Open access peer-reviewed chapter

Chaos on Set-Valued Dynamics and Control Sets

Written By

Heriberto Román-Flores and Víctor Ayala

Submitted: April 25th, 2017 Reviewed: November 6th, 2017 Published: March 28th, 2018

DOI: 10.5772/intechopen.72232

From the Edited Volume

Chaos Theory

Edited by Kais A. Mohamedamen Al Naimee

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Abstract

The aim of this chapter is threefold. First, we show some advances in complexity dynamics of set-valued discrete systems in connection with the Devaney’s notion of chaos. Secondly, we start to explore some relationships between control sets for the class of linear control systems on Lie groups with chaotic sets. Finally, through several open problems, we invite the readers to give a contribution to this beauty theory.

Keywords

  • chaos
  • set-valued maps
  • dynamic
  • Devaney
  • control sets

1. Introduction

Relevant classes of real problems are modelled by a discrete dynamical system

xn+1=fxn,n=0,1,2,E1

where Xd is a metric space and f:XX is a continuous function. The basic goal of this theory is to understand the nature of the orbit Oxf=fnx/n=012 for any state xX, as n becomes large and, in general this is a hard task. The study of orbits says us how the initial states are moving in the base space X and, in many cases, these orbits present a chaotic structure. In 1989 in [1], Devaney isolates three main conditions which determine the essential features of chaos.

Definition 1 Let X be a metric space and f:XX a continuous map. Hence, f.

  1. is transitive if for any couple of non-empty open subsets U and V of X there exists a natural number k such that fkUV.

  2. is periodically dense if the set of periodic points of f is a dense subset of X.

  3. has sensitive dependence on initial conditions if there is a positive number δ (a sensitivity constant) such that for every point xX and every neighbourhood N of x there exists a point yN and a non-negative integer number n such that dfnxfnyδ.

Next, we mention a remarkable characterisation of transitive maps. In fact, as a consequence of the Birkhoff Transitivity Theorem (see [2] for details), it is possible to prove.

Proposition 2 Let X be a complete metric space which is also perfect (closed and without isolated points). If f:XX is continuous, then f is transitive if and only if there exists at least one orbit Oxf dense in X.

Remark 3 Also, other concepts very useful in this work are the following: i) f is weakly mixing iff for any non-empty open subsets U and V of X there exists a natural number k such that fkUV and fkVV. ii) f is mixing iff given two non-empty open subsets U and V of X there exists a natural number k such that fnUV for all nk. iii) f is exact iff given a non-empty open subsets U there exists a natural number k such that fkU=X. It is clear that f exact f mixing f weakly mixing f transitive.

It is worth to point out that sensitivity dependence on initial conditions was widely understood as being the central idea in chaos for many years. However, in a surprising way, Banks et al. has proved that transitivity and periodically density imply sensitivity dependence (for details see [3]). Furthermore, for continuous functions on real intervals, Vellekoop and Berglund in [4] show that transitivity by itself is sufficient to get chaos. This last result is not necessarily true in other type of metric spaces (see Example 4.1 in [5]).

However, sometimes we need to know information about the collective dynamics, i.e. how are moved subsets of X via iteration or dynamics induced by f. For example, if X denotes an ecosystem and xX, then, by using radio telemetry elements, we can obtain information about the movement of x in the ecosystem X. In this form, it is possible to build an individual displacement function f:XX. Of course, this function could be chaotic or not. Eventually, we could also be interested to get information about the collective dynamics induced by f, means, to follow the dynamics of a group of individuals. Thus, in a natural way the following question appears: what is the relationship between individual and collective dynamics? This is the main topic of this chapter.

Given the system (1), consider the set-valued discrete system associated to f defined by

An+1=f¯An,n=0,1,2,E2

where f¯ is the natural extension of f to the metric space KXH of the non-empty compact subsets of X endowed with the Hausdorff metric H induced by the original distance d of X.

In a more general set up, this work is strictly related with the following fundamental question: what is the relationship between individual and collective chaos?

As a partial response to this question, in this chapter we search the transitivity of a continuous function f on X in relation to the transitivity of its extension f¯ to KX. Our main result here establishes that f¯ transitive implies f transitive. That is to say, collective chaos implies individual chaos under the dynamics of f¯.

On the other hand, we propose a new approach to this problem: to study the dynamics induced by f on the subextension KcX of KX. Precisely, on the class of non-empty compact-convex subsets of X. We prove that the induced dynamics is less chaotic than the original one!

Finally, we mention that some relevant problems in the theory of control systems can be also approached by the theory of set-valuated map. In fact, to any initial state x of the system, one can associate its reachable set Ax. In other words, Ax contains all the possible states of the manifold that starting from x you can reach in non-negative time by using the admissible control functions U of the system. The aim of this section is twofold. First of all, to apply to the class of linear control systems on Lie groups, the existent relationship between control sets of an affine control system Σ on a Riemannian manifold M with chaotic sets of the shift flow induced by Σ on M×U, [6]. In particular, we are looking for the consequences of this relation on the controllability property. At the very end, we propose a challenge to the readers to motivate the research on this topic through some open problem relatives to the mentioned relationship.

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2. Preliminaries

In this section, we mention some notions and fundamental results we use through the chapter.

2.1. Extensions

If Xd is a metric space and f:XX continuous, then we can consider the space KXH of all non-empty and compact subsets of X endowed with the Hausdorff metric induced by d and f¯:KXKX,f¯A=fA}, the natural extension of f to KX. Also, we denote by KcX=AKX/Aisconvex. If AKX we define the “ϵ -dilatation of A” as the set NAϵ=xX/dxA<ϵ, where dxA=infaAdxa.

The Hausdorff metric on KX is given by

HAB=infϵ>0/ANBϵandBNAϵ.

We know that KXH is a complete (separable, compact) metric space if and only if Xd is a complete (separable, compact) metric space, respectively, (see [3, 7, 8]).

Also, if AKX, the set BAϵ=BKX/H(AB)<ϵ denotes the ball centred in A and radius ϵ in the space KXH.

Furthermore, given a continuous function IdfId on a real interval I, we also consider the extension KcIHf¯cKcIH, where f¯c is the restriction f¯KcI.

2.2. Baire spaces

In this section, we review some properties of Baire spaces.

Definition 4 A topological space X is a Baire space if for any given countable family of closed sets An:nN covering X, then intAn for at least one n.

Definition 5 In any Baire space X,

  1. DX is called nowhere dense if intclD=.

  2. Any countable union of nowhere dense sets is called a set of first category.

  3. Any set not of first category is said to be of second category.

  4. The complement of a set of first category is called a residual set.

Remark 6 It is important to note that:

  1. Any complete metric space is a Baire space.

  2. Every residual set is of second category in X.

  3. Every residual set is dense in X.

  4. The complement of a residual set is of first category.

  5. If B is of first category and AB, then A is of first category.

(For details, see [8, 9, 10])

In particular, if X=I is an interval, then CX and CXR, endowed with the respective supremum metrics, are Baire spaces.

In a Baire space X, we say that “most elements of X” verify the property (P) if the set of all xX that do not verify property (P) is of first category in X. In this form, sets of second category can be regarded as “big” sets. A relevant area of the real analysis is to estimate the “size” of some sets associated to a continuous interval function f such as the set Pf of periodic points of f, or the set Ff of fixed points of f. Typically, continuous interval functions have a first category set of periodic points (see [11]) and, in particular, a first category set of fixed points. It has also been recently proved that a typical continuously differentiable interval function has a finite set of fixed points and a countable set of periodic points (see [12] and references therein). It is also well-known that the class of nowhere differentiable functions NDI is a residual set in CI (see [13, 14]). Also, a special class of functions in CI is the class CNLI of all continuous functions whose graphs “cross no lines” defined in a negative way as follows (see [10]):

Definition 7 Let f:abab a continuous map and L:RR a function whose graph is a straight line. We say that L crosses f (or f crosses L) if there exists x0ab and δ>0 such that fx0=Lx0 and either.

(a) Lxfx for all xx0δx0ab and Lxfx for all xx0x0+δab; or.

(b) Lxfx for all xx0δx0ab and new Lxfx for all xx0x0+δab.

The following result can be found in [10]:

Theorem 8 ([10]) The set CNLI=fCI/fcrossesnolines is residual in CI.

The set CNLI will play an important role in the next sections.

2.3. The dynamics of control theory

In Section 7, we propose some challenges through the relationship between the notion of chaotic sets in the Devaney sense and control sets for the class of Linear Control Systems on Lie Groups, [15]. In particular, we explicitly show some results concerning the controllability property in terms of chaotic dynamics.

In the sequel, we follow the relevant book The Dynamics of Control by Colonius and Kliemann, [6]. Let M be a d dimensional smooth manifold. By an affine control system Σ in M, we understand the family of ordinary differential equations:

Σ:ẋt=Xxt+j=1mujtYjxt,u=u1umUE3

where X,Yj, j=0,1,,m are arbitrary C vector fields on M. The set ULRΩRm is the class of restricted admissible control functions where ΩRm with 0intΩ, is a compact and convex set.

Assume Σ satisfy the Lie algebra rank condition, i.e.

foranyxMSpanLAXY1Ymx=d.

Of course, LA means the Lie algebra generated by the vector fields through the usual notion of Lie bracket. Furthermore, the ad -rank condition for Σ is defined as follows:

foranyxMSpanadiYj:j=1mandi=01x=d.

For each uU and each initial value xM, there exists an unique solution φtxu defined on an open interval containing t=0, satisfying φ0xu=x. Since we are concerned with dynamics on Lie Groups, without loss of generality we assume that the vector fields X, Y1,,Ym are completes. Then, we obtain a mapping Φ satisfying the cocycle property

Φ:R×M×UM,txuΦtxuandΦt+sxu=ΦtΦsxuΘsu

for all t,sR, xM, uU. Where, for any tR, the map Θt is the shift flow on U defined by Θsutut+s. Hence, Φ is a skew-product flow. The topology here is given by the product topology between the topology of the manifold and the weak* topology on U.

It turns out the following results.

Lemma 9 [6] Consider the set U equipped with the weak* topology associated to LRRm=(L1RRm as a dual vector space. Therefore,

  1. Ud is a compact, complete and separable metric space with the distance given by

    du1u2=n=112nR<u1tu2tvnt>dt1+R<u1tu2tvnt>dt.

    Here, vn:nNL1RRm is a dense set of Lebesgue integrable functions.

  2. The map Θ :R×UU defines a continuous dynamical systems on U. Its periodic points are dense and the shift is topologically mixing (and then topologically transitive).

  3. The map Φ defines a continuous dynamical system on M×U.

On the other hand, the completely controllable property of Σ, i.e. the possibility to connect any two arbitrary points of M through a Σ-trajectory in positive time, is one of the most relevant issue for any control system. But, few systems have this property. A more realistic approach comes from a Kliemann notion introduced in [16].

Definition 10 A non-empty set CM is called a control set of (3) if.

  1. for every xM there exists uU such that φtxu:t0C

  2. for every xC, CclAx

  3. C is maximal with respect to the properties i and ii.

Ax denotes the states that can be reached from x by Σ in positive time and cl its closure

Ax=yM:uUandt>0withy=φtxu.

Moreover, for an element xM, the set of points that can be steered to x through a Σ-trajectory in positive time is denoted by

Ax=τ>0yM:uUe=φτ,ux.

Finally, we mention that the Lie algebra rank condition warranty that the system is locally accessible, which means that for every τ>0,

intAτxandintAτxarenon empty,foranyxM.
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3. f¯ transitive implies f transitive

As we explain, in terms of the original dynamics and its extensions a natural question arises: what are the relations between individual and collective chaos? As a partial response to this question, in the sequel, we show that the transitivity of the extension f¯ implies the transitivity of f. For that, we need to describe some previous results.

Lemma 11 [5] Let A be a non-empty open subset of X. If KKX and KA, then there exists ϵ>0 such that NKϵA..

Definition 12 Let AX be. Then the extension of A to KX is given by eA=KKX/KA.

Remark 13 eA=A=..

Lemma 14 [5] Let AX be, A, an open subset of X. Then, eA is a non-empty open subset of KX.

Lemma 15 [5] If A,BX, then: i) eAB=eAeB, ii) f¯eAefA, and iii) f¯p=fp¯, for every pN.

Now, we are in a position to prove the following results

Theorem 16 Let f:XX be a continuous function. Then, f¯ transitive implies f transitive.

Proof: Let A,B be two non-empty open sets in X. Due to Lemma 13, eA and eB are non-empty open sets in KX. Thus, by transitivity of f¯, there exists some kN such that

f¯keAeB=fk¯eAeB

and, from Lemma 14, we obtain

efkAeB=efkAB

which implies fkAB and, consequently, f is a transitive function.

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4. Two examples

Now we show that, in general, the converse of Theorem 15 is not true.

Example 4.1 (Translations of the circle). If λR is an irrational number and we define Tλ:S1S1 by Tλe=eiθ+2πλ, then it was shown by Devaney [1] that each orbit Tλne/nN is dense in S1 and, due Proposition 2, Tλ is transitive. Nevertheless, Tλ has no periodic points and, because Tλ is isometric, it does not exhibit sensitive dependence on initial conditions either.

If KKS1, because Tλ¯ preserves diameter, then diamK=diamTλ¯nK, for all nN.

Now, let KKS1 such that diamK=1, and let ϵ>0 sufficiently small. Then

FU=BKϵdiamF1GV=B1ϵdiamG0.

Thus, diamTλ¯nF1nN and, consequently, Tλ¯nUV= for all nN, which implies that Tλ¯ is not transitive on KS1.

Example 4.2 Define the “tent” function f:0101 as fx=2xif0x1/2 and fx=21xif1/2x1.

It is not difficult to show that f is an exact function on [0,1]. In fact, intuitively we can see that, after each iteration, the number of tent in the graphics is increasing, whereas the base of each tent is decreasing and they are uniformly distributed over the interval 01.

Thus, if U is an arbitrary non-empty open subset of 01, then U contains an open interval J and, after certain number of iterations, there exists a tent, with height equal to one, whose base is contained in J, which implies that fU=01 and, according to Remark 3, f is an exact mapping and, consequently, f is a mixing function.

The conclusions in Examples 4.1 and 4.2 come from the next result, Banks [17] in 2005.

Theorem 17 If f:XX is continuous, then the following conditions are equivalent:

i) f is weakly mixing, ii) f¯ is weakly mixing, iii) f¯ is transitive.

Hitherto, we have used the strong topology induced by the H-metric on KX. However, considering the we-topology on KX generated by the sets eA with A an open set in X, we obtain the following complementary result, see [5]:

Theorem 18 For a continuous map f:XX the following conditions are equivalent:

i) f is transitive in Xd, ii) f¯ is transitive in the we-topology.

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5. Sensitivity and periodic density of f¯

Let f:XX be a continuous function and let f¯ be its corresponding extension to the hyperspace KX. Then, the study of sensitivity of f in the base space in relation to the sensitivity of f¯ on KX has been very exhaustively analysed in the last years. Román and Chalco published the first result in this direction [18] in 2005, where the authors prove

Theorem 19 f¯ sensitively dependent implies f sensitively dependent.

Proof: If f¯ has sensitive dependence, then there exists a constant δ>0 such that for every KKX and every ϵ>0 there exists GBKϵ and nN such that HfnKfnGδ.

Now, let xX be and ϵ>0. Then, taking K=xKX, we have that there exists GBxϵ and nN such that HfnxfnG=HfnxfnGδ.

Thus, HfnxfnG=supyGdfnxfnyδ and, due to the compactness of G and the continuity of f, there exists y0G such that HfnxfnG=dfnxfny0δ.

But, GBxϵ implies GBxϵ and, consequently, y0Bxϵ. This proves that f is sensitively dependent (with constant δ).

The reverse of this theorem is not true. In fact, recently Sharma and Nagar [19] show an example where Xd is sensitive but KXH is not. Now, in order to overcome that shortcoming, the authors in [19] introduce the following notion of sensitivity:

Definition 20 (Stronger sensitivity [19]). Let f:XX be a continuous function. Then f is strongly sensitive if there exists δ>0 such that for each xX and each ϵ>0, there exists n0N such that for every nn0, there is a yX with dxy<ϵ and dfnxfny>δ.

Obviously, the notion of stronger sensitivity is more restrictive than sensitivity, and the authors in [19] obtain the following results:

Theorem 21 If f:XX is a continuous function and KXHf¯ is strongly sensitive then Xdf is strongly sensitive.

In the compact case, it is possible to obtain a characterization as follows.

Theorem 22 Let Xd be a compact metric space and f:XX a continuous function. Then KXHf¯ is strongly sensitive if and only if Xdf is strongly sensitive.

In connection with these results, recently Subrahmomian ([20], 2007) has been shown that most of the important sensitive dynamical systems are all strongly sensitive (the author here calls them cofinitely sensitive). Hence, we can say that for most cases, sensitivity is equivalent in both cases Xd and KXH. It turns out that, strongly sensitivity and sensitivity are equivalent on the class of interval functions, which implies that

Theorem 23 If f:II is a continuous function, the following conditions are equivalent.

a) Idf is sensitive, b) KIHf¯ is sensitive.

We finish this section assuming the existence of a dense set of periodic points for f¯, we have

Theorem 24 Let Xd be a compact metric space and f:XX a continuous function. If f:XX has a dense set of periodic points then f¯:KXKX has the same property.

Proof: Let KKX and ϵ>0. Then there exists a ϵ/2-net covering K, That is to say, there are x1,,xp in K such that KBx1ϵ/2Bxpϵ/2. Because f has periodic density, there are yiX and niN such that:

yiBxiϵ/2,i=1,,pandfniyi=yi,i=1,,p.

Now, take G=y1yp. By construction, we have HKG<ϵ and, moreover, fn1n2npyi=yi, for all i=1,,p. Therefore, fn1n2npG=G, which implies that f¯ has periodic density.

The converse of this theorem is no longer true (for a counterexample, see Banks [17]). However, to find conditions on f¯ warranting the existence of a dense set of periodic points for f is a very hard problem which still remains open.

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6. The dynamics on the KcIH extension

In the previous sections, we have studied the diagram

KXHf¯KXHXdfXdE4

and the chaotic relationships between f and f¯. However, in the setting of mathematical modelling of many real-world applications, it is necessary to take into account additional considerations such as vagueness or uncertainty on the variables. This implies the use of interval parameters and, consequently, to deal with interval systems. That is, it is necessary to consider an interval X=I and to study the following new diagram:

KcIHf¯cKcIHIdfIdE5

along with the analysis of the connection between their respective dynamical relationships. Here f¯c denotes the restriction of f¯ to KcI, the class of all compact subintervals of I. For A=ab,B=cdKcI, the Hausdorff metric can be explicitly computed as

HAB=maxacbd.E6

The aim of this section is to show that the Devaney complexity of the extension f¯c on KcI is less or equal than the complexity of f on the base space I. More precisely, f¯c is never transitive for any continuous function fCI. Also, we will show that f¯c has no dense set of periodic points for most functions fCI. Finally, we prove that f¯c has no sensitive dependence for most functions fCI.

As a motivation, we present the following examples.

Example 6.1 Consider the “tent” function f:0101 defined by

fx=2xif0x1221xif12x1.

Then it is well known that f is D-chaotic on 01 (see [1]). Moreover, because f is a mixing function on 01, then f¯ is transitive on K01 (see [17]). Also, we observe that x=23 is a fixed point of f. On the other hand, it is clear that if K is a compact and convex subset of X=01, then f¯K is also a compact and convex subset of X. Consequently, if we let Kc01 denote the class of all closed subintervals of 01, then we can consider f¯c as a mapping f¯c:Kc01Kc01. We recall that Kc01 is a closed subspace of K01 (see [21]). Now, considering the open balls B01110 and B0110 in Kc01, we have.

KB0111023K which implies 23f¯cpK,pN..

On the other hand, if FB0110, then F01/10 . Consequently, Hf¯cpKF1730 for every KB01110 and FB0110.

Therefore,

f¯cpB01110B0110=,pN.

Thus, f¯c is not transitive on Kc01.

Example 6.1 shows a function f which is transitive on the base space X=01 and f¯ is also transitive on the total extension K01, but f¯c is not transitive on the subextension Kc01.

The following example shows a function f:010,1] with a dense set of periodic points, and where the total extension of f to K01 also has a dense set of periodic points, whereas f¯c does not have a dense set of periodic points on Kc01.

Example 6.2. Let X=01 and consider the “logistic” function f:0101 defined by fx=4x1x. It is well known that f is D-chaotic on 01 (see [1]). Moreover, f is a mixing function. Thus, in particular, f has a dense set of periodic points and, therefore, f¯ also has a dense set of periodic points on the total extension K01) (see Theorem 24).

However, f¯c has no a dense set of periodic points on KcX.

In order to see this, we claim that the open ball B183818 in Kc01H does not contain periodic points of f¯c.

In fact, if K=cdB183818, then c18<18 and d38<18, which implies that 0<c<14 and 14<d<12.

Thus, we obtain that 14K34fKfKK.

On the other hand,

34fK34fnK,n2fnKK,n1

and, consequently, f¯c has no periodic points in the ball B183814Kc01H, which implies that f¯c has no dense set of periodic points on Kc01H.

Lemma 25 f¯c transitive on Kcab implies f transitive on ab.

Proof. Let U,V non-empty open subsets of X=ab. We can choose xU, yV and ϵ>0 such that BxϵU and ByϵV. Now, in Kcab consider the open balls Bxϵ and Byϵ with respect to the H-metric. Due to the transitivity of f¯c on Kcab, there exists nN such that f¯cnBxϵByϵ.

Therefore, there exists an interval JBxϵ such that f¯cnJ=fnJByϵ. However, JBxϵ and, analogously, fnJByϵ, which implies that fnBxϵByϵ and, consequently, fnUV. And f is a transitive function on ab.

It is well-known that if X=I is an interval, then most functions fCI has no dense orbits, that is to say, there exists a residual set DCI such that every function fD has no point whose orbit is dense in I (see [22]) and, consequently, most functions fCI are not transitive. From Lemma 24, we can conclude that f¯c is not transitive for most functions fCI.

The next theorem provides a stronger result.

Theorem 26 Let f:abab be continuous. Then f¯c is not transitive on Kcab.

Proof. By Schauder Theorem, f has at least one fixed point pab.

Case 1. Suppose that pab and let r=maxpabp. Without loss of generality, we can suppose that r=pa and, because a<b, it is clear that r>0.

Now, let r=bp>0 and let ϵ=r2. If we consider the open balls Babϵ,BaϵKcab, it follows that KBabϵpKpf¯nK for any nN.

On the other hand,

FBaϵHFa<ϵFa,a+ϵ].

Because r<r we get

Hf¯nKFpaϵ=rr2>0

for each KBabϵ, FBaϵ and for any nN. Thus,

f¯nB(abϵ)Baϵ=,nN.

Consequently, f¯ is not transitive on Kcab.

Case 2. Suppose that f has no fixed points in ab. From the continuity of f, we have that fx>x for all xab or fx<x for all xab. This clearly implies that f is not a transitive function, and consequently, due to Lemma 24, f¯c is not transitive on Kcab.

An important question to answer is what about the size of the set of periodic points of f¯c. It is clear that there are some functions fCI with a dense set of periodic points on I, and such that their extensions f¯c also has a dense set of periodic points on KcI (for instance, fx=x). Therefore, an analogous result to Theorem 26, but for periodic density of f¯c, cannot be obtained. However, as we will see, most functions fCI do not have an extension f¯c with a dense set of periodic points on KcI. To prove it, we need the following lemma.

Lemma 27 Let I be a compact interval in R, and f:II be a continuous function. If we suppose that f¯c has periodic density on KcI, then f has periodic density on I.

Proof. If x0I and ϵ>0 then x0KcI and, consequently, there exists KKcI and nN such that

  1. Hx0K<ϵ

  2. f¯cnK=K.

Combining a. and b. we get

dx0fnx<ϵ,forallxK.E7

Because f¯nK=fn¯K=fnK=K and fn is continuous on K then, by the Schauder’s Fixed Point Theorem, there exists xpK such that fnxp=xp. Thus, xp is a periodic point of f and, due to (7), we obtain dx0xp<ϵ. Hence, f has periodic density on I.

Theorem 28 Let I=ab be a compact interval in R. Then f¯c does not have a dense set of periodic points in KcI, for most functions fCI.

Proof. The proof is based on an exhaustive analysis of the behaviour of the fixed points of f. We connect this analysis with an adequate residual set in CI. The analysis of each fixed point of f is fundamental to decide whether the function f allows or not an extension f¯c that has a dense set of periodic points. More precisely, the behaviour of each fixed point will imply only two (mutually exclusive) options:

  1. f¯c does not have a dense set of periodic points, or.

  2. fCNLIc, which is a set of first category in CI.

Towards this end, let f:abab be a continuous function. By the Schauder’s Fixed Point Theorem, f has at least one fixed point pab. The proof is divided in.

Case1. f has no fixed points in ab.

In this case, we have the following three subcases:

1i) p=a is the unique fixed point of f.

We have, either

fx>x,xabx<fx<f2x<<fnx<,or
fx<x,xabx>fx>f2x>>fnx>.

In both cases it follows that f has no periodic points in ab.

1ii) p=b is the unique fixed point of f.

This case is analogous to the case 1i).

1iii) p=a and p=b are the unique fixed points of f.

This case is also analogous to the cases 1i) and 1ii).

Therefore, in case 1 the function f does not have a dense set of periodic points in ab. Due to Lemma 24, f¯c does not have a dense set of periodic points in Kcab.

Case2. f has at least one fixed point pab.

We have the following subcases:

2i) qab,qp such that fq=p.

Without loss of generality, suppose that qap. Then, taking 0<ϵ<minqa2pq2, we can consider the open ball Bqϵq+ϵϵ in the space Kcab. If J=cdBqϵq+ϵϵ, from (6) we have

cqϵ<ϵanddq+ϵ<ϵ

which implies that a<c<q and q<d<p and, consequently, qJ whereas pJ. Thus,

qJfq=pfJfJJ.E8

On the other hand, pfJ implies that

pfnJ,n2fnJJ,n2,E9

and, consequently, f¯c has no periodic points in the ball Bqϵq+ϵϵKcabH, which implies that f¯c does not have a dense set of periodic points on KcabH.

2ii) q=a,qp, is the unique point such that fa=p.

Without loss of generality, we can suppose that fx>p, for all xap.

Now, in addition to hypothesis 2ii), we have two subcases:

2iia1) f does not cross the line y=p and fx>p for all xap.

In this situation, fxp for all xa,b]. Thus, choosing qap and 0<ϵ<maxqa2pq2, we can consider the open ball Bqϵ to have

K=cdBqϵKap.E10

From our hypothesis, we obtain

fnz>p,zK,nN,E11

which implies that fnKK, nN. Consequently, f¯c has no periodic points in the ball Bqϵ. In other words, f¯c does not have a dense set of periodic points in KcI.

2iia2) f does not cross the line y=p and fx<p for all xap.

In this case, fxp for all xab. Thus, choosing qpb and 0<ϵ<maxqp2bq2, we can consider the open ball Bqϵ to obtain

K=cdBqϵKpb.E12

Again, from our hypothesis, we get

fnz<p,zK,nN,E13

which implies that fnKK, nN and, consequently, f¯c has no periodic points in the ball Bqϵ. In other words, f¯c does not have a dense set of periodic points in KcI.

2iib) f crosses the line y=p.

It is clear that, in this case, fCNLIc which, due to Theorem 8 and Remark 6, is a set of first category in Cab.

2iii) q=b,qp, is the unique point such that fb=p.

This case is analogous to case 2ii) and, consequently, if f does not cross the line y=p then f¯c does not have a dense set of periodic points in KcI, whereas if f crosses the line y=p, then fCNLIc.

2iv) q1=a and q2=b, q1,q2p, are the unique points such that fa=fb=p.

In this case, we have the following subcases:

2iva1) f does not cross the line y=p and fx>p and fx>p for all xab\p.

This case is analogous to the case 2iia1) and the same is true for 2iva2) when f does not cross the line y=p and fx<p for all xab\p which is analogous to the case 2iia2) Finally, there only remains two subcases:

2ivb1) f crosses the line y=p and fx>p in ap and fx<p in pb, and.

2ivb2) f crosses the line y=p and fx<p in ap and fx>p in pb.

It is clear that in both cases fCNLIc.

Thus, as a direct consequence of the analysis of the behaviour of the set of fixed points of f, it turns out that the unique cases in which f could have an extension f¯c with a dense set of periodic points on KcI are when there exists a fixed point p of f such that f crosses the line y=p at x=p. In other words, we obtain

HDSI=fCI/f¯chasadensesetofperiodicpointsinKcIHDSICNLIc,

But, CNLI is a residual set in CI, therefore from Remark 6, we conclude that HDSI is of first category in CI. Equivalently, f¯c does not have a dense set of periodic points, for most functions fCI, which ends the proof.

Finally based on the following result,

Theorem 29 ([23]) For most functions fCI, the set of all points where f is sensitive is dense in the set of all periodic points of f.

we show an analogous result for the sensitivity property, as follows.

Theorem 30 For most functions fCI, the extension f¯cCKcI is not sensitive.

Proof. This is a direct consequence of Theorem 28 and Theorem 29.

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7. Control sets of linear systems and chaotic dynamics

The aim of this section is twofold. First of all, to start to apply to the class of linear control systems on Lie groups, the existent relationship between control sets of an affine control system Σ on a Riemannian manifold M with chaotic sets of the shift flow induced by Σ on M×U, [6]. In particular, we are looking for the consequences of this relation on the controllability property The second part is intended to motivate the research on this topic to writing down some open problems relatives to this relationship.

7.1. Linear control systems on lie groups

Let G be a connected d dimensional Lie group with Lie algebra g. A linear control system ΣL on G is an affine system determined by

ΣL:ẋt=Xxt+j=1mujtYjxt,u=u1umUE14

where X is linear, that is, its flow XttR is a one-parameter group of G-automorphism, the control vectors Yj, j=1,,m are invariant vector fields, as elements of g. The restricted class of admissible control U is the same as before.

Certainly, the drift vector field X is complete and the same is true for every invariant vector field Yj, j=1,,m. As usual, we assume that ΣL satisfy the Lie algebra rank condition, i.e.

foranyxMSpanLAXY1Ymx=d.

The system is said to be controllable if Ae=A is G.

The class of systems ΣL is huge and contains many relevant algebraic systems as the classical linear and bilinear systems on Euclidean spaces [6], and the class of invariant systems on Lie groups, [24]. Furthermore, according to the Jouan Equivalence Theorem [25], ΣL is also relevant in applications. It approaches globally any affine non-linear control system Σ on a Riemannian manifold when the Lie algebra of the dynamics of Σ is finite dimensional.

One can associate to X a derivation D of g defined by DY=XYe,Yg. Indeed, the Jacobi identity shows DXY=DXY+XDY is in fact a derivation. The relation between φt and D is given by the formula

φtexpY=expetDY,foralltR,Yg.

Consider the generalised eigenspaces of D defined by

gα=Xg:DαnX=0forsomen1

where αSpecD. Then, gαgβgα+β when α+β is an eigenvalue of D and zero otherwise. Therefore, it is possible to decompose g as g=g+g0g, where

g=g+g0g,where
g+=α:Reα>0gα,g0=α:Reα=0gαandg=α:Reα<0gα.

Actually, g+,g0,g are Lie algebras and g+, g are nilpotent. Denote by G+, G and G0 the connected and closed Lie subgroups of G with Lie algebras g+, g and g0 respectively.

Despite the fact that for an invariant system the global controllability property is local, this class has been studied for more than 50 years, see [24] and the references there in. The important point to note here is: for an invariant system the reachable set from the identity is a semigroup. However, in [26] the authors show that this is not the case for a linear system which turns the problem more complicated. Therefore, we would like to explore the mentioned connection between control sets and the Devaney and Colonius-Kliemann ideas. This section is the starting point for the ΣL class. We begin with a fundamental result.

Theorem 31 Assume the system ΣL satisfy the Lie algebra rank condition. Therefore, there exists a control set

Ce=clAeAe

which contains the identity element e in its interior. Here, Ae is the set of states of G that can be sent by ΣL to e in positive time.

For a proof in a more general set up, see [6].

Recently, we were able to establish some algebraic, topological, and dynamical conditions on ΣL to study uniqueness and boundness of control sets and it consequences on controllability . But, the state of arts is really far from being complete. In order to approach this problem for ΣL, as in [27] we assume here that G has finite semisimple centre, i.e. all semisimple Lie subgroups of G have finite center. We notice that any nilpotent and solvable Lie group, and any semisimple Lie group with finite centre has the finite semisimple centre property. But also, the product between groups with finite semisimple centre have the same property. We also assume that A is open. This is true if for example, the system satisfy the ad -rank condition. About the uniqueness and boundness of control sets of a linear systems, we know few things [27].

Theorem 32 Let ΣL a linear control system on the Lie group G.

1. If G= GG0G+ is decomposable, Ce is the only control set with non-empty interior. In particular, this is true for any solvable Lie group.

2. Suppose that G is semisimple or nilpotent, it turns out that

ifclAG,clAG+andG0arecompact setsCis bounded.

3. If G is a nilpotent simply connected Lie group, it follows that

Cis boundedclAGandclAG+arecompact sets andDis hyperbolic.

Furthermore, it is possible to determine algebraic sufficient conditions to decide when C is bounded. Actually, in a forthcoming paper we show that

Theorem 33 Let ΣL be a linear control system on the Lie group G. Assume that G is decomposable and G+,0 is a normal subgroup of G. Hence, clGA is compact.

A analogous result is obtained for G+A assuming that G,0 is normal. Of course, G+,0 is a normal subgroup of G if and only if g+g0 is an ideal of g. On the other hand,

g+g0andg+g0areideals ofgg+g0=0andg+gg0.

7.2. Chaos and control sets

We start with an explicitly relationship between chaotic subsets of M×U and the Σ-control sets.

Theorem 34 Let M×U and the canonical projection πM:M×U M. Hence,

πM=xM:there existsuUwithxu

is compact and its non-void interior consists of locally accessible points. Then,

  1. is a maximal topologically mixing set if and only if there exists a control C such that

    =clxuM×U:φ(txu)intCfor everytR

    In this case, C is unique and intC=intπM, clC=clπM.

  2. The periodic points of Φ are dense in .

  3. Φ restrict to is topologically mixing, topologically transitive and has sensitive dependence on initial conditions.

In order to apply this fundamental result for a non-controllable linear control system, the boundness property of its control set is crucial. Let us assume that C is a bounded control set with non-empty interior of ΣL and define =πM1C=clC×UC where

UC=uU:existxCwithφtxuintCfor everytR.

The Lie group G is finite dimensional and UC is a closed subset of the compact class of admissible control ULRΩRm with the weak* topology. Since the projection is a continuous map, it turns out that πM is compact and , C are uniquely defined.

On the other hand, we are assuming that ΣL satisfy the Lie algebra rank condition, hence the system is locally accessible at any point of the state space. Therefore, we are in a position to apply Theorem 32, first, for some classes of controllable linear systems, as follows.

Theorem 35 Let ΣL be a linear control system on a Lie group G. Any condition.

  1. G is compact, or

  2. G is Abelian, or

  3. G has the finite semisimple centre property and the Lyapunov spectrum of D is 0 implies that the skew flow Φ is chaotic in G×U.

Proof. Under the hypothesis in 1, any control set is bounded. Furthermore, if G is compact, the Lie algebra rank condition assures that the linear control system ΣL is controllable on G, see [15]. Hence, Φ is topologically mixing, topologically transitive and the periodic points of Φ are dense in G×U, which give us the desired conclusion.

It is well known that any Abelian Lie group is a product G=Rm×Tn between the Euclidean space Rm and the torus Tn=S1×× S1 (n times), for some m,nN. In this case, ΣL is also controllable [15]. Indeed, since the automorphism group of Tn is discrete, any linear vector field on the torus is trivial. But, we are assuming the Lie algebra condition on G which coincides with the Kalman rank condition in Rm. And, on the compact part, we apply 1. Hence, the skew flow Φ is chaotic in G×U. In fact, πM1C=G×U and the hypothesis of the compacity on the projection in Theorem 32 is not necessary for the lifting, see Proposition 4.3.3 in [6]. The same is true for 3. Actually, for this more general set up, we recently prove that the system is also controllable, [28, 29].

In the sequel, we use some topological properties of Ce to translate these properties to its associated chaotic set , as follows.

Theorem 36 Let ΣL be a linear control system on a Lie group G. It holds.

  1. If G= GG0G+ there exists one and only one chaotic set = πM1Ce in G×U given by

    =clxu:φ(txu)intCefor everytRM×U

  2. If G is nilpotent and D has only eigenvalues with non-positive real parts, then the only chaotic set = πM1C in G×U is closed

  3. If G is nilpotent and D has only eigenvalues with non-negative real parts then the only chaotic set = πM1C in G×U is open

Proof. If G is decomposable, we know that there exists just one control set: the one which contains the identity element. Hence, = πM1Ce is the only chaotic set of Φ on G×U which proves 1. To prove 2 and 3, we observe that the Lyapunov spectrum condition on the derivation D associated to the drift vector field X is equivalent to the control set Ce be closed or open, respectively. Since the projection πG:G×U G is a continuous map with the weak* topology on U, the lifting πG1Ce is both closed and open, respectively.

7.3. Challenge

In this very short section, we would like to invite the readers to work on the relationship between chaotic and control sets. We suggest to go further in this research through some specific examples on low-dimensional Lie groups. For that, we give some relevant information about two groups of dimension three: the simply connected nilpotent Heisenberg Lie group H and the special linear group SL2R. We finish by computing an example on H.

1. The nilpotent Lie algebra h= R3+, has the basis E12E23E13 with E12E23=E13. Here, Eij denotes the real matrix of order 3 with zero everywhere except 1 in the position ij. The associated Heisenberg Lie group has the matrix representation

G=g=1xz01y001:xyzRφ:gxyzR3.

As invariant vector fields, the basis elements of g has the following description

E12=x,E23=y+xzandE13=z.

The canonical form of any g-derivation is given by

D=ad0be0cfa+e:a,b,c,d,e,fR.

Any linear vector field X reads as

Xxyz=ax+dyx+bx+eyy+b2x2+d2y2+cx+fy+a+ezz.

2. The vector space g=sl2R of all real matrices of order three and trace zero is the Lie algebra of the Lie group G=SL2R=det11. Let us consider the following generators of g: Y1=0110, Y2=0100 and Y3=1001. The Lie group G is semisimple, then any g derivation is inner which means that there exists an invariant vector field Y such that adY represents . Thus, a general form of a derivation reads as

αadY1+βadY2+γadY3.

Example 7.1 On the Heisenberg Lie group, consider the system

ΣL:gt=Xgt+u1tE12gt+u2tE23gt,u=u1u2UE15

where X is determined by the derivation D=adE12=E32. Since the group is nilpotent, it has the semisimple finite centre property. The Lyapunov spectrum of D reduces to zero. Finally, the reachable set from the identity A is open. In fact, the ad-rank condition is obviously true because DE12=E13. It turns out that the skew flow Φ is chaotic in H×U.

References

  1. 1. Devaney RL. An Introduction to Chaotic Dynamical Systems. Redwood City: Addison-Wesley; 1989
  2. 2. Robinson C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. New York: CRC Press; 1999
  3. 3. Banks J, Brooks J, Cairns G, Stacey P. On the Devaney’s definition of chaos. The American Mathematical Monthly. 1992;99:332-334
  4. 4. Vellekoop M, Berglund R. On intervals, transitivity=chaos. The American Mathematical Monthly. 1994;101:353-355
  5. 5. Román-Flores H. A note on transitivity in set-valued discrete systems. Chaos, Solitons & Fractals. 2003;17:99-104
  6. 6. Colonius F, Kliemann C. The Dynamics of Control, Systems & Control: Foundations & Applications. Boston, MA: Birkäuser Boston, Inc.; 2000
  7. 7. Román-Flores H, Barros LC, Bassanezi RCA. Note on the Zadeh’s extensions. Fuzzy Sets and Systems. 2001;17:327-331
  8. 8. Román-Flores H, Chalco-Cano Y. A note on dynamics of interval extensions of interval function. 2015 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS) held jointly with 2015 5th World Conference on Soft Computing (WConSC), 2015
  9. 9. Dugundji J. Topology. Boston: Allyn and Bacon Inc.; 1966
  10. 10. Thomson B, Bruckner J, Bruckner A. Elementary Real Analysis. Prentice Hall: Upper Saddle River; 2001
  11. 11. Agronski S, Bruckner A, Laczkovich M. Dynamics of typical continuous functions. Journal of the London Mathematical Society. 1989;40:227-243
  12. 12. Alikhani-Koopaei A. On the set of fixed points and periodic points of continuously differentiable functions. International Journal of Mathematics and Mathematical Sciences. 2013. Article ID: 929475
  13. 13. Banach S. Über die Baire’sche Kategorie gewisser Funktionenmengen. Studia Math. 1931;3:174-179
  14. 14. Mazurkiewicz S. Sur les fonctions non-dérivables. Studia Mathematica. 1932;3:92-94
  15. 15. Ayala V, Tirao J. Linear control systems on lie groups and controllability. American Mathematical Society, Series: Symposia in Pure Mathematics. 1999;64:47-64
  16. 16. Kliemann W. Analysis of nonlinear stochastic systems. Schiehlen W, Wedig W, eds. In: Analysis and Estimation of Stochastic Mechanical Systems: Springer-Verlag; 1988
  17. 17. Banks J. Chaos for induced hyperspace maps. Chaos, Solitons & Fractals. 2005;25:681-685
  18. 18. Román-Flores H, Chalco-Cano Y. Robinson’s chaos in set-valued discrete systems. Chaos, Solitons & Fractals. 2005;25:33-42
  19. 19. Sharma P, Nagar A. Inducing sensitivity on hyperspaces. Topology and its Applications. 2010;157:2052-2058
  20. 20. Subramonian TK. Stronger forms of sensitivity for dynamical systems. Nonlinearity. 2007;20:2115-2126
  21. 21. Klein E, Thompson A. Theory of Correspondences. New York: Wiley-Interscience; 1984
  22. 22. Jones SH. Topical survey: Applications of the Baire category theorem. Real Analysis Exchange. 1997;23:363-394
  23. 23. Bernardes N. On the predictability of discrete dynamical systems II. Proceedings of American Mathematical Society. 2005;133:3473-3483
  24. 24. Sachkov Y. Survey on controllability of invariant control systems on solvable lie groups. AMS Proceedings of Symposia in Pure Mathematics. 1999;64:297-317
  25. 25. Jouan P. Equivalence of control systems with linear systems on lie groups and homogeneous spaces. ESAIM: Control Optimization and Calculus of Variations. 2010;16:956-973
  26. 26. Ayala V, San Martin L. Controllability properties of a class of control systems on lie groups. Lecture Notes in Control and Information Sciences. 2001;258:83-92
  27. 27. Ayala V, Da Silva A, Zsigmond G. Control sets of linear systems on lie groups. Nonlinear Differential Equations and Applications. 2017;24:8, 99.1-99.15
  28. 28. Ayala V, Da Silva A. Controllability of linear systems on Lie groups with finite semisimple center. SIAM Journal on Control and Optimization. 2017;55(2):1332-1343
  29. 29. Ayala V, Da Silva A. Control systems on flag manifolds and their chain control sets. Discrete and Continuous Dynamical Systems. 2017;37(5):2301-2313

Notes

  • Partially supported by Conicyt, Chile through Regular Fondecyt Projects no. 1151159 and no. 1150292 respectively.

Written By

Heriberto Román-Flores and Víctor Ayala

Submitted: April 25th, 2017 Reviewed: November 6th, 2017 Published: March 28th, 2018