small δ are ‘‘inside’’ the intervals Ij ). Theorem 3. Denote by µ ˆ the Feigenbaum constant. For any x, y ∈ [0, 1] and (µ, ε) ∈ R we have: (1) if µ ≥ µ ˆ then lim dist(F n (x, y), ∆(Iµ ) ∪ {(0, 0)}) = 0,

n→∞

Fig. 3. Topological entropy h(fµ ) in the family of maps fµ (with log := log2 ).

(2) if µ ∈ (2, µ) ˆ and (x, y) ∈ [0, 1]2 then either (x, y) ∈ {0, 1}2 or there is z ∈ Iµ ∩ Per(fµ ) such that limn→∞ ∥F n (x, y) − F n (z , z )∥ = 0. (3) if µ ∈ (1, 2] and (x, y) ∈ [0, 1]2 then either (x, y) ∈ {0, 1}2 or limn→∞ F n (x, y) = (pµ , pµ ). (4) if µ ≤ 1 then limn→∞ F n (x, y) = (0, 0). Proof. The case µ ≤ 1 is very simple so we consider it first. Note that is this case limn→∞ fµn (1/2) = 0 and fµn (1/2) ≥ fµn (x) for every n and every x ∈ [0, 1/2]. But it is easy to verify that F (x, y) ∈ [0, 1/2] × [0, 1/2] and hence F n+1 (x, y) ∈ [0, fµn (1/2)] × [0, fµn (1/2)]

Fig. 4. Set R defined by (3.1).

By the above remark, F is chaotic when µ = 4 and many other values of µ ∈ [µ, ˆ 4), where µ ˆ ≈ 3.56994 . . . is the Feigenbaum constant (smallest µ where fµ has point of period 2n for every n ∈ N). It can be seen on the bifurcation diagram (see Fig. 2), which is also supported by the results on conjugacy with the tent family [7]. For the readers convenience, in Fig. 3, we present a graph of approximate values of topological entropy of fµ (with logarithm with base 2) calculated with the method of Block, Keesling, Li, and Peterson [27]. Intuitively it seems that dynamics of F should be much richer than can be observed over the diagonal. One of our aims will be to search for interesting off the diagonal sub-dynamics. 3. Diagonal as an attractor Let us consider the set (see Fig. 4): R = {(µ, ε) ∈ [0, 4] × [0, 1] : 1 > µ|1 − 2ε|}.

(3.1)

Recall that we denote by ∆ the diagonal in [0, 1]2 , that is ∆ = {(x, x) : x ∈ [0, 1]}. For J ⊂ [0, 1] we denote the diagonal over J by ∆(J ) = ∆ ∩ (J × J ). There are known situations when most of the points approach the diagonal, provided that µ, ε are well chosen (e.g. see [21]). In

for every n, which shows (4). It remains to consider the case µ > 1. First, we are going to prove that limn→∞ dist(F n (x, y), ∆(Iµ ) ∪ {(0, 0)}) = 0 when µ > 2. This will prove (2) and will be used in the proof of (3). It is well known that limn→∞ dist(F n (x, y), ∆) = 0 when µ > 0 (and (µ, ε) ∈ R), however we have to repeat these calculations, since we will need them later. Fix any x, y ∈ [0, 1] and assume that x ̸= y. Observe that by (3.1) there is λ such that 1 > λ > µ|1 − 2ε|. Then by the Mean Value Theorem we get

|Fx (x, y) − Fy (x, y)| ≤ µ|1 − 2ε| · |x − y| < λ|x − y|. Hence, either F n (x, y) ∈ ∆ for some integer n > 0 or lim |Fxn (x, y) − Fyn (x, y)| ≤ lim λn |x − y| = 0.

n→∞

n→∞

But this means that for any η > 0 there is N > 0 such that for every n > N and every (x, y) ∈ [0, 1]2 we have dist(F n (x, y), ∆) ≤ ∥(Fxn (x, y), Fxn (x, y)) − F n (x, y)∥

≤ |Fxn (x, y) − Fyn (x, y)| < η.

(3.2)

Now we are ready for the proof of (1). If µ = 4 then ∆ = ∆(Iµ ) and so by (3.2) there is nothing to prove. Therefore, we assume that µ < 4. Note that in this case fµ (x) ∈ (0, 1) for every x ∈ (0, 1), hence we have Fx (x, y) ∈ {0, 1} only for x, y ∈ {0, 1}. In other words, F n (x, y) = (0, 0) for some n only when (x, y) ∈ {(0, 0), (0, 1), (1, 0), (1, 1)}. Assume that (x, y) ̸∈ {(0, 0), (0, 1), (1, 0), (1, 1)}. For every η > 0 and every m > 0 there is δ > 0 such that ∥F i (p, q) − F i (s, t )∥ < η for i = 0, 1, . . . , m provided that ∥(p, q) − (s, t )∥ < δ . Additionally note that F 2 (x, y) ∈ [0, fµ (1/2)] × [0, fµ (1/2)].

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Now let us analyze the dynamics at (0, 0). We see that eigenvalues of derivative DF (0, 0) are λ1 = µ and λ2 = µ(1 − 2ε) < 1. Eigenvectors are v1 = (1, 1) and v2 = (−1, 1), respectively. Since µ > 1, (0, 0) is a saddle point, and the location of (local) stable and unstable manifold implies that there is a small neighborhood U of (0, 0) such that for every (p, q) ∈ U ∩ [0, 1]2 there is n > 0 such that F n (p, q) ̸∈ U, provided that (p, q) ̸= (0, 0). Observe that there is m > 0 such that F m (z , z ) ∈ Iµ × Iµ for every

(z , z ) ∈ ∆ \ (U ∪ {(1, 1)}). The above two observations combined with the fact that limn→∞ dist(F n (x, y), ∆) = 0 show that there is an integer n > 0 such that F n (x, y) ∈ Iµ × Iµ , provided that 2 < µ < 4. Obviously, when µ > 2 we have F (Iµ × Iµ ) ⊂ Iµ × Iµ so we obtain that lim dist(F n (x, y), ∆(Iµ ) ∪ {(0, 0)}) = 0

(3.3)

n→∞

for every µ > 2, (µ, ε) ∈ R. In particular (1) holds. Fix any z ∈ [0, 1]2 . Identify F |∆ = fµ and observe that F k |∆ = (F |∆ )k = fµk . If µ < µ ˆ , then for some k we have Per(fµk ) = Fix(fµk ), because by Sharkovsky theorem (e.g. see [24]) and definition of µ ˆ map fµ has only periodic points of period 2i where i is bounded from the above by a constant i < k = k(µ). Fix any ε > 0 and let (0, 0) = d0 < d1 < · · · < dn = (1, 1) be a partition of ∆ into intervals Ij of diameter diamIj < ε/3. Let δ be provided by Lemma 2. By (3.2) there exists N > 0 such that for every i > N and every w ∈ [0, 1]2 we have ∥F ki (w) − (Fxki (w), Fxki (w))∥ < ε/3 and

∥F k (Fxki (w), Fxki (w)) − (Fxk(i+1) (w), Fxk(i+1) (w))∥ = ∥(fµk (Fxki (w)), fµk (Fxki (w))) − (Fxk(i+1) (w), Fxk(i+1) (w))∥ < δ. Cover ∆ by finitely many open sets Jj with diamJj < δ . There exists at least one j such that # i : Fxk(i+1) (w) ∈ Jj = ∞.





Furthermore, since endpoints of Ji are not periodic points by definition, we also have # i : Fxk(i+1) (w) ∈ intJj = ∞.





k(i1 +1)

Let i1 < i2 be such that Fx

k(i2 +1)

(w), Fx

k(i+1) Fx

(w) ∈ intJj and hence

(w) ∈ Jj′ for i = i1 , i1 + by Lemma 2 there is j such that 1, . . . , i2 . But then j′ = j, since otherwise intJj ∩ Jj′ ̸= ∅ which is a contradiction. By the above observation, Lemma 2 implies that for some M > N there is j such that Fxki (w) ∈ Ij for all i ≥ M and therefore ′

∥F ki (w) − F k(i+1) (w)∥ ≤ ∥F ki (w) − (Fxki (w), Fxki (w))∥ + ∥(Fxk(i+1) (w), Fxk(i+1) (w)) − F k(i+1) (w)∥ < ε/3 + diamIj + ε/3 < ε.

≤ lim ∥F n (x, y) − (zn , zn )∥ + ∥F n (z , z ) − (zn , zn )∥ = 0. n→∞

But it is known that there are numerous values of µ such that fµ is conjugated on the core to a tent map without shadowing [7,29] while it was recently proved that transitive maps with limit shadowing also possess shadowing property [30]. In the view of the above, the following question arises: Question 1. Let (µ, ε) ∈ R, µ ≥ µ ˆ and fix any (x, y) ∈ [0, 1]2 \ ∆. Does there always exist z ∈ [0, 1] such that limn→∞ ∥F n (x, y) − F n (z , z )∥ = 0? 4. Coupling and horseshoes A chance for more interesting dynamics can occur when we are outside the set R with coupling parameters, see Fig. 5. In such situations there can exist ‘‘trapping regions’’ outside diagonal where trajectories remain forever. Some attempts to analytic analysis of situations when this occurs were undertaken in [31] by a clever change of parameters. Here we present an alternative approach for finding invariant regions, when coupling is small (but positive). Note that when ε = 0 then F = fµ × fµ . Therefore, it is natural to expect that for small values of ε dynamics of F will be in some sense similar to dynamical independence of coordinates fµ × fµ . At least we may hope for two effects. First of all, the invariant subsets outside the diagonal should survive. Second, entropy should increase over the value htop (fµ ) supported on the diagonal.

j =0

(3.4)

which shows that there is a fixed point for F k such that limi→∞ F ki (z ) = p. Combining the above with (3.3) we see that p ∈ ∆(Iµ ), finishing the proof of (2). When µ ∈ (1, 2] then Per(fµ ) = Fix(fµ ) = {0, pµ } and as we have seen (0, 0) is a saddle point (unstable fixed point of fµ ) hence (3) easily follows by (3.4).  Remark 4. By Theorem 3 we see that for (µ, ε) ∈ R the nonwandering set Ω (F ) ⊂ ∆, in particular htop (F ) = htop (F |Ω (F ) ) = htop (F |∆ ) = htop (fµ ).

lim ∥F n (x, y) − F n (z , z )∥

n→∞

n−1

But ε > 0 can be arbitrarily small, hence i→∞

By Theorem 3, when (µ, ε) ∈ R and µ is below Feigenbaum constant, there is z ∈ [0, 1] such that limn→∞ ∥F n (x, y) − F n (z , z )∥ = 0, that is dynamics of every point is asymptotically reflected by the dynamics of a point on the diagonal. When µ ≥ µ ˆ the situation becomes much more complex. If limn→∞ dist(F n (x, y), ∆(Iµ )) = 0 it is easy to construct a sequence zn ⊂ Iµ such that limn→∞ ∥F (zn , zn ) − (zn+1 , zn+1 )∥ = 0 and limn→∞ ∥F n (x, y) − (zn , zn )∥ = 0. If fµ has the so-called limit shadowing property on the core, then there is z ∈ Iµ such that

Definition 6. We say that a map f : X → X acting on a compact metric space has an n-horseshoe if there are closed sets U0 , . . . , n−1 Un−1 such that j=0 Uj ⊂ f (Ui ) for every i. Horseshoe is strong if

+ ∥(Fxki (w), Fxki (w)) − (Fxk(i+1) (w), Fxk(i+1) (w))∥

lim ∥F ki (z ) − F k(i+1) (z )∥ = 0

Remark 5. There are two types of periodic points of F in ∆ when µ < µ ˆ : sinks and saddles. Locally, stable manifold of saddle is one dimensional, while for sink the whole neighborhood is attracted. This suggests that we have partition of [0, 1]2 into onedimensional sets consisting of points attracted by saddles and two dimensional basins of attraction for sinks.

Uj ⊂ intf (Ui ).

Lemma 7. Fix an integer n > 0 and a parameter 2 < µ ≤ 4. Let J , K be closed intervals and let V be a closed set, such that V ⊂ int(fµn (J ) × fµn (K )). Then there is ε > 0 such that V ⊂ F n (J × K ). Proof. Without loss of generality we may assume that V is a square in [0, 1]2 . Observe that J and K can be decomposed into intervals J1 , . . . , Js and K1 , . . . , Kt , respectively such that fµn is monotone on each of them and they are maximal (i.e., common point of Ji , Ji+1 is a local extremum). Then it is not hard to verify (projecting V onto the first and second coordinate, respectively)that there  are closed  sets Vi ⊂ intJi and Wj ⊂ intKj such that V = i,j fµn ( Vi ) × fµn ( Wj ). Note that fµn × fµn is a homeomorphism on its image over each square Ji × Kj and then we can apply [32, Lemma 2.3]. Hence,

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Fig. 5. Global attractor for F and inner points of [0, 1]2 when µ = 3.8 and: (A) ε = 0, (B) ε = 0.07, (C) ε = 0.1, (D) ε = 0.17, (E) ε = 0.2 and (F) ε = 0.3.

for each pair i, j there exists γ (i, j) > 0 such that if ε < γ (i, j) then fµn (Vi ) × fµn (Wj ) ⊂ intF n (Ji × Kj ). In particular, if we fix  n   any ε < mini,j γ (i, j) then V = Vi ) × fµn ( Wj ) ⊂ i,j fµ (  n n  i,j intF (Ji × Kj ) ⊂ intF (J × K ) which completes the proof. Theorem 8. Suppose that fµn has a strong k-horseshoe J0 , . . . , Jk−1 for ε > 0 such that  some iterate n > 0. Then there exists Ji × Jj : i, j = 0, . . . , k − 1 is a strong k2 -horseshoe for F n . Proof. It is an immediate consequence of Lemma 7.



The above theorem allows us to estimate entropy of F in cases when entropy of fµ is positive and ε is sufficiently small. Corollary 9. Fix any µ ∈ [0, 4] and any δ > 0. Then there exists γ > 0 such that for any ε < γ associated map F satisfies htop (F ) > 2htop (fµ ) − δ . Proof. If htop (fµ ) = 0 then desired inequality is trivially satisfied, so let us assume that htop (fµ ) > 0. By Misiurewicz’s theorem [33] there are integers n > 0 and k > 0 such that 1n log(k) > htop (fµ ) − δ/2 and fµn has strong k-horseshoe. But then by Theorem 8 if ε > 0 is sufficiently small (say ε < γ for some γ > 0) then F n has a strong k2 -horseshoe. But then htop (F ) ≥

1 n

htop (F n ) ≥

1 n

log(k2 ) ≥ 2htop (fµ ) − δ. 

By Theorem 8 we know that F has a horseshoe for sufficiently small ε > 0 related to parameter µ, provided that fµ has a horseshoe. This is also seen by numerical evidence (for small ε invariant regions outside the diagonal arise). Now we will try to locate some of these horseshoes. For this purpose, we need some estimates of error introduced to trajectory of fµ on each coordinate of trajectory of F arising as an effect of coupling. Theorem 10. Denote by Fxn , Fyn , respectively, the first and second coordinate of F n . For any x, y ∈ [0, 1] and (µ, ε) ∈ [0, 4] × [0, 1]

we have that

|Fxn (x, y) − fµn (x)| <

n  ε i =1

|Fyn (x, y) − fµn (y)| <

4

n  ε i=1

4

µi , µi .

Proof. Fix any (µ, ε) ∈ [0, 4]×[0, 1]. We prove the theorem using induction on n. Fix n = 1 and notice that for any p ∈ [0, 1] we have that 0 ≤ fµ (p) ≤ fµ (1/2) = µ/4. Fix any x, y ∈ [0, 1] and observe that by the above inequality we have |fµ (x) − fµ (y)| ≤ µ/4. This immediately gives that

|Fx (x, y) − fµ (x)| = ε|fµ (x) − fµ (y)| ≤

µ 4

ε.

The first step of induction is completed. Now fix any n ≥ 1 and assume that

|Fxn (x, y) − fµn (x)| <

n  ε i =1

|Fyn (x, y) − fµn (y)| <

4

n  ε i=1

4

µi , µi

for every x, y ∈ [0, 1]. This implies that

|Fxn+1 (x, y) − fµn+1 (x)| = |Fx (Fxn (x, y), Fyn (x, y)) − fµ (fµn (x))| ≤ |Fx (Fxn (x, y), Fyn (x, y)) − fµ (Fxn (x, y))| + |fµ (Fxn (x, y)) − fµ (fµn (x))| ≤

≤

ε 4

µ + µ|Fxn (x, y) − fµn (x)| ≤

n +1  ε

4 i=1

µi .

Similar calculations yield that

|Fyn (x, y) − fµn (y)| <

n+1  ε

4 i=1

µi .

ε 4

µ+µ

n  ε

4 i =1

µi

50

M. Lampart, P. Oprocha / Physica D 335 (2016) 45–53

Fig. 6. A horseshoe stable under small perturbation for fµ2 with (A) µ = 3.8 and (B) µ = 3.95.

This completes the induction finishing the proof.



Then for µ ≥ 2 we get a more rough, however more useful in practice formula, by noting that:

εµ(µn − 1) ε µ = ≤ µn+1 . 4 4(µ − 1) 4

n  ε i =1

i

Corollary 11. For any x, y ∈ [0, 1] and (µ, ε) ∈ [2, 4] × [0, 1] we have that

|Fxn (x, y) − fµn (x)| < |Fyn (x, y) − fµn (y)| <

ε 4

ε

4

µn+1 , µn + 1 .

As it was said, if fµ has positive entropy then by Misiurewicz’s theorem [33], fµn has a horseshoe for some n > 0. But then, using Theorem 10 we can find admissible values of ε > 0 such that F n has also a horseshoe. Recall (e.g. √see Fig. 3) that there are values of µ such that htop (fµ ) ∈ (0, log 2), in particular fµ2 cannot have a horseshoe. In what follows we will focus on constructing a horseshoe for fµ2 . Then, using estimates from Theorem 10 we will present a graph of regions in (µ, ε) parameters space, where horseshoe of F 2 exists. In our considerations, the following fact will be especially useful (the following lemma is known [31], an alternative proof is presented for completeness).

Recall that when µ > 1 then fµ has in (0, 1) fixed point pµ = 1 − 1/µ and by symmetry fµ (1/µ) = pµ . Observe (see Fig. 6) that when fµ2 (1/2) ≤ 1/µ then fµ2 has a 2-horseshoe. Note that

parameter µ for which fµ2 (1/2) = 1/µ is a solution of the equation

µ4 − 4µ3 + 16 = 0. It is easy to check that µ = 2 is a solution of this equation (but it is not µ we need, since it is too small), and hence desired µ is a solution of the equation µ3 − 2µ2 − 4µ − 8 = 0. Using Cardano’s formula we see that there are two complex roots and exactly one real which could be written in algebraic form. For us it is enough to take µ > µ0 ≈ 3.6786 for which there is a 2horseshoe for fµ2 and hence 4-horseshoe for F 2 . Let us try to locate a 2-horseshoe which will be maintained under the largest perturbation. Let z ∈ [fµ2 (1/2), 1/µ] be the point such that fµ2 (z ) = max fµ2 (y), y ∈ [fµ2 (1/2), 1/µ] .





Fix a ∈ (fµ2 (1/2), 1/2) and notice that if J = [a, 1/2] and a > z then max fµ2 (J ) = fµ2 (a) while if a ≤ z then max fµ2 (J ) = fµ2 (z ). Now, if we put



max

zˆ ∈[f 2 (1/2),z ] µ

(4.1)

max

zˆ ∈[z ,1/µ]

where Gx (x, y) = ε fµ (x) + (1 − ε)fµ (y), Gy (x, y) = ε fµ (y) + (1 − ε)fµ (x). Then for every x, y ∈ [0, 1] we have F 2 (x, y) = G2 (x, y). Proof. First note that Fx (x, y) = (1 − ε)fµ (x) + ε fµ (y) = Gy (x, y) = Gx (y, x) and similarly Fy (x, y) = Gx (x, y) = Gy (y, x). Hence Fx (F (x, y)) = Gx (Fy (x, y), Fx (x, y))

= Gx (Gx (x, y), Gy (x, y)) = Gx (G(x, y)) and analogously Fy (F (x, y)) = Gy (G(x, y)). Consequently, F 2 (x, y) 

= G2 (x, y).

Remark 14. By Corollary 13, if we detect a horseshoe for F and small coupling constant ε > 0, then a ‘‘twin’’ horseshoe exists for F and large ε < 1 (e.g. see Fig. 7).

εˆ 0 = max

Lemma 12. Let ε ∈ [0, 1] and let G be the function defined by G(x, y) = (Gx (x, y), Gy (x, y))

Corollary 13. Let ε ∈ [0, 1] and let G be the function defined by Eq. (4.1). Then F 2 has a horseshoe if and only if G2 has a horseshoe.





 |fµ2 (1/2) − zˆ |, |fµ2 (z ) − 1 + zˆ | ,

 |fµ2 (1/2) − zˆ |, |fµ2 (z ) − 1 + zˆ |

and if zˆ is the point for which the above maximum is achieved, then J0 = [ˆz , 1/2] and J1 = [1/2, 1 − zˆ ] form a 2-horseshoe for fµ2 and this horseshoe will survive under perturbation smaller than εˆ 0 of 4εˆ ε fµ2 . But if we fix ε0 such that 40 µ2+1 = εˆ 0 that is ε0 = µ30 , then for

every ε < ε0 the map F 2 has 4-horseshoe by Corollary 11 (we can get even wider admissible range of ε using Theorem 10). If we fix ε < ε0 then there exists ξ > 0 such that J0′ = [ˆz , 1/2 −ξ ] and J1′ = [1/2, 1 − zˆ −ξ ] is a strong 2-horseshoe which will survive under perturbation, forming a strong 4-horseshoe for F 2 with ε fixed above (and by Corollary 13 also for F 2 with 1 − ε ). By these observations we can calculate regions for which F 2 has 4horseshoe (see Fig. 7). These regions can be slightly enlarged, since horseshoe for some iterate fµk exists in each case when entropy of fµ is positive. Clearly k > 2 when 0 < htop (fµ ) < log

√

2

M. Lampart, P. Oprocha / Physica D 335 (2016) 45–53

51

Fig. 7. Values of µ and ε where there mathematically strict evidence of 4-horseshoe for F 2 (and hence there is also an invariant set for F outside the diagonal).

Fig. 8. Off-diagonal invariant sets for F when µ = 3.8 and: (A) ε = 0.02, (B) ε = 0.054, (C) ε = 0.06, (D) ε = 0.09.

(fµ2 cannot have a horseshoe for that case) so admissible values of ε will be much smaller than in Fig. 7, nevertheless they will be strictly positive. Simply, the upper bound on ε detected by Theorem 10 decreases exponentially with respect to k. In Fig. 8 there is a numerical evidence of evolution of invariant set for µ = 0.38 visible when coupling constant ε increases. In the case of Fig. 8(A) we have the analytic evidence of invariant set by previous considerations (see Fig. 7). By the method of construction (see Fig. 6) it is also clear that off-diagonal invariant set can be quite a large, totally disconnected set, which is confirmed by Fig. 8(A). When ε is increased, our arguments do not work anymore, however, it is visible that the invariant set remains in the dynamics. Only its size and shape become different. Analyzing Fig. 8(C) it seems that there is an invariant ‘‘solid’’ region, say a connected two-dimensional invariant set. Fig. 8(D) suggests that there may be an invariant simple closed curve. If it is the case, then these sets are not the effect of existence of a horseshoe. It is rather a

kind of invariant curve that emerges from point of period two, as an effect of bifurcation. We leave the formal explanation of observed phenomenon for further research. 5. Conclusion In this paper, the dynamics of Laplacian-type coupling initiated by the logistic family on periodic lattice was investigated. Our approach was to split the region of parameters (the coupling strength and logistic family parameter) into several areas. As the first step the area of parameters R for which dynamics can be almost completely characterized in terms of trajectories for logistic map fµ on the diagonal (see Theorem 3). Secondly, the case where the model contains rich chaotic motion outside the diagonal was partly described. A region of parameters for which the systems containing topological horseshoes were analytically

52

M. Lampart, P. Oprocha / Physica D 335 (2016) 45–53

Fig. 9. Off-diagonal fixed points of F where: (left) µ = 3.3 and ε ∈ (0.883, 1) and (right) µ = 4.0 and ε ∈ (0.75, 1).

described (see Theorem 8 and Fig. 7). The remaining part of the range of parameters is exhibiting the periodic doubling effect. We also observed some interesting patterns forming outside the diagonal and changes in the number of fixed points depending on range of coupling and family of logistic maps. Acknowledgments The authors would like to thank the anonymous referees for comments which led to improvements of this paper. This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project ‘‘IT4Innovations excellence in science — LQ1602’’. This paper has been elaborated in the framework of the project ‘‘Support research and development in the Moravian–Silesian Region 2014 DT 1 — Research teams’’ (RRC/07/2014), financed from the budget of the Moravian-Silesian Region, and the Polish Ministry of Science and Higher Education from sources for science (AGH local grant 11.11.420.04).

Fig. 10. Union of all fixed points of the map F for all parameters (µ, ε) ∈ [0, 4] × [0, 1]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Appendix As it was mentioned in Section 2, the set of all fixed points of map F contains at most four points. For example, if µ = 7/2 and ε = 9/10 then

 Fix(F ) =

(0, 0), 



√ √    19 19 19 19 , , − , + ,

5 5 7 7

28

√ 19 28

+

√

19 19 28

,

28

−

19

28

28

28



28

.

Formulas for these points can be stated explicitly, by solving equations Fx (x, y) = x and Fy (x, y) = y with respect to parameters µ and ε . Using mathematical software (e.g., Matlab MuPAD) it can be verified that these fixed points are:

 {A}    {A, B, C , D}    Fix(F ) = {A, B, E , F }       {A, B}

if µ ∈ [0, 1], if µ > 1 and ε = 0, if µ ∈ (3, 4], ε ̸∈

A = (0, 0),

B=

1 2

c1 , c2 ∈ R, and c3 ≤ 0, otherwise

where points A, B, C , D, E , F are given by



0,

 µ−1 µ−1 , , µ µ



,

(A.1)

Fig. 11. Region of parameters (blue) with ε ∈ (0.75, 1) where there are four fixed points of the map F and region R (gray) for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

   µ−1 µ−1 , D= ,0 , µ µ   r − c2 − 1 s1 E= , , r 2r (1 − 2ε)   r − c1 − 1 s2 F = , r 2r (1 − 2ε) 

C =

0,

M. Lampart, P. Oprocha / Physica D 335 (2016) 45–53

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