Open access peer-reviewed chapter

Chaos on Set-Valued Dynamics and Control Sets

By Heriberto Román-Flores and Víctor Ayala

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

DOI: 10.5772/intechopen.72232

Downloaded: 286

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 Xdis a metric space and f:XXis a continuous function. The basic goal of this theory is to understand the nature of the orbit Oxf=fnx/n=012for any state xX,as nbecomes 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 Xand, 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 Xbe a metric space and f:XXa continuous map. Hence, f.

  1. is transitive if for any couple of non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fkUV.

  2. is periodically dense if the set of periodic points of fis 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 xXand every neighbourhood Nof xthere exists a point yNand a non-negative integer number nsuch 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 Xbe a complete metric space which is also perfect (closed and without isolated points). If f:XXis continuous, then fis transitive if and only if there exists at least one orbit Oxfdense in X.

Remark 3 Also, other concepts very useful in this work are the following: i) fis weakly mixing iff for any non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fkUVand fkVV. ii) fis mixing iff given two non-empty open subsets Uand Vof Xthere exists a natural number ksuch that fnUVfor all nk. iii) fis exact iff given a non-empty open subsets Uthere exists a natural number ksuch that fkU=X. It is clear that fexact fmixing fweakly mixing ftransitive.

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 Xvia iteration or dynamics induced by f. For example, if Xdenotes an ecosystem and xX, then, by using radio telemetry elements, we can obtain information about the movement of xin 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 fdefined by

An+1=f¯An,n=0,1,2,E2

where f¯is the natural extension of fto the metric space KXHof the non-empty compact subsets of Xendowed with the Hausdorff metric Hinduced by the original distance dof 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 fon Xin relation to the transitivity of its extension f¯to KX.Our main result here establishes that f¯transitive implies ftransitive. 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 fon the subextension KcXof 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 xof the system, one can associate its reachable set Ax. In other words, Axcontains all the possible states of the manifold that starting from xyou can reach in non-negative time by using the admissible control functions Uof 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 Σon a Riemannian manifold Mwith chaotic sets of the shift flow induced by Σon 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.

2. Preliminaries

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

2.1. Extensions

If Xdis a metric space and f:XXcontinuous, then we can consider the space KXHof all non-empty and compact subsets of Xendowed with the Hausdorff metric induced by dand f¯:KXKX,f¯A=fA}, the natural extension of fto KX. Also, we denote by KcX=AKX/Aisconvex. If AKXwe define the “ϵ-dilatation of A” as the set NAϵ=xX/dxA<ϵ, where dxA=infaAdxa.

The Hausdorff metric on KXis given by

HAB=infϵ>0/ANBϵandBNAϵ.

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

Also, if AKX, the set BAϵ=BKX/H(AB)<ϵdenotes the ball centred in Aand radius ϵin the space KXH.

Furthermore, given a continuous function IdfIdon a real interval I, we also consider the extension KcIHf¯cKcIH, where f¯cis the restriction f¯KcI.

2.2. Baire spaces

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

Definition 4 A topological space Xis a Baire space if for any given countable family of closed sets An:nNcovering X, then intAnfor at least one n.

Definition 5 In any Baire space X,

  1. DXis 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 Bis of first category and AB, then Ais of first category.

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

In particular, if X=Iis an interval, then CXand 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 xXthat 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 fsuch as the set Pfof periodic points of f, or the set Ffof 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 NDIis a residual set in CI(see [13, 14]). Also, a special class of functions in CIis the class CNLIof all continuous functions whose graphs “cross no lines” defined in a negative way as follows (see [10]):

Definition 7 Let f:ababa continuous map and L:RRa function whose graph is a straight line. We say that Lcrosses f(or fcrosses L) if there exists x0aband δ>0such that fx0=Lx0and either.

(a) Lxfxfor all xx0δx0aband Lxfxfor all xx0x0+δab; or.

(b) Lxfxfor all xx0δx0aband new Lxfxfor all xx0x0+δab.

The following result can be found in [10]:

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

The set CNLIwill 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 Mbe a ddimensional smooth manifold. By an affine control system Σin M, we understand the family of ordinary differential equations:

Σ:ẋt=Xxt+j=1mujtYjxt,u=u1umUE3

where X,Yj, j=0,1,,mare arbitrary Cvector fields on M.The set ULRΩRmis the class of restricted admissible control functions where ΩRmwith 0intΩ,is a compact and convex set.

Assume Σsatisfy the Lie algebra rank condition, i.e.

foranyxMSpanLAXY1Ymx=d.

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

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

For each uUand each initial value xM, there exists an unique solution φtxudefined 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,,Ymare completes. Then, we obtain a mapping Φsatisfying the cocycle property

Φ:R×M×UM,txuΦtxuandΦt+sxu=ΦtΦsxuΘsu

for all t,sR, xM, uU.Where, for any tR, the map Θtis the shift flow on Udefined by Θsutut+s.Hence, Φis 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 Uequipped with the weak* topology associated to LRRm=(L1RRmas a dual vector space. Therefore,

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

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

Here, vn:nNL1RRmis a dense set of Lebesgue integrable functions.

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

  • The map Φdefines 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 Mthrough 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 CMis called a control set of (3) if.

    1. for every xMthere exists uUsuch that φtxu:t0C

    2. for every xC, CclAx

    3. Cis maximal with respect to the properties iand ii.

    Axdenotes the states that can be reached from xby Σin positive time and clits closure

    Ax=yM:uUandt>0withy=φtxu.

    Moreover, for an element xM, the set of points that can be steered to xthrough 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.

    3. f¯transitive implies ftransitive

    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 Abe a non-empty open subset of X. If KKXand KA,then there exists ϵ>0such that NKϵA..

    Definition 12 Let AXbe. Then the extension of Ato KXis given by eA=KKX/KA.

    Remark 13 eA=A=..

    Lemma 14 [5] Let AXbe, A, an open subset of X. Then, eAis 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:XXbe a continuous function. Then, f¯transitive implies ftransitive.

    Proof: Let A,Bbe two non-empty open sets in X. Due to Lemma 13, eAand eBare non-empty open sets in KX. Thus, by transitivity of f¯, there exists some kNsuch that

    f¯keAeB=fk¯eAeB

    and, from Lemma 14, we obtain

    efkAeB=efkAB

    which implies fkABand, consequently, fis a transitive function.

    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 λRis an irrational number and we define Tλ:S1S1by Tλe=eiθ+2πλ, then it was shown by Devaney [1] that each orbit Tλne/nNis dense in S1and, due Proposition 2, Tλis transitive. Nevertheless, Tλhas no periodic points and, because Tλis 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 KKS1such that diamK=1,and let ϵ>0sufficiently small. Then

    FU=BKϵdiamF1GV=B1ϵdiamG0.

    Thus, diamTλ¯nF1nNand, consequently, Tλ¯nUV=for all nN, which implies that Tλ¯is not transitive on KS1.

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

    It is not difficult to show that fis 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 Uis an arbitrary non-empty open subset of 01, then Ucontains an open interval Jand, after certain number of iterations, there exists a tent, with height equal to one, whose base is contained in J, which implies that fU=01and, according to Remark 3, fis an exact mapping and, consequently, fis 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:XXis continuous, then the following conditions are equivalent:

    i) fis 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 KXgenerated by the sets eAwith Aan open set in X, we obtain the following complementary result, see [5]:

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

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

    5. Sensitivity and periodic density of f¯

    Let f:XXbe a continuous function and let f¯be its corresponding extension to the hyperspace KX. Then, the study of sensitivity of fin the base space in relation to the sensitivity of f¯on KXhas 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 fsensitively dependent.

    Proof: If f¯has sensitive dependence, then there exists a constant δ>0such that for every KKXand every ϵ>0there exists GBKϵand nNsuch that HfnKfnGδ.

    Now, let xXbe and ϵ>0. Then, taking K=xKX, we have that there exists GBxϵand nNsuch that HfnxfnG=HfnxfnGδ.

    Thus, HfnxfnG=supyGdfnxfnyδand, due to the compactness of Gand the continuity of f, there exists y0Gsuch that HfnxfnG=dfnxfny0δ.

    But, GBxϵimplies GBxϵand, consequently, y0Bxϵ. This proves that fis sensitively dependent (with constant δ).

    The reverse of this theorem is not true. In fact, recently Sharma and Nagar [19] show an example where Xdis sensitive but KXHis 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:XXbe a continuous function. Then fis strongly sensitive if there exists δ>0such that for each xXand each ϵ>0, there exists n0Nsuch that for every nn0, there is a yXwith dxy<ϵand 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:XXis a continuous function and KXHf¯is strongly sensitive then Xdfis strongly sensitive.

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

    Theorem 22 Let Xdbe a compact metric space and f:XXa continuous function. Then KXHf¯is strongly sensitive if and only if Xdfis 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 Xdand KXH. It turns out that, strongly sensitivity and sensitivity are equivalent on the class of interval functions, which implies that

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

    a) Idfis 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 Xdbe a compact metric space and f:XXa continuous function. If f:XXhas a dense set of periodic points then f¯:KXKXhas the same property.

    Proof: Let KKXand ϵ>0. Then there exists a ϵ/2-net covering K, That is to say, there are x1,,xpin Ksuch that KBx1ϵ/2Bxpϵ/2.Because fhas periodic density, there are yiXand niNsuch that:

    yiBxiϵ/2,i=1,,pandfniyi=yi,i=1,,p.

    Now, take G=y1yp.By construction, we have HKG<ϵand, 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 fis a very hard problem which still remains open.

    6. The dynamics on the KcIHextension

    In the previous sections, we have studied the diagram

    KXHf¯KXHXdfXdE4

    and the chaotic relationships between fand 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=Iand to study the following new diagram:

    KcIHf¯cKcIHIdfIdE5

    along with the analysis of the connection between their respective dynamical relationships. Here f¯cdenotes 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¯con KcIis less or equal than the complexity of fon the base space I. More precisely, f¯cis never transitive for any continuous function fCI. Also, we will show that f¯chas no dense set of periodic points for most functions fCI.Finally, we prove that f¯chas no sensitive dependence for most functions fCI.

    As a motivation, we present the following examples.

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

    fx=2xif0x1221xif12x1.

    Then it is well known that fis D-chaotic on 01(see [1]). Moreover, because fis a mixing function on 01, then f¯is transitive on K01(see [17]). Also, we observe that x=23is a fixed point of f. On the other hand, it is clear that if Kis a compact and convex subset of X=01, then f¯Kis also a compact and convex subset of X. Consequently, if we let Kc01denote the class of all closed subintervals of 01, then we can consider f¯cas a mapping f¯c:Kc01Kc01. We recall that Kc01is a closed subspace of K01(see [21]). Now, considering the open balls B01110and B0110in Kc01, we have.

    KB0111023Kwhich implies 23f¯cpK,pN..

    On the other hand, if FB0110, then F01/10. Consequently, Hf¯cpKF1730for every KB01110and FB0110.

    Therefore,

    f¯cpB01110B0110=,pN.

    Thus, f¯cis not transitive on Kc01.

    Example 6.1 shows a function fwhich is transitive on the base space X=01and f¯is also transitive on the total extension K01, but f¯cis 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 fto K01also has a dense set of periodic points, whereas f¯cdoes not have a dense set of periodic points on Kc01.

    Example 6.2. Let X=01and consider the “logistic” function f:0101defined by fx=4x1x. It is well known that fis D-chaotic on 01(see [1]). Moreover, fis a mixing function. Thus, in particular, fhas 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¯chas no a dense set of periodic points on KcX.

    In order to see this, we claim that the open ball B183818in Kc01Hdoes not contain periodic points of f¯c.

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

    Thus, we obtain that 14K34fKfKK.

    On the other hand,

    34fK34fnK,n2fnKK,n1

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

    Lemma 25 f¯ctransitive on Kcabimplies ftransitive on ab.

    Proof. Let U,Vnon-empty open subsets of X=ab. We can choose xU, yVand ϵ>0such that BxϵUand ByϵV. Now, in Kcabconsider the open balls Bxϵand Byϵwith respect to the H-metric. Due to the transitivity of f¯con Kcab, there exists nNsuch that f¯cnBxϵByϵ.

    Therefore, there exists an interval JBxϵsuch that f¯cnJ=fnJByϵ. However, JBxϵand, analogously, fnJByϵ, which implies that fnBxϵByϵand, consequently, fnUV. And fis a transitive function on ab.

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

    The next theorem provides a stronger result.

    Theorem 26 Let f:ababbe continuous. Then f¯cis not transitive on Kcab.

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

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

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

    On the other hand,

    FBaϵHFa<ϵFa,a+ϵ].

    Because r<rwe get

    Hf¯nKFpaϵ=rr2>0

    for each KBabϵ,FBaϵand for any nN. Thus,

    f¯nB(abϵ)Baϵ=,nN.

    Consequently, f¯is not transitive on Kcab.

    Case 2. Suppose that fhas no fixed points in ab. From the continuity of f, we have that fx>xfor all xabor fx<xfor all xab. This clearly implies that fis not a transitive function, and consequently, due to Lemma 24, f¯cis 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 fCIwith a dense set of periodic points on I, and such that their extensions f¯calso 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 fCIdo not have an extension f¯cwith a dense set of periodic points on KcI. To prove it, we need the following lemma.

    Lemma 27 Let Ibe a compact interval in R, and f:IIbe a continuous function. If we suppose that f¯chas periodic density on KcI, then fhas periodic density on I.

    Proof. If x0Iand ϵ>0then x0KcIand, consequently, there exists KKcIand nNsuch that

    1. Hx0K<ϵ

    2. f¯cnK=K.

    Combining a. and b. we get

    dx0fnx<ϵ,forallxK.E7

    Because f¯nK=fn¯K=fnK=Kand fnis continuous on Kthen, by the Schauder’s Fixed Point Theorem, there exists xpKsuch that fnxp=xp. Thus, xpis a periodic point of fand, due to (7), we obtain dx0xp<ϵ. Hence, fhas periodic density on I.

    Theorem 28 Let I=abbe a compact interval in R. Then f¯cdoes 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 fis fundamental to decide whether the function fallows or not an extension f¯cthat has a dense set of periodic points. More precisely, the behaviour of each fixed point will imply only two (mutually exclusive) options:

    1. f¯cdoes 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:ababbe a continuous function. By the Schauder’s Fixed Point Theorem, fhas at least one fixed point pab. The proof is divided in.

    Case1.fhas no fixed points in ab.

    In this case, we have the following three subcases:

    1i)p=ais 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 fhas no periodic points in ab.

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

    This case is analogous to the case 1i).

    1iii)p=aand p=bare the unique fixed points of f.

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

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

    Case2.fhas at least one fixed point pab.

    We have the following subcases:

    2i)qab,qpsuch that fq=p.

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

    cqϵ<ϵanddq+ϵ<ϵ

    which implies that a<c<qand q<d<pand, consequently, qJwhereas pJ. Thus,

    qJfq=pfJfJJ.E8

    On the other hand, pfJimplies that

    pfnJ,n2fnJJ,n2,E9

    and, consequently, f¯chas no periodic points in the ball Bqϵq+ϵϵKcabH, which implies that f¯cdoes 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)fdoes not cross the line y=pand fx>pfor all xap.

    In this situation, fxpfor all xa,b]. Thus, choosing qapand 0<ϵ<maxqa2pq2, we can consider the open ball Bqϵto have

    K=cdBqϵKap.E10

    From our hypothesis, we obtain

    fnz>p,zK,nN,E11

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

    2iia2)fdoes not cross the line y=pand fx<pfor all xap.

    In this case, fxpfor all xab. Thus, choosing qpband 0<ϵ<maxqp2bq2, we can consider the open ball Bqϵto obtain

    K=cdBqϵKpb.E12

    Again, from our hypothesis, we get

    fnz<p,zK,nN,E13

    which implies that fnKK, nNand, consequently, f¯chas no periodic points in the ball Bqϵ. In other words, f¯cdoes not have a dense set of periodic points in KcI.

    2iib)fcrosses the line y=p.

    It is clear that, in this case, fCNLIcwhich, 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 fdoes not cross the line y=pthen f¯cdoes not have a dense set of periodic points in KcI, whereas if fcrosses the line y=p, then fCNLIc.

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

    In this case, we have the following subcases:

    2iva1)fdoes not cross the line y=pand fx>pand fx>pfor all xab\p.

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

    2ivb1)fcrosses the line y=pand fx>pin apand fx<pin pb, and.

    2ivb2)fcrosses the line y=pand fx<pin apand fx>pin 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 fcould have an extension f¯cwith a dense set of periodic points on KcIare when there exists a fixed point pof fsuch that fcrosses the line y=pat x=p. In other words, we obtain

    HDSI=fCI/f¯chasadensesetofperiodicpointsinKcIHDSICNLIc,

    But, CNLIis a residual set in CI,therefore from Remark 6, we conclude that HDSIis of first category in CI. Equivalently, f¯cdoes 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 fis 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¯cCKcIis not sensitive.

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

    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 Σon a Riemannian manifold Mwith chaotic sets of the shift flow induced by Σon 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 Gbe a connected ddimensional Lie group with Lie algebra g. A linear control system ΣLon Gis an affine system determined by

    ΣL:ẋt=Xxt+j=1mujtYjxt,u=u1umUE14

    where Xis linear, that is, its flow XttRis a one-parameter group of G-automorphism, the control vectors Yj,j=1,,mare invariant vector fields, as elements of g. The restricted class of admissible control Uis the same as before.

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

    foranyxMSpanLAXY1Ymx=d.

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

    The class of systems ΣLis 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], ΣLis also relevant in applications. It approaches globally any affine non-linear control system Σon a Riemannian manifold when the Lie algebra of the dynamics of Σis finite dimensional.

    One can associate to Xa derivation Dof gdefined by DY=XYe,Yg.Indeed, the Jacobi identity shows DXY=DXY+XDYis in fact a derivation. The relation between φtand Dis given by the formula

    φtexpY=expetDY,foralltR,Yg.

    Consider the generalised eigenspaces of Ddefined by

    gα=Xg:DαnX=0forsomen1

    where αSpecD. Then, gαgβgα+βwhen α+βis an eigenvalue of Dand zero otherwise. Therefore, it is possible to decompose gas g=g+g0g, where

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

    Actually, g+,g0,gare Lie algebras and g+, gare nilpotent. Denote by G+, Gand G0the connected and closed Lie subgroups of Gwith Lie algebras g+, gand g0respectively.

    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 ΣLclass. We begin with a fundamental result.

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

    Ce=clAeAe

    which contains the identity element ein its interior. Here, Aeis the set of states of Gthat can be sent by ΣLto ein 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 ΣLto 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 Ghas finite semisimple centre, i.e. all semisimple Lie subgroups of Ghave 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 Ais 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 ΣLa linear control system on the Lie group G.

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

    2. Suppose that Gis semisimple or nilpotent, it turns out that

    ifclAG,clAG+andG0arecompact setsCis bounded.

    3. If Gis 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 Cis bounded. Actually, in a forthcoming paper we show that

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

    A analogous result is obtained for G+Aassuming that G,0is normal. Of course, G+,0is a normal subgroup of Gif and only if g+g0is 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×Uand the Σ-control sets.

    Theorem 34 Let M×Uand the canonical projection πM:M×UM.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 Csuch that

      =clxuM×U:φ(txu)intCfor everytR

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

  • The periodic points of Φare dense in .

  • Φrestrict 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 Cis a bounded control set with non-empty interior of ΣLand define =πM1C=clC×UCwhere

    UC=uU:existxCwithφtxuintCfor everytR.

    The Lie group Gis finite dimensional and UCis a closed subset of the compact class of admissible control ULRΩRmwith the weak* topology. Since the projection is a continuous map, it turns out that πMis compact and ,Care uniquely defined.

    On the other hand, we are assuming that ΣLsatisfy 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 ΣLbe a linear control system on a Lie group G. Any condition.

    1. Gis compact, or

    2. Gis Abelian, or

    3. Ghas the finite semisimple centre property and the Lyapunov spectrum of Dis 0implies that the skew flow Φis chaotic in G×U.

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

    It is well known that any Abelian Lie group is a product G=Rm×Tnbetween the Euclidean space Rmand the torus Tn=S1××S1(ntimes), for some m,nN.In this case, ΣLis also controllable [15]. Indeed, since the automorphism group of Tnis discrete, any linear vector field on the torus is trivial. But, we are assuming the Lie algebra condition on Gwhich coincides with the Kalman rank condition in Rm.And, on the compact part, we apply 1.Hence, the skew flow Φis chaotic in G×U. In fact, πM1C=G×Uand the hypothesis of the compacity on the projection in Theorem 32is 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 Ceto translate these properties to its associated chaotic set ,as follows.

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

    1. If G=GG0G+there exists one and only one chaotic set =πM1Cein G×Ugiven by

      =clxu:φ(txu)intCefor everytRM×U

  • If Gis nilpotent and Dhas only eigenvalues with non-positive real parts, then the only chaotic set =πM1Cin G×Uis closed

  • If Gis nilpotent and Dhas only eigenvalues with non-negative real parts then the only chaotic set =πM1Cin G×Uis open

  • Proof. If Gis decomposable, we know that there exists just one control set: the one which contains the identity element. Hence, =πM1Ceis the only chaotic set of Φon G×Uwhich proves 1.To prove 2and 3, we observe that the Lyapunov spectrum condition on the derivation Dassociated to the drift vector field Xis equivalent to the control set Cebe closed or open, respectively. Since the projection πG:G×UGis a continuous map with the weak* topology on U,the lifting πG1Ceis 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 Hand the special linear group SL2R. We finish by computing an example on H.

    1. The nilpotent Lie algebra h=R3+,has the basis E12E23E13with E12E23=E13.Here, Eijdenotes the real matrix of order 3with zero everywhere except 1in 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 ghas 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 Xreads as

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

    2. The vector space g=sl2Rof 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=0100and Y3=1001.The Lie group G is semisimple, then any gderivation is inner which means that there exists an invariant vector field Ysuch that adYrepresents .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 Xis determined by the derivation D=adE12=E32.Since the group is nilpotent, it has the semisimple finite centre property. The Lyapunov spectrum of Dreduces to zero. Finally, the reachable set from the identity Ais open. In fact, the ad-rank condition is obviously true because DE12=E13. It turns out that the skew flow Φis chaotic in H×U.

    Notes

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

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    Heriberto Román-Flores and Víctor Ayala (March 28th 2018). Chaos on Set-Valued Dynamics and Control Sets, Chaos Theory, Kais A. Mohamedamen Al Naimee, IntechOpen, DOI: 10.5772/intechopen.72232. Available from:

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