# On problems of Topological Dynamics in non-autonomous discrete systems

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## Abstract

Most of problems in Topological Dynamics in the theory of general autonomous discrete dynamical systems have been addressed in the non-autonomous setting. In this paper we will review some of them, giving references and stating open questions.

## Abstract

Most of problems in Topological Dynamics in the theory of general autonomous discrete dynamical systems have been addressed in the non-autonomous setting. In this paper we will review some of them, giving references and stating open questions.

## 1 Introduction, definitions and first results

The interest for non-autonomous discrete systems (n.a.d.s.) or simply (na) has been increasing in last years because they are adequate to model some phenomena in applied sciences, such as biology [32, 53], physics [38], economy [55], etc., and to solve problems generated in mathematics (see [41]).

By other hand, more realistic models in the setting of dynamical systems are those where the trajectories of all points in the phase state are affected by small random perturbations. Most of such situations can be studied following the methodology of non-autonomous systems. In the autonomous case, we have a phase space and a unique continuous map where the trajectories of points are obtained iterating such map. For non-autonomous systems, the trajectories are produced using iteration methods by changing the map in each step.

Keeping the above ideas in mind, we are introducing precisely the general setting of (na). Let $(Xi)i=0∞$ = X0,∞ be a sequence of Hausdorff topological spaces and $(fi)i=0∞$ = f0,∞ a sequence of continuous maps, where fi: XiXi+1 for i ∈ ℕ = ℕ ⋃ {0}. For any pair of positive integers (i, n), we set

$fin=fi+(n−1)∘fi+(n−2)∘⋯∘fi+1∘fi.$

We also state $fi0$ = Identity|Xi and $fi−n=(fin)−1$ (taken in the sense of inverse images when the maps are not invertible).

The pair (X0,∞, f0,∞) is a (na) in which the sequence $(f1n(x))n=0∞=(xn)n=0∞,$ where x0 = x is the trajectory of the point xX0. The set of points of that trajectory is the orbit of xX1. In some problems, we will denote by $(X1,∞[n],f1,∞[n])$ the nth iterate of the system, that is, $Xi[n]=X(i−1)n+1$ and $fi[n]=f(i−1)+1[n].$

In order to have more concrete results, in particular in applications to real models in sciences, we will be restricted to the case when all spaces Xi are compact or compact and metric (in the last case, we will denote by $(dn)n=1∞$ the corresponding sequence of metrics). It is evident that when all spaces coincide with X and all maps with f, then we simply have the autonomous discrete dynamical system denoted by the pair (X, f).

In the literature on (na) there are a lot of results in the case when all spaces Xi are real compact intervals and the continuous maps fi are of a particular type, for example, piecewise linear maps (see for example [44]).

Of interest is also the case when the spaces and maps, components of the non-autonomous system, fulfill some periodic conditions.

Definition 1

A (na) is pperiodic if Xn+p = Xn and fn+p(x) = fn(x) for xXn, n ≥ 0, and p ≥ 1 being a positive integer. If p = 1, then we have the autonomous case.

Such non-autonomous systems have deserved special interest to many researchers in the theory of dynamical systems trying to extend to them the topics of Topological Dynamics considered in the autonomous case. For some of them, see [50] and the references therein.

In most cases, it is supposed we have only a topological space X = Xi for all n ∈ ℕ but a sequence of distinct maps. The resulting system will be denoted by (X, f0,∞). At most all applications deal with this case. Moreover, in order to obtain more concrete results, we will take X as a compact metric space.

The rest of the paper will be devoted to the consideration of the well known topics on Topological Dynamics for autonomous systems but now applied to the non-autonomous setting.

The introduction of non-autonomous systems in mathematics has been motivated by the computation of the topological entropy for skew product or triangular discrete dynamical systems in the unit square [0, 1]2 = I2 = Q, that is, discrete dynamical systems (Q, F), where F(x, y) = (f(x),g(x, y)) = (f(x),gx(y)) and F : QQ is a continuous self-map (written FC(Q)). The notion of (na) was formulated in [41] as an extension of that of autonomous system and it was studied the topological entropy. The introduction was made considering the above notion of trajectory or orbit of points of the spaces. The extension of the notion of topological entropy can be made using covers in the way of [1] or using the Bowen’s formula (see [23]).

$max{h(f),hf(F)}≤h(F)≤h(f)+hf(F),$

where h(F),h(f) denote respectively the topological entropy of F and f, hf = sup{h(F|Ix)}, and Ix = {x} × I. But to compute hf(F) is necessary to obtain the trajectory of every point (x, y) ∈ Ix, and this implies the knowledge of the sequence (gx(y), gf(x)(y), …, gfn(x)(y), …), that is, in every step of the iteration, the map to be taken is different and must follow the sequence of maps (Identity on I, gx, gf(x), …, gfn(x), …).

S. Kolyada and Ľ. Snoha introduced in [41] the notion of topological entropy for (na) adapting the original definition of [1], denoted by h(f0,∞), using covers, and when X is metrizable that one in [23], using spanning and separating sets. For compact metric spaces both notions are the same. If X is a compact topological space, then it is proved that $h(f0,∞n)$nh(f0,∞) for every n ≥ 1, where $f0,∞n=(fin+1∞)i=0∞.$ When f0,∞ is periodic of period p, (fn+p = fn for every n) then $h(f0,∞p)$ = ph(f0,∞). If X is a compact metric space and f0,∞ is a sequence of equicontinuous self-maps, then $h(f0,∞n)$ = nh(f0,∞) for every n ≥ 1. In [41] it was also introduced the notion of asymptotically topological entropy, h(f0,∞), as limn→∞h(fn,∞) where fn,∞ is the tail from n of the sequence f0,∞. It is proved that such a limit always exists. Many results on usual topological entropy of autonomous systems are held by the asymptotically topological entropy, proving that in different settings the two notions are analogous. In [41], it was also proved that for compact metric spaces, if the sequence f0,∞ converges uniformly to f or it is an equicontinuous family, then h(f) ≤ h(f).

One relevant consequence of the entropy in the non-autonomous case is the proof that in autonomous cases, the topological entropy is commutative for the composition of two continuous maps in compact metric spaces, that is, h(fg) = h(gf) for f, gC(X). Additionally, in [41] some other results were proved.

## 2 Topological entropy

We introduce the notion of topological entropy in the setting of non-autonomous systems of the form (X, f0,∞), where (X, d) is a compact metric space. We follow the Bowen’s line of introduction of the notion and the notation above considered and also [15].

For x,yX and n ≥ 0,

$ρn(x,y)=maxi=0,…,n−1d(f0i(x),f0i(y)).$

The set EX is said to be (n,ϵ,f0,∞)−separate if ρn(x, y) > ϵ for every distinct x,yE. Now denote by sn(f0,∞) the maximal cardinality of (n,ϵ,f0,∞)−separate sets. Then the topological entropy of (X, f0,∞) is

$h(f0,∞)=limϵ→0limsupn→∞1nlog(sn(f0,∞,ϵ)).$

This definition is just an extension of the topological entropy for autonomous systems in compact metric spaces.

### 2.1 Topological entropy and limits

We state the question of what is the behaviour of the entropy of a (na) given by the pair (X, f0,∞) if the sequence $(fn)n=1∞$ converges to a continuous map fC(X). We will consider uniform or piecewise convergences. They are expected different behaviours in these cases.

In the next result, proved in [41], we consider the case where the convergence is uniform.

Theorem 1

Let X be a compact metric space and $(fi)i=0∞$ be a sequence of continuous maps converging uniformly to a continuous map fC(X). Then h(f0,∞) ≤ h(f).

In the following examples we see that if the convergence of the sequence is piecewise to f but not uniform, the above statement is not true in general and it is possible to contruct some examples. In the case that h(f) = ∞, the previous formula is true. In [15], we proved the following result.

Proposition 2

For every continuous interval map f, there is a non-autonomous system (I, f0,∞) such that $(fn)n=1∞$ converges pointwise to a continuous map f and h(f0,∞) = ∞.

Proof

Choose in [0, 1] an infinite sequence of closed intervals [an, bn], for example, take $an=12nandbn=an+14n.$ From this, $an−bn+1=12n+1−14n+1>0$ for all n ∈ ℕ, which means that the above election is possible. Then in each interval [an, bn] we choose nsubintervals [ck, dk] from k = 1, …, n with $dk=ak+14n+k−1andcnk=dnk+1.$ Then inside every subinterval $Ink=[cnk,dnk],$ we choose another subinterval $[αnk,βnk]$ taking $αnk=cnk+110(dnk−cnk)$ and $βnk=cnk+910(dnk−cnk).$

Given fC(I), for every n ∈ ℕ we define fn(x) = f(x) for all $x∉⋃k=1n(Ink),fn(cnk)=f(cnk),fn(dnk)=f(dnk),$ $fn(αnk)=1,fn(βnk)=0,$ and in the rest of In we define fn connecting the dots.

In fact, what has been done is to introduce in every subinterval $Ink$ a linear perturbation in such a way that fn results continuous and fn(In) = I for every n, that is, fn is surjective.

First, it is evident that in I, limn→∞fn(x) = f(x) is point-wisely since the perturbation is acting on In only for the index n but not for the rest of indexed of the limit. Besides, the perturbation is moving to the left when n increases.

Now consider a fixed m and any n > m. It is evident that fm(Im) ⊂ $⋃k=1n$ Ik creating an infinite number of horseshoes. As a consequence, by applying Theorem 3 of [2], we have h(f1,∞) ≥ log m. But since m is arbitrarily large, then we conclude that h(f1,∞) = ∞.

Remark 1

Using the above result and construction, choosing the continuous map f with zero, positive or infinite topology, we have examples of three types with h(f0,∞) = ∞.

• Let $(fn)n=1∞$ be pointwise convergent to f.

1. Construct an example of a non-autonomous system for which h(f0,∞) > 0 and h(f) = 0.

2. Construct an example of a non-autonomous system such that h(f0,∞) = 0 and h(f) = 0.

### 2.2 Topological entropy and Li-Yorke chaos

Using the definition of trajectories for non-autonomous systems, next we state the defintion of Li-Yorke chaos for (na) in the same sense as in the autonomous case.

Definition 2

Let (X, f0,∞) and x, yX. The pair (x, y) is Li-Yorke chaotic if

1. $liminfn→∞d(f1n,f1n)=0,$ and

2. $limsupn→∞d(f1n,f1n)>0.$

Definition 3

A set SX is a scrambled set if it is uncountable and every pair of distinct points x, yS is Li-Yorke chaotic.

Definition 4

fC(X, X) is Li-Yorke chaotic if it possesses an uncountable scrambled set.

For autonomous systems in compact metric spaces, it is proved in [18] that positive topological entropy implies the existence of Li-Yorke chaos. In the following result from [15], it is proved that in general it is not true in (na). This was proved by constructing an interval example composed only by two different maps such that h(f0,∞) > 0 and the sequence $(fi)i=1∞$ converges to a map f which is not Li-Yorke chaotic.

• Using the methodology and approaches of [37, 52], try to extend the results of these papers to general metric spaces.

Theorem 3

There exists a (na) on the interval, (I, f0,∞), such that the sequence $(fi)i=0∞$ converges pointwisely to a non-continuous map f and satisfying that

1. h(f0,∞) > 0.

2. f is not Li-Yorke chaotic.

Proof

According to [15], take the interval [0, 1] and divide it into three subintervals of length $13.$ Denote the central subinterval by J and consider the two piecewise linear maps f1 and f2 (see [15]). Consider now the sequence of maps composed by f1 and f2 where the map f1 appears infinitely many times. With such a distribution, the points 0, 1 are fixed and the rest of points of [0, 1] are asymptotic to 0. As a consequence, the pointwise limit of the initial sequence is a non-continuous map.

The behavior of g in the sequence f0,∞ is described as follows. Take m0 = 1, mn = 2n2, and put fn = g for n = mn and fi = h for any other index. The autonomous system ([0, 1], h) has a 2−strong horseshoe in the subinterval J. Then there is ϵ such that for every n, there exists an (n,ϵ,h)−separate set EJ holding card(E) = 2n.

For every n there is an interval Kn such that $f1mn(Kn)=J.$ We state K = gn−1(J), let ln = mn+1mn − 1, and Fn be an (ln,ϵ,h)−separate set of h having maximal cardinality. Then Kn = gn−1(Fn) ⊂ K is (mn+1,ϵ,f0,∞)−separate. That is, for mnj < mn+1, we have $f1j(Kn)=gmnj−mn=hj−mn(Fn).$ Then

$h(f0,∞)≥limsupn→∞1mn+1−1log⁡cardFn≥limn→∞lnmn+1log⁡2≥limn→∞mn+1−mn−1mm+1log⁡2≥(1−limn→∞2(n2+1)−(n+1)2)log⁡2≥log⁡2.$

1. Prove that for a (na) system of the form of the form (I, f0,∞) composed by onto maps converging uniformly to f, it holds that h(f) = h(f0,∞).

2. In other spaces different from the interval I, construct examples of onto continuous maps fn converging uniformly to f and such that

• h(f) = ∞ and h(f0,∞) = ∞.

• h(f) = ∞ and h(f0,∞) > 0.

• h(f) = ∞ and h(f0,∞) = 0.

Similar results to above have been obtained in [52] for (Im, f0,∞) where the sequence of maps converges uniformly to a map in Im and all trajectories are subjected to small random perturbations. In fact, it is proved that if f is the limit map of the sequence of maps and PIm is recurrent in the autonomous dynamical system (Im,f), then P is also recurrent in the non-autonomous case (see definitions of recurrence in [34]) affected by small random perturbatons. It is also proved that under some sufficient conditions, a non-autonomous system (I, f0,∞) subjected to small perturbations can be non-chaotically converted in the Li-Yorke sense.

### 2.3 Minimal sets

We say that an autonomous system (X, f) is minimal if there is no proper subset MX which is non-empty, closed, and f−invariant (f(M) ⊆ M). Then we also say that the map f is minimal. It is immediate that f is minimal, if and only if, the forward orbit of every point xX is dense on X (see [34]).

We say that (X,f1,∞) is minimal if every trajectory is dense in X. Some properties of minimal autonomous systems, such as f being feebly open (the map transforms open sets into sets with non-empty interiors) or almost one-to-one (a typical point has just one pre-image) are not held in the setting of non-autonomous systems (see [43]). For example, the former properties cannot be obtained neither for the maps fn nor fnfn−1 ∘ ⋅ s∘ f1f0. But this is not the unique property: in fact, there is a wider variety of dynamical behaviours in the non-autonomous systems than in the autonomous cases.

In [42] it is proved that it is equivalent for (X,f0,∞) not being minimal to the fact that there is a non-empty open set BX such that the system has arbitrarily long finite trajectories disjoint with B. This has as a corollary a sufficient condition for metric spaces without isolated points to be non-minimal. That condition holds whether there is a nonempty open set BX and n0 ∈ ℕ satisfying the two following:

1. $f1n0−1$ is onto.

2. The non-autonomous system has arbitrarily long finite trajectories disjoint with B.

The same happens with the conditions $f1n0−1$ as well as the maps fn for nn0 which are onto and for every nn0, fn(B) ⊆ fn(XB). Under the former conditions for (X,f1,∞), suppose that the sequence $(fn)n=1∞$ converges uniformly to f. If f is not onto, then the system is not minimal, and even more, no trajectory is dense.

In [42], there is a discussion using examples in X = [0,1] to check the validity of the former conditions and to prove that even if fnf and for every n ∈ ℕ, fn is onto, then f is only monotonic (not necessarily strict). Theorem 3.2 in [42] proves the existence of (I,f0,∞) such that fn converges uniformly to the Identity on I, for every n, fn is onto and can be chosen piecewise linear with non-zero slopes, and such that (I,fn,∞) is a minimal system. The arguments stated in the referred results are used to construct and improve some examples introduced in [13] in the setting of skew product maps on Q with the property that almost all orbits in Q have the second projection dense in I and whose omega-limit sets are {0} × I.

### 2.4 Topological entropy of non-autonomous systems on the square and on ℝ2

We have remarked previously that the computation of topological entropy in triangular systems on the square given by F(x, y) = (f(x), gx(y)) are related to the consideration of trajectories of points y ∈ [0, 1], given by (gx(y), gf(x)(y), …, gfn(x)(y), …). As a consequence, some results on entropy of autonomous triangular systems can be obtained from non-autonomous systems as defined above. In [44], it is developed a theory of the topological entropy for non-autonomous piecewise monotonic systems on the interval. It is made with additonal conditions in the system, namely, for systems (I1,∞, f0,∞) being bounded and long-lapped or Markov (see [44] for such notions). In the next result, we denote by c1,n the number of pieces of monotonocity of the map f1,n.

Theorem 4

Let (I, f0,∞) be finite piecewise monotonic or Markov. Then the non-autonomous system satisfies

$h(f0,∞)=limsupn→∞1nlog⁡c1,n.$

As an application of that result, in [44] it is constructed a large class of triangular maps on the square of type 2 (such maps have periodic trajectories of all periods, all being powers of two) of class C, extending a previous result appeared in [12].

1. Prove or disprove if those triangular maps may be obtained in the class of real analytic or polynomial maps.

With respect to minimality, in [42] it is proved the existence of minimal non-autonomous systems on the interval, (I, f0,∞), such that the sequence $(fn)n=0∞$ converges uniformly to the identity map and all maps fn are onto. Even more, all fn can be choosen piecewise linear in I with non-zero slopes, having at most three pieces of linearity, and for every n, the (na), (I, fn,∞) being minimal.

Such results are used to prove a result on autonomous triangular systems on the square (see [13, 42]).

Theorem 5

There is a triangular map F defined on the square I2 satisfying that

1. All points of the form (0, y) are fixed.

2. limn→∞fn(x) = 0 for every x.

3. Every point in I2 not being of the form (0, y) has a trajectory whose second projection is dense in I.

4. h(F) = 0.

A relevant fact in the proof of this theorem [42] consists of using an Extension Lemma (see [35] or [40]) which allows to carry out adequate constructions and to obtain properties in subsets of I2 which can be extended to the total square keeping the properties.

1. Try to obtain an example of the previous theorem in the class C.

2. Is there a triangular (na) on the square such that $(fn)n=0∞$ converges pointwise to f and h(f0,∞) = 0 but h(f) > 0?

One of the most known general two-dimensional map defined in ℝ2 is the Hénon map which is given by

$Ha,b(x,y)=(a+by−x2,x),$

where a and b are real parameters. When b ≠ 0, then the map has an inverse given by

$Ha,b−1(x,y)=(y,x−a+y2b).$

If b = 0, then we have essentially the map Ha(x, y) = (ax2, x) which behaves similarly to a one dimensional map. Therefore, we will deal with the case with parameter b ≠ 0. It can be proved that there exists a Cantor invariant set K ⊂ ℝ2 (H(K) = K) where the map is topologically conjugate to a shift map defined on a finite number of symbols. Therefore, in K the map is Devaney chaotic (a proof can be seen in [54]). That proof applies sufficient conditions called Conley and Moser conditions (see [47]).

In [16], it is considered a non-autonomous version of the Hénon map when b = −1. For this value, the map is area preserving. The other parameter is allowed to vary for each n by

$fn(x,y)=(a(n)−y−x2,x)fn−1=(y,a(n)−x−y2).$

The sequence a(n) will be taken as a trigonometric perturbation of the number $192,$ that is

$a(n)=192+ϵcos⁡(n),ϵ=110.$

The choosing of the two values are to be able to apply a new version of the Conley-Moser condition (see [16]). The domains Dn are introduced as

$Dn=D=[−R,R]×[−R,R]R=1+a(0).$

A similar approach has been recently applied in [46] to the Lozi map on ℝ2.

1. Solve the same problem of Hénon (na) for b ∉ {−1,0}, that is when the system is not conservative.

2. Consider a perturbation for a(n) of the form

$a(n)=a0+ϵcs(n,m),$

where cs denotes the Jacobi cosam elliptic maps. Alternately, it can be done also using the Jacobi senam elliptic map. See all previous results and statements in [8].

3. Solve the previous questions for Lozi system.

## 3 Difference and systems of difference non-autonomous equations

### 3.1 Examples in difference equations

For a wide range of types of difference equations, autonomous or non-autonomous, deterministic or stochastic, discrete or continuous, it has been proved that the asymptotic distribution of trajectories hold very often the so called Benford’s law, which we introduce in this subsection. Frequently numerical data got from dynamical systems follows such law.

Firstly, we are dealing with dynamical properties associated to Benford sequences. It is known (see [21]) that frequently, trajectories from discrete dynamical systems satisfy the Benford’s law of logarith mantissa distributions. This law is the probability distribution of the mantissa function or simply mantissa with respect to a base b ∈ ℕ∖ {1}. This is given by

$P(mantissab≤t)=logb⁡t,t∈[1,b).$

The mantissa function, denoted by < ⋅ >, is a function from ℝ+ to [1,b) given by < x >b = b[logbx]. With this, we state the following

Definition 5

A sequence $(xn)n=0∞$ of real numbers is b−Benford if

$limn→∞card{jb≤t}n=logb⁡t,t∈[1,b),$

and it is strictly Benford if it is b−Benford for every b ∈ ℕ ∖ {1}.

It is well-known (see for example, those of [21] and other references therein) the following result which compares the Benford property of a sequence and the same for the sequence of the log of the absolute value of their terms.

Proposition 6

A sequence of real numbers $(xn)n=0∞$ is BBenford, if and only if, the sequence $(log⁡|xn|)n=0∞$ is uniformly distributed modulus 1.

Using the above result and others from uniformly distribution, in many examples can be proved the property of trajectories of dynamical systems starting in an initial point x0 or simply general sequences of real numbers.

Example 7

1. The sequence (n!) is Benford.

2. The trajectories constructed from the Fibonacci equation Fn+2 = Fn + Fn+1 for n ≥ 0 are Bedford, except for the starting point (0,0).

3. For almost Lebesgue initial point x0 ∈ ℝ, the corresponding trajectoy is Benford.

4. The sequence $(2n)n=0∞$ is Benford if logb 2 is irrational, that is, if and only if, b ≠ 2j for some j ∈ ℕ.

5. Not all sequences are Benford. For example, (nα) for α ∈ ℝ+ and the sequence of prime numbers are not Benford for any b.

In the case of (na) there are a few results in the literature. We will refer here to those from [19] concerning non-autonomous linear systems

$xn=Anxn−1,n∈N,$

where for every n, An is a real m −-matrix, and where the problem is to study under what conditions the mantissa distribution generated for the trajectories with initial conditions x0 ∈ ℝd satisfy the Benford law. The results we obtain are related to the b-resonance condition introduced in [19].

Definition 6

1. A set Λ ⊂ ℂ is b−resonant if there exists a finite non-empty subset Λ0 = {λ1, …, λq ⊂ Λ with |λ1| = ⋅ s = |λq| such that either card(Λ0 ∩ ℝ) = 2 or the numbers 1, logb|λ1| as well as the elements of ${12πarg⁡λ1,…,12πarg⁡λq}∖{0,12π}$ are ℚ−dependent.

2. The matrix A (real or complex) has b−resonant spectrum if the set σ+(A) is b−resonant.

With this in mind, in [19] it is proved the following result.

Theorem 8

Let $(An)n=1∞$ be pperiodic for some p ≥ 1 and assume that the matrices A1, …, Aq do not have bresonant spectrums. Then for every c,x0 ∈ ℝd, the sequence c, Orb(x0) is either finite or bBenford.

1. Given the one dimensional dynamical system (I, f), study the points in I whose trajectories satisfy the Benford’s law. This means try to state the properties of these trajectories to have such property.

2. Consider the sequences composed by distances of pair of points and relate the above results with existence or not of distributional chaos (for definitions, see [14]).

3. Consider nonlinear non-autonomous systems and study their behaviour concerning the same property.

### 3.2 On forbidden sets

In recent literature, there are an increasing number of papers where the forbidden sets of difference equations are computed. We review and complete different attempts to describe the forbidden set and propose new perspectives for further research and a list of open problems in this field.

The study of difference equations (DE) is an interesting and useful branch of discrete dynamical systems due to their variety of behaviours and their ability to model phenomena of applied sciences (see [24, 26, 36, 45] and references therein). The standard framework for this study is to consider iteration functions and sets of initial conditions in such a way that the values of the iterates belong to the domain of definition of the iteration function and therefore, the solutions are always well-defined. For example, in rational difference equations (RDE), a common hypothesis is to consider positive coefficients and initial conditions, see [36, 45].

Such restrictions are also motivated by the use of (DE) as applied models, where negative initial conditions and/or parameters are usually meaningless [48].

But there is a recent interest to extend the known results to a new framework where initial conditions can be taken to be arbitrary real numbers and no restrictions are imposed to iteration functions. In this setting the forbidden set of a (DE) appears, namely, the set of initial conditions for which after a finite number of iterates we reach a value outside the domain of definition of the iteration function. Indeed, the central problem of the theory of (DE) is reformulated in the following terms:

Given a (DE), determine the good 𝔊 and forbidden 𝔉 sets of initial conditions. For points in the good set, describe the dynamical properties of the solutions generated by them: boundedness, stability, periodicity, asymptotic behavior, etc.

Here, we are interested in the first part of the former problem: how to determine the forbidden set of a given (DE) of order k. In the previous literature to describe such sets, when it is achieved, it is usually interpreted as to be able to write a general term of a sequence of hypersurfaces in ℝk. But in those cases are precisely the corresponding to (DE) when it is also possible to give a general term for the solutions. Unfortunately, there are a little number of (DEs) with explicitly defined solutions. Hence, we claim that new qualitative perspectives must be assumed to deal with the problem above. Therefore, the goals in this subsection are the following: to organize several techniques used in the literature for the explicit determination of the forbidden set, revealing their resemblance in some cases, and giving some hints about how they can be generalized. Thus, we get a large list of (DEs) with known forbidden set that can be used as a frame to deal with the more ambitious problem to describe the forbidden set of a general (DE). We review and introduce some methods to work also in that case. Finally, we propose some future directions of research.

The difference equation of Riccati plays a key role in this theory since as far as we know, almost all the literature where the forbidden set is described using a general term includes some kind of semiconjugacy involving such an equation. The (DE) obtained via a change of variables or topological semiconjugacy is a relevant tool. In the following, we will discuss how algebraic invariants can be used to transform a given equation into a Riccati or linear one depending upon a parameter, and therefore, determining its forbidden set.

After that, we will deal with an example of description, found in [25], where the elements of the forbidden set are given recurrently but explicitly.

It can be introduced a symbolic description of complex and real points of F, where we study some additional ways to deal with the forbidden set without an explicit formula.

Now we are concentrating in some problems stated in the recent literature concerning the above problems, in particular, in the non-autonomous Riccati difference equation of first order given by

$xn+1=anxn+bncnxn+dn,n=0,1,…,$

where the sequences $(an)n=0∞,(bn)n=0∞,(cn)n=0∞,and(dn)n=0∞$ are q−periodic and x0 ∈ ℝ.

In [5], it is given a geometric description of the forbidden sets in terms of the coefficients of the equation in the general case and also in the following particular cases:

1. bn = 0 and dn = 1 for all n = 0,1, … in both cases, when all the parameters are positive real numbers and they are general real numbers without restriction (see also [51]).

2. an = 1 for n = 0,1, … and the sequence $(cn)n=0∞$ is not periodic.

1. In (), describe the forbidden set in the cases when the sequences of coefficients are bounded and none of them is periodic.

2. The same in the cases when the maps $fn=anx+bncnx+dn$ are uniformly convergent to a map f, or alternatively, converge pointwise to f.

3. Solve the same problems that above through by another general nonlinear rational difference equations.

4. Face the same problems in the setting of systems of difference equations.

## 4 Lyapunov exponents in non-autonomous systems

During years, a powerful tool to understand the behaviour and predictability in nonlinear discrete dynamical systems and time series obtained from them have been Lyapunov exponents. They have been used to decide when orbits are stable or instable in the Lyapunov sense. First, it is necessary to remark that while stability in the Lyapunov sense is a notion of topological character, Lyapunov exponents have a numerical nature and are calculated using the derivative of maps in the points of orbits.

It is an extended practice, particularly in experimental dynamics, to associate having trajectories with positive Lyapunov exponents with their instability and negative Lyapunov exponents with their stability. However, such interpretation has no firm mathematical foundation if some restrictions on the maps describing the dynamical systems are not introduced. In connection with such statement, in [31], they have been constructed two examples of autonomous dynamical systems defined by interval maps, one having a trajectory with positive Lyapunov exponent but stable and other having a trajectory with negative Lyapunov exponent but instable. But such maps are highly non-differentiable and therefore, we wonder if it is possible to obtain the same results via differentiable maps. In [11], they have been obtained such examples in the semi-open interval [0, 1). We wonder if the above example can be constructed in the setting of non-autonomous systems.

In [8], it is introduced and applied for them an immediate extension of the definition of the Lyapunov exponents in the autonomous case (if the limit exists), when X = I. This definition is as follows.

Definition 7

$λ(x)=limn→∞1nlog⁡|(fn−1∘⋯∘f1∘f0)′(x)|=limn→∞1n∑j=0n−1log⁡|fj′(xj)|,$

and similarly for the strong Lyapunov exponent.

Stability and instability of trajectories are stated as the condition of being or not sensitive to initial conditions, namely,

Definition 8

The forward orbit $(xn)n=0∞$ is sensitive to initial conditions or Lyapunov instable if there exists ϵ > 0 such that for any δ > 0, there exists y with d(x0,y) < δ and N ≥ 0 such that d(fN(y), fN(x0))≥ ϵ.

Definition 9

The forward orbit $(xn)n=0∞$ is not sensitive to initial conditions or it is Lyapunov stable if for any ϵ > 0 there is δ > 0 such that whenever d(x0,y) < δ, then d(fn(y),fn(x0)) < ϵ for all n ≥ 0.

One relevant situation is when f0,∞ is composed by a periodic sequence of maps of minimal period m, namely fn+m = fn for every n ≥ 0. Then the non-autonomous system is called periodic of minimal period m or simply periodic of period m (see [3, 27]).

To non-autonomous systems it is immediately extended the notion of instable trajectory or orbit in the Lyapunov sense.

Using the two maps f, g introduced in Section 4 of [11], we compute the Lyapunov exponent for the periodic case in the non-autonomous situation. Consider a periodic non-autonomous case of minimal period m composed by periodic blocks of repetitions of the former maps in any ordering. Let the map f be applied 0 < p < m times and q = mp times the map g. The case of alternating maps (see [3, 27]) holds when p = q = 1 and the block is {f, g, f, g, …}.

Proposition 9

For the periodic non-autonomous system in I given by repetition of the block composed by p times f and mp = q times g following any ordering, the trajectory of 0 has a strong Lyapunov exponent of value

$Λ(0)=p−qmlog⁡2$

and the orbit is instable.

Proof

Given any block of p times the map f and pq times g in any ordering and pq, we use the former definitions to obtain the value of the Lyapunov exponent of the orbit of 0. The partial sums Sn of the corresponding series are given by

$Sn=k(p−q)+i−jkm+i+jlog⁡2$

where 0 < ip, 0 < jq, and n = k m + i + j.

When n → ∞, that series is convergent and it is immediate that its value is $p−qm$ log 2, which is also its Λ (0).

Instability of the orbit is due to instability of the trajectory of g(0).

1. We claim that the orbit of initial condition x0 = 0 would be instable if q > 0 and independently of this value. Moreover, we think that it is the case for non-periodic non-autonomous systems when the map g appears an infinite number of times, such as in the sequence f0,∞ = {f, g, f, f, g, f, f, f,g, …}.

2. Another way to choose the maps f and g is using the Thue-Morse sequence given by {0,1,1,0,1,0,0,1,0, …} (see [6] for more details). We choose the elements of the sequence. When we find one 0 we choose f and with 1, we choose g.

3. Solve the problem with Thue-Morse sequence.

4. Solve the problem for non-autonomous Hénon transformation (defined above).

5. Do the same in non-autonomous systems generated by perturbations of trigonometric and Jacobi types (see [8]).

Communicated by Juan L.G. Guirao

Acknowledgements

This research has been supported by Grant MTM2014-51891-P from Spanish MINECO.

## References

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R.L. Adler, A.G. Konheim and M.H. McAndrew, (1965), Topological entropy, Transactions of the American Mathematical Society, 114, No 2, 309-319.

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L. Alsedà, J. Llibre and M.Misiurewicz, (2000), Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing Co., River Edge.

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J.F. Alves, (2009), What we need to find out the periods of a periodic difference equation, Journal of Difference Equations and Applications, 15, No 8-9, 833-847.

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Z. AlSharawi, J. Angelos, S. Elaydi and L. Rakesh, (2006), An extension of Sharkovsky’s theorem to periodic difference equations, Journal of Mathematical Analysis and Applications, 316, No 1, 128-141.

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R. Azizi, Nonautonomous Riccati difference equation with real k–periodic (k ≥ 1) coefficients, submitted to Journal of Difference Equations and Applications.

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F. Balibrea, (2016), Los secretos de algunas sucesiones de números enteros, MATerials MATemàtics. Publicaciò electrònica de divulgació del Departament de Matemàtiques de la Universitat Autònoma de Barcelona, treball no. 2, 32 pp.

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F. Balibrea and A. Cascales, (2015), On forbidden sets, Journal of Difference Equations and Applications, 21, No 10, 974-996.

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F.Balibrea and R.Chacón, (2011), A simple non-autonomous system with complicated dynamics, Journal of Difference Equations and Applications, 17, No 2, 131-136.

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F. Balibrea and A. Linero, (2003), On the periodic structure of delayed difference equations of the form xn = f(xn - k) on 𝕀 and 𝕊1, Journal of Difference Equations and Applications, 9, No 3-4, 359-371.

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F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, (2010), Recent developments in Dynamical Systems: three perspectives, International Journal of Bifurcation and Chaos, 20, No 9, 2591-2636.

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F. Balibrea and M. Victoria Caballero, (2013), Stability of orbits via Lyapunov exponents in autonomous and nonautonomous systems, International Journal of Bifurcation and Chaos, 23, No 7, 1350127 [11 pages].

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F. Balibrea, F. Esquembre and A.Linero, (1995), Smooth triangular maps of type 2 with positive topological entropy, International Journal of Bifurcation and Chaos, 5, No 5, 1319-1324.

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F. Balibrea, J. L. G. Guirao and J.I. Muñoz Casado, (2001), Description of ω–limit sets of a triangular map on I2, Far East Journal of Dynamical Systems, 3, 87-101.

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F. Balibrea, J. Smítal and M.Stefánková, (2005), The three versions of distributional chaos, Chaos, Solitons and Fractals, 23, No 5, 1581-1583.

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F. Balibrea and P. Oprocha, (2012), Weak mixing and chaos in nonautonomous discrete systems, Applied Mathematics Letters, 25, No 8, 1135-1141.

• [16]

F. Balibrea-Iniesta, C. Lopesino, S. Wiggins and A. Mancho, (2015), Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautonomous Hénon Map, International Journal of Bifurcation and Chaos, Vol 25, No 12, 1550172 [14 pages].

• [17]

F. Benford, (1938), The Law of Anomalous Numbers, Proceedings of the American Philosophical Society, 78, No 4, 551-572.

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F. Blanchard, E. Glasner, S. Kolyada and A. Maass, (2002), On Li-Yorke pairs, Journal für die reine und angewandte Mathematik (Crelles Journal), 547, 51-68.

• [19]

A. Berger, (2005), Multi-dimensional dynamical systems and Benford’s Law, Discrete and Continuous Dynamical Systems - Series A, 13, 1, 219-237.

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A. Berger and S. Siegmund, (2007), On the distribution of mantissae in nonautonomous difference equations, Journal of Difference Equations and Applications, 13, No 8-9, 829-845.

• [21]

A. Berger, (2011), Some dynamical properties of Benford sequences, Journal of Difference Equations and Applications, 17, No 2, 137-159.

• [22]

Bogenschütz, (1992), Entropy, pressure, and a variational principle for random dynamical systems, Random Comp., 1, 219-227.

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R. Bowen, (1971), Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society, 153, 401-414.

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E. Camouzis and G. Ladas, (2007), Periodically forced Pielou’s equation, Journal of Mathematical Analysis and Applications, 333, No 1, 117-127.

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E. Camouzis and R. DeVault, (2003), The Forbidden Set of xn+1=p+xn1xn,$\begin{array}{} x_{n+1} = p + \frac{x_{n-1}}{x_{n}}, \end{array}$ Journal of Difference Equations and Applications, 9, No. 8, 739-750.

• [26]

E. Camouzis and G. Ladas, (2007), Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC.

• [27]

J. S. Cánovas and A. Linero, (2006), Periodic structure of alternating continuous interval maps, Journal of Difference Equations and Applications, 12, No 8, 847-858.

• [28]

J. S. Cánovas, (2006), On ω–limit sets of non-autonomous discrete systems, Journal of Difference Equations and Applications, 12, No 1, 95-100.

• [29]

J. S. Cánovas, (2013), Recent results on non-autonomous discrete systems, Boletín de la Sociedad Española de Matemática Aplicada, 51, No 1, 33-40.

• [30]

J.M. Cushing and S.M. Henson, (2002), A Periodically Forced Beverton-Holt Equation, Journal of Difference Equations and Applications, 8, No 12, 1119-1120.

• [31]

B. Demir and S. Koçak, (2001), A note on positive Lyapunov exponent and sensitive dependence on initial conditions, Chaos, Solitons and Fractals, 12, No 11, 2119-2121.

• [32]

M. De la Sen, (2008), The generalized Beverton-Holt equation and the control of populations, Applied Mathematical Modelling, 32, No 11, 2312-2328.

• [33]

S. Elaydi, (2005), An Introduction to Difference Equations, Springer-Verlag, Berlin.

• [34]

H. Furstenberg, (1981), Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, New Jersey.

• [35]

M. Grinc and L’. Snoha (2000), Jungck theorem for triangular maps and related results, Applied General Topology, 1, No 1, 83-92.

• [36]

E. A. Grove and G. Ladas, (2004), Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC.

• [37]

K. Janková and J. Smítal, (1995), Maps with random perturbations are generically not chaotic, International Journal of Bifurcation and Chaos, 5, No 5, 1375-1378.

• [38]

N. Joshi, D. Burtonclay and R.G.Halburd, (1992), Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem, Letters in Mathematical Physics, 26, No 2, 123-131.

• [39]

R. Kempf, (2002), On Ω–limit Sets of Discrete-time Dynamical Systems, Journal of Difference Equations and Applications, 8, No 12, 1121-1131.

• [40]

S. Kolyada and L. Snoha, (1992), On ω-limit sets of triangular maps, Real Analysis Exchange 18, 115-130.

• [41]

S. Kolyada and Ľ. Snoha, (1996), Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4, 205-233.

• [42]

S. Kolyada, Ľ. Snoha and S.Trofimchuk, (2004), On Minimality of Nonautonomous Dynamical Systems, Nonlinear Oscillations, 7, No 1, 83-89.

• [43]

S. Kolyada, Ľ. Snoha and S.Trofimchuk, (2001), Noninvertible minimal maps, Fundamenta Mathematicae, 168, No 2, 141-163.

• [44]

S. Kolyada, M. Misiurewicz and Ľ. Snoha, (1999), Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fundamenta Mathematicae, 160, No 2, 161-181.

• [45]

M.R.S. Kulenović and G. Ladas, (2001), Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/CRC.

• [46]

C. Lopesino, F. Balibrea-Iniesta, S. Wiggins and A.M. Mancho, (2015), The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions, International Journal of Bifurcation and Chaos, 25, No 13, 1550184 [18 pages].

• [47]

J. Moser, (1973), Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (AM-77), Princeton University Press, Princeton.

• [48]

J. Rubió-Massegú and V. Mañosa, (2007), Normal forms for rational difference equations with applications to the global periodicity problem, Journal of Mathematical Analysis and Applications, 332, No 2, 896-918.

• [49]

H. Sedaghat, (2003), Nonlinear Difference Equations. Theory and Applications to Social Science Models, Springer-Verlag, Berlin.

• [50]

Y. Shi, L. Zhang, P. Yu and Q. Huang, (2015),Chaos in Periodic Discrete Systems, International Journal of Bifurcation and Chaos, 25, No 1, 1550010 [21 pages].

• [51]

S. Stević, (2013), Domains of undefinable solutions of some equations and systems of difference equations, Applied Mathematics and Computation, 219, No 24, 11206-11213.

• [52]

L. Szała, (2015), Chaotic behaviour of uniformly convergent non-autonomous systems with randomly perturbed trajectories, Journal of Difference Equations and Applications, 21, No 7, 592-605.

• [53]

J. Wright, (2913), Periodic systems of population models and enveloping functions, Computers & Mathematics with Applications, 66, No 11, 2178-2195.

• [54]

S. Wiggins, (2003), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer.

• [55]

W.-B. Zhang, (2006), Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Elsevier Science, Amsterdam.

[1]

R.L. Adler, A.G. Konheim and M.H. McAndrew, (1965), Topological entropy, Transactions of the American Mathematical Society, 114, No 2, 309-319.

[2]

L. Alsedà, J. Llibre and M.Misiurewicz, (2000), Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publishing Co., River Edge.

[3]

J.F. Alves, (2009), What we need to find out the periods of a periodic difference equation, Journal of Difference Equations and Applications, 15, No 8-9, 833-847.

[4]

Z. AlSharawi, J. Angelos, S. Elaydi and L. Rakesh, (2006), An extension of Sharkovsky’s theorem to periodic difference equations, Journal of Mathematical Analysis and Applications, 316, No 1, 128-141.

[5]

R. Azizi, Nonautonomous Riccati difference equation with real k–periodic (k ≥ 1) coefficients, submitted to Journal of Difference Equations and Applications.

[6]

F. Balibrea, (2016), Los secretos de algunas sucesiones de números enteros, MATerials MATemàtics. Publicaciò electrònica de divulgació del Departament de Matemàtiques de la Universitat Autònoma de Barcelona, treball no. 2, 32 pp.

[7]

F. Balibrea and A. Cascales, (2015), On forbidden sets, Journal of Difference Equations and Applications, 21, No 10, 974-996.

[8]

F.Balibrea and R.Chacón, (2011), A simple non-autonomous system with complicated dynamics, Journal of Difference Equations and Applications, 17, No 2, 131-136.

[9]

F. Balibrea and A. Linero, (2003), On the periodic structure of delayed difference equations of the form xn = f(xn - k) on 𝕀 and 𝕊1, Journal of Difference Equations and Applications, 9, No 3-4, 359-371.

[10]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, (2010), Recent developments in Dynamical Systems: three perspectives, International Journal of Bifurcation and Chaos, 20, No 9, 2591-2636.

[11]

F. Balibrea and M. Victoria Caballero, (2013), Stability of orbits via Lyapunov exponents in autonomous and nonautonomous systems, International Journal of Bifurcation and Chaos, 23, No 7, 1350127 [11 pages].

[12]

F. Balibrea, F. Esquembre and A.Linero, (1995), Smooth triangular maps of type 2 with positive topological entropy, International Journal of Bifurcation and Chaos, 5, No 5, 1319-1324.

[13]

F. Balibrea, J. L. G. Guirao and J.I. Muñoz Casado, (2001), Description of ω–limit sets of a triangular map on I2, Far East Journal of Dynamical Systems, 3, 87-101.

[14]

F. Balibrea, J. Smítal and M.Stefánková, (2005), The three versions of distributional chaos, Chaos, Solitons and Fractals, 23, No 5, 1581-1583.

[15]

F. Balibrea and P. Oprocha, (2012), Weak mixing and chaos in nonautonomous discrete systems, Applied Mathematics Letters, 25, No 8, 1135-1141.

[16]

F. Balibrea-Iniesta, C. Lopesino, S. Wiggins and A. Mancho, (2015), Chaotic Dynamics in Nonautonomous Maps: Application to the Nonautonomous Hénon Map, International Journal of Bifurcation and Chaos, Vol 25, No 12, 1550172 [14 pages].

[17]

F. Benford, (1938), The Law of Anomalous Numbers, Proceedings of the American Philosophical Society, 78, No 4, 551-572.

[18]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, (2002), On Li-Yorke pairs, Journal für die reine und angewandte Mathematik (Crelles Journal), 547, 51-68.

[19]

A. Berger, (2005), Multi-dimensional dynamical systems and Benford’s Law, Discrete and Continuous Dynamical Systems - Series A, 13, 1, 219-237.

[20]

A. Berger and S. Siegmund, (2007), On the distribution of mantissae in nonautonomous difference equations, Journal of Difference Equations and Applications, 13, No 8-9, 829-845.

[21]

A. Berger, (2011), Some dynamical properties of Benford sequences, Journal of Difference Equations and Applications, 17, No 2, 137-159.

[22]

Bogenschütz, (1992), Entropy, pressure, and a variational principle for random dynamical systems, Random Comp., 1, 219-227.

[23]

R. Bowen, (1971), Entropy for group endomorphisms and homogeneous spaces, Transactions of the American Mathematical Society, 153, 401-414.

[24]

E. Camouzis and G. Ladas, (2007), Periodically forced Pielou’s equation, Journal of Mathematical Analysis and Applications, 333, No 1, 117-127.

[25]

E. Camouzis and R. DeVault, (2003), The Forbidden Set of xn+1=p+xn1xn,$\begin{array}{} x_{n+1} = p + \frac{x_{n-1}}{x_{n}}, \end{array}$ Journal of Difference Equations and Applications, 9, No. 8, 739-750.

[26]

E. Camouzis and G. Ladas, (2007), Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC.

[27]

J. S. Cánovas and A. Linero, (2006), Periodic structure of alternating continuous interval maps, Journal of Difference Equations and Applications, 12, No 8, 847-858.

[28]

J. S. Cánovas, (2006), On ω–limit sets of non-autonomous discrete systems, Journal of Difference Equations and Applications, 12, No 1, 95-100.

[29]

J. S. Cánovas, (2013), Recent results on non-autonomous discrete systems, Boletín de la Sociedad Española de Matemática Aplicada, 51, No 1, 33-40.

[30]

J.M. Cushing and S.M. Henson, (2002), A Periodically Forced Beverton-Holt Equation, Journal of Difference Equations and Applications, 8, No 12, 1119-1120.

[31]

B. Demir and S. Koçak, (2001), A note on positive Lyapunov exponent and sensitive dependence on initial conditions, Chaos, Solitons and Fractals, 12, No 11, 2119-2121.

[32]

M. De la Sen, (2008), The generalized Beverton-Holt equation and the control of populations, Applied Mathematical Modelling, 32, No 11, 2312-2328.

[33]

S. Elaydi, (2005), An Introduction to Difference Equations, Springer-Verlag, Berlin.

[34]

H. Furstenberg, (1981), Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, New Jersey.

[35]

M. Grinc and L’. Snoha (2000), Jungck theorem for triangular maps and related results, Applied General Topology, 1, No 1, 83-92.

[36]

E. A. Grove and G. Ladas, (2004), Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC.

[37]

K. Janková and J. Smítal, (1995), Maps with random perturbations are generically not chaotic, International Journal of Bifurcation and Chaos, 5, No 5, 1375-1378.

[38]

N. Joshi, D. Burtonclay and R.G.Halburd, (1992), Nonlinear nonautonomous discrete dynamical systems from a general discrete isomonodromy problem, Letters in Mathematical Physics, 26, No 2, 123-131.

[39]

R. Kempf, (2002), On Ω–limit Sets of Discrete-time Dynamical Systems, Journal of Difference Equations and Applications, 8, No 12, 1121-1131.

[40]

S. Kolyada and L. Snoha, (1992), On ω-limit sets of triangular maps, Real Analysis Exchange 18, 115-130.

[41]

S. Kolyada and Ľ. Snoha, (1996), Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4, 205-233.

[42]

S. Kolyada, Ľ. Snoha and S.Trofimchuk, (2004), On Minimality of Nonautonomous Dynamical Systems, Nonlinear Oscillations, 7, No 1, 83-89.

[43]

S. Kolyada, Ľ. Snoha and S.Trofimchuk, (2001), Noninvertible minimal maps, Fundamenta Mathematicae, 168, No 2, 141-163.

[44]

S. Kolyada, M. Misiurewicz and Ľ. Snoha, (1999), Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fundamenta Mathematicae, 160, No 2, 161-181.

[45]

M.R.S. Kulenović and G. Ladas, (2001), Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/CRC.

[46]

C. Lopesino, F. Balibrea-Iniesta, S. Wiggins and A.M. Mancho, (2015), The Chaotic Saddle in the Lozi Map, Autonomous and Nonautonomous Versions, International Journal of Bifurcation and Chaos, 25, No 13, 1550184 [18 pages].

[47]

J. Moser, (1973), Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (AM-77), Princeton University Press, Princeton.

[48]

J. Rubió-Massegú and V. Mañosa, (2007), Normal forms for rational difference equations with applications to the global periodicity problem, Journal of Mathematical Analysis and Applications, 332, No 2, 896-918.

[49]

H. Sedaghat, (2003), Nonlinear Difference Equations. Theory and Applications to Social Science Models, Springer-Verlag, Berlin.

[50]

Y. Shi, L. Zhang, P. Yu and Q. Huang, (2015),Chaos in Periodic Discrete Systems, International Journal of Bifurcation and Chaos, 25, No 1, 1550010 [21 pages].

[51]

S. Stević, (2013), Domains of undefinable solutions of some equations and systems of difference equations, Applied Mathematics and Computation, 219, No 24, 11206-11213.

[52]

L. Szała, (2015), Chaotic behaviour of uniformly convergent non-autonomous systems with randomly perturbed trajectories, Journal of Difference Equations and Applications, 21, No 7, 592-605.

[53]

J. Wright, (2913), Periodic systems of population models and enveloping functions, Computers & Mathematics with Applications, 66, No 11, 2178-2195.

[54]

S. Wiggins, (2003), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer.

[55]

W.-B. Zhang, (2006), Discrete Dynamical Systems, Bifurcations and Chaos in Economics, Elsevier Science, Amsterdam.

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