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Open access peer-reviewed chapter
By Heriberto Román-Flores and Víctor Ayala
Submitted: April 25th 2017Reviewed: November 6th 2017Published: March 28th 2018
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.
Relevant classes of real problems are modelled by a discrete dynamical system
where is a metric space and is a continuous function. The basic goal of this theory is to understand the nature of the orbit for any state as 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 and, in many cases, these orbits present a chaotic structure. In 1989 in , Devaney isolates three main conditions which determine the essential features of chaos.
Definition 1 Let be a metric space and a continuous map. Hence, .
is transitive if for any couple of non-empty open subsets and of there exists a natural number such that .
is periodically dense if the set of periodic points of is a dense subset of .
has sensitive dependence on initial conditions if there is a positive number (a sensitivity constant) such that for every point and every neighbourhood of there exists a point and a non-negative integer number such that .
Next, we mention a remarkable characterisation of transitive maps. In fact, as a consequence of the Birkhoff Transitivity Theorem (see  for details), it is possible to prove.
Proposition 2 Let be a complete metric space which is also perfect (closed and without isolated points). If is continuous, then is transitive if and only if there exists at least one orbit dense in .
Remark 3 Also, other concepts very useful in this work are the following: i) is weakly mixing iff for any non-empty open subsets and of there exists a natural number such that and . ii) is mixing iff given two non-empty open subsets and of there exists a natural number such that for all . iii) is exact iff given a non-empty open subsets there exists a natural number such that . It is clear that exact mixing weakly mixing 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 ). Furthermore, for continuous functions on real intervals, Vellekoop and Berglund in  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 ).
However, sometimes we need to know information about the collective dynamics, i.e. how are moved subsets of via iteration or dynamics induced by . For example, if denotes an ecosystem and , then, by using radio telemetry elements, we can obtain information about the movement of in the ecosystem . In this form, it is possible to build an individual displacement function . Of course, this function could be chaotic or not. Eventually, we could also be interested to get information about the collective dynamics induced by , 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 defined by
where is the natural extension of to the metric space of the non-empty compact subsets of endowed with the Hausdorff metric induced by the original distance of .
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 on in relation to the transitivity of its extension to Our main result here establishes that transitive implies transitive. That is to say, collective chaos implies individual chaos under the dynamics of .
On the other hand, we propose a new approach to this problem: to study the dynamics induced by on the subextension of Precisely, on the class of non-empty compact-convex subsets of . 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 of the system, one can associate its reachable set . In other words, contains all the possible states of the manifold that starting from you can reach in non-negative time by using the admissible control functions 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 on a Riemannian manifold with chaotic sets of the shift flow induced by on . 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.
In this section, we mention some notions and fundamental results we use through the chapter.
If is a metric space and continuous, then we can consider the space of all non-empty and compact subsets of endowed with the Hausdorff metric induced by and , the natural extension of to . Also, we denote by . If we define the “-dilatation of ” as the set , where .
The Hausdorff metric on is given by
Also, if , the set denotes the ball centred in and radius in the space .
Furthermore, given a continuous function on a real interval , we also consider the extension , where is the restriction .
In this section, we review some properties of Baire spaces.
Definition 4 A topological space is a Baire space if for any given countable family of closed sets covering , then for at least one .
Definition 5 In any Baire space ,
is called nowhere dense if
Any countable union of nowhere dense sets is called a set of first category.
Any set not of first category is said to be of second category.
The complement of a set of first category is called a residual set.
Remark 6 It is important to note that:
Any complete metric space is a Baire space.
Every residual set is of second category in .
Every residual set is dense in .
The complement of a residual set is of first category.
If is of first category and , then is of first category.
In particular, if is an interval, then and , endowed with the respective supremum metrics, are Baire spaces.
In a Baire space , we say that “most elements of ” verify the property (P) if the set of all that do not verify property (P) is of first category in . 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 such as the set of periodic points of , or the set of fixed points of . Typically, continuous interval functions have a first category set of periodic points (see ) 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  and references therein). It is also well-known that the class of nowhere differentiable functions is a residual set in (see [13, 14]). Also, a special class of functions in is the class of all continuous functions whose graphs “cross no lines” defined in a negative way as follows (see ):
Definition 7 Let a continuous map and a function whose graph is a straight line. We say that crosses (or crosses ) if there exists and such that and either.
(a) for all and for all ; or.
(b) for all and new for all .
The following result can be found in :
Theorem 8 () The set is residual in .
The set will play an important role in the next sections.
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, . 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, . Let be a dimensional smooth manifold. By an affine control system in , we understand the family of ordinary differential equations:
where , are arbitrary vector fields on The set is the class of restricted admissible control functions where with is a compact and convex set.
Assume satisfy the Lie algebra rank condition, i.e.
Of course, means the Lie algebra generated by the vector fields through the usual notion of Lie bracket. Furthermore, the -rank condition for is defined as follows:
For each and each initial value , there exists an unique solution defined on an open interval containing satisfying . Since we are concerned with dynamics on Lie Groups, without loss of generality we assume that the vector fields are completes. Then, we obtain a mapping satisfying the cocycle property
for all , , Where, for any , the map is the shift flow on defined by Hence, 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
It turns out the following results.
Lemma 9  Consider the set equipped with the weak* topology associated to as a dual vector space. Therefore,
is a compact, complete and separable metric space with the distance given by
Here, is a dense set of Lebesgue integrable functions.
The map defines a continuous dynamical systems on . Its periodic points are dense and the shift is topologically mixing (and then topologically transitive).
The map defines a continuous dynamical system on
On the other hand, the completely controllable property of i.e. the possibility to connect any two arbitrary points of through a -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 .
Definition 10 A non-empty set is called a control set of (3) if.
for every there exists such that
for every ,
is maximal with respect to the properties and
denotes the states that can be reached from by in positive time and its closure
Moreover, for an element , the set of points that can be steered to through a -trajectory in positive time is denoted by
Finally, we mention that the Lie algebra rank condition warranty that the system is locally accessible, which means that for every ,
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 implies the transitivity of For that, we need to describe some previous results.
Lemma 11  Let be a non-empty open subset of . If and then there exists such that .
Definition 12 Let be. Then the extension of to is given by .
Remark 13 .
Lemma 14  Let be, , an open subset of . Then, is a non-empty open subset of .
Lemma 15  If , then: i) , ii) , and iii) for every .
Now, we are in a position to prove the following results
Theorem 16 Let be a continuous function. Then, transitive implies transitive.
Proof: Let be two non-empty open sets in . Due to Lemma 13, and are non-empty open sets in . Thus, by transitivity of , there exists some such that
and, from Lemma 14, we obtain
which implies and, consequently, is a transitive function.
Now we show that, in general, the converse of Theorem 15 is not true.
Example 4.1 (Translations of the circle). If is an irrational number and we define by , then it was shown by Devaney  that each orbit is dense in and, due Proposition 2, is transitive. Nevertheless, has no periodic points and, because is isometric, it does not exhibit sensitive dependence on initial conditions either.
If , because preserves diameter, then , for all .
Now, let such that and let sufficiently small. Then
Thus, and, consequently, for all , which implies that is not transitive on .
Example 4.2 Define the “tent” function as and .
It is not difficult to show that 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 .
Thus, if is an arbitrary non-empty open subset of , then contains an open interval and, after certain number of iterations, there exists a tent, with height equal to one, whose base is contained in , which implies that and, according to Remark 3, is an exact mapping and, consequently, is a mixing function.
The conclusions in Examples 4.1 and 4.2 come from the next result, Banks  in 2005.
Theorem 17 If is continuous, then the following conditions are equivalent:
i) is weakly mixing, ii) is weakly mixing, iii) is transitive.
Hitherto, we have used the strong topology induced by the -metric on . However, considering the -topology on generated by the sets with an open set in , we obtain the following complementary result, see :
Theorem 18 For a continuous map the following conditions are equivalent:
i) is transitive in , ii) is transitive in the -topology.
Let be a continuous function and let be its corresponding extension to the hyperspace . Then, the study of sensitivity of in the base space in relation to the sensitivity of on has been very exhaustively analysed in the last years. Román and Chalco published the first result in this direction  in 2005, where the authors prove
Theorem 19 sensitively dependent implies sensitively dependent.
Proof: If has sensitive dependence, then there exists a constant such that for every and every there exists and such that .
Now, let be and . Then, taking , we have that there exists and such that .
Thus, and, due to the compactness of and the continuity of , there exists such that .
But, implies and, consequently, . This proves that is sensitively dependent (with constant ).
The reverse of this theorem is not true. In fact, recently Sharma and Nagar  show an example where is sensitive but is not. Now, in order to overcome that shortcoming, the authors in  introduce the following notion of sensitivity:
Definition 20 (Stronger sensitivity ). Let be a continuous function. Then is strongly sensitive if there exists such that for each and each , there exists such that for every , there is a with and .
Obviously, the notion of stronger sensitivity is more restrictive than sensitivity, and the authors in  obtain the following results:
Theorem 21 If is a continuous function and is strongly sensitive then is strongly sensitive.
In the compact case, it is possible to obtain a characterization as follows.
Theorem 22 Let be a compact metric space and a continuous function. Then is strongly sensitive if and only if is strongly sensitive.
In connection with these results, recently Subrahmomian (, 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 and . It turns out that, strongly sensitivity and sensitivity are equivalent on the class of interval functions, which implies that
Theorem 23 If is a continuous function, the following conditions are equivalent.
a) is sensitive, b) is sensitive.
We finish this section assuming the existence of a dense set of periodic points for , we have
Theorem 24 Let be a compact metric space and a continuous function. If has a dense set of periodic points then has the same property.
Proof: Let and . Then there exists a -net covering , That is to say, there are in such that Because has periodic density, there are and such that:
Now, take By construction, we have and, moreover, , for all . Therefore, , which implies that has periodic density.
The converse of this theorem is no longer true (for a counterexample, see Banks ). However, to find conditions on warranting the existence of a dense set of periodic points for is a very hard problem which still remains open.
In the previous sections, we have studied the diagram
and the chaotic relationships between and . 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 and to study the following new diagram:
along with the analysis of the connection between their respective dynamical relationships. Here denotes the restriction of to , the class of all compact subintervals of . For , the Hausdorff metric can be explicitly computed as
The aim of this section is to show that the Devaney complexity of the extension on is less or equal than the complexity of on the base space . More precisely, is never transitive for any continuous function . Also, we will show that has no dense set of periodic points for most functions Finally, we prove that has no sensitive dependence for most functions .
As a motivation, we present the following examples.
Example 6.1 Consider the “tent” function defined by
Then it is well known that is D-chaotic on (see ). Moreover, because is a mixing function on , then is transitive on (see ). Also, we observe that is a fixed point of . On the other hand, it is clear that if is a compact and convex subset of , then is also a compact and convex subset of . Consequently, if we let denote the class of all closed subintervals of , then we can consider as a mapping . We recall that is a closed subspace of (see ). Now, considering the open balls and in , we have.
which implies .
On the other hand, if , then . Consequently, for every and .
Thus, is not transitive on .
Example 6.1 shows a function which is transitive on the base space and is also transitive on the total extension , but is not transitive on the subextension .
The following example shows a function with a dense set of periodic points, and where the total extension of to also has a dense set of periodic points, whereas does not have a dense set of periodic points on .
Example 6.2. Let and consider the “logistic” function defined by . It is well known that is D-chaotic on (see ). Moreover, is a mixing function. Thus, in particular, has a dense set of periodic points and, therefore, also has a dense set of periodic points on the total extension ) (see Theorem 24).
However, has no a dense set of periodic points on .
In order to see this, we claim that the open ball in does not contain periodic points of .
In fact, if , then and , which implies that and .
Thus, we obtain that .
On the other hand,
and, consequently, has no periodic points in the ball , which implies that has no dense set of periodic points on .
Lemma 25 transitive on implies transitive on .
Proof. Let non-empty open subsets of . We can choose , and such that and . Now, in consider the open balls and with respect to the -metric. Due to the transitivity of on , there exists such that .
Therefore, there exists an interval such that . However, and, analogously, , which implies that and, consequently, . And is a transitive function on .
It is well-known that if is an interval, then most functions has no dense orbits, that is to say, there exists a residual set such that every function has no point whose orbit is dense in (see ) and, consequently, most functions are not transitive. From Lemma 24, we can conclude that is not transitive for most functions .
The next theorem provides a stronger result.
Theorem 26 Let be continuous. Then is not transitive on .
Proof. By Schauder Theorem, has at least one fixed point .
Case 1. Suppose that and let . Without loss of generality, we can suppose that and, because , it is clear that .
Now, let and let . If we consider the open balls , it follows that for any .
On the other hand,
Because we get
for each and for any . Thus,
Consequently, is not transitive on .
Case 2. Suppose that has no fixed points in . From the continuity of , we have that for all or for all . This clearly implies that is not a transitive function, and consequently, due to Lemma 24, is not transitive on .
An important question to answer is what about the size of the set of periodic points of . It is clear that there are some functions with a dense set of periodic points on , and such that their extensions also has a dense set of periodic points on (for instance, ). Therefore, an analogous result to Theorem 26, but for periodic density of , cannot be obtained. However, as we will see, most functions do not have an extension with a dense set of periodic points on . To prove it, we need the following lemma.
Lemma 27 Let be a compact interval in , and be a continuous function. If we suppose that has periodic density on , then has periodic density on .
Proof. If and then and, consequently, there exists and such that
Combining a. and b. we get
Because and is continuous on then, by the Schauder’s Fixed Point Theorem, there exists such that . Thus, is a periodic point of and, due to (7), we obtain . Hence, has periodic density on .
Theorem 28 Let be a compact interval in . Then does not have a dense set of periodic points in , for most functions .
Proof. The proof is based on an exhaustive analysis of the behaviour of the fixed points of . We connect this analysis with an adequate residual set in . The analysis of each fixed point of is fundamental to decide whether the function allows or not an extension that has a dense set of periodic points. More precisely, the behaviour of each fixed point will imply only two (mutually exclusive) options:
does not have a dense set of periodic points, or.
, which is a set of first category in .
Towards this end, let be a continuous function. By the Schauder’s Fixed Point Theorem, has at least one fixed point . The proof is divided in.
has no fixed points in .
In this case, we have the following three subcases:
is the unique fixed point of .
We have, either
In both cases it follows that has no periodic points in .
is the unique fixed point of .
This case is analogous to the case .
and are the unique fixed points of .
This case is also analogous to the cases and .
Therefore, in case the function does not have a dense set of periodic points in . Due to Lemma 24, does not have a dense set of periodic points in .
has at least one fixed point .
We have the following subcases:
such that .
Without loss of generality, suppose that . Then, taking , we can consider the open ball in the space . If , from (6) we have
which implies that and and, consequently, whereas . Thus,
On the other hand, implies that
and, consequently, has no periodic points in the ball , which implies that does not have a dense set of periodic points on .
, is the unique point such that .
Without loss of generality, we can suppose that , for all .
Now, in addition to hypothesis , we have two subcases:
does not cross the line and for all .
In this situation, for all . Thus, choosing and , we can consider the open ball to have
From our hypothesis, we obtain
which implies that , . Consequently, has no periodic points in the ball . In other words, does not have a dense set of periodic points in .
does not cross the line and for all .
In this case, for all . Thus, choosing and , we can consider the open ball to obtain
Again, from our hypothesis, we get
which implies that , and, consequently, has no periodic points in the ball . In other words, does not have a dense set of periodic points in .
crosses the line .
It is clear that, in this case, which, due to Theorem 8 and Remark 6, is a set of first category in .
, is the unique point such that .
This case is analogous to case and, consequently, if does not cross the line then does not have a dense set of periodic points in , whereas if crosses the line , then .
and , , are the unique points such that .
In this case, we have the following subcases:
does not cross the line and and for all .
This case is analogous to the case and the same is true for when does not cross the line and for all which is analogous to the case Finally, there only remains two subcases:
crosses the line and in and in , and.
crosses the line and in and in .
It is clear that in both cases .
Thus, as a direct consequence of the analysis of the behaviour of the set of fixed points of , it turns out that the unique cases in which could have an extension with a dense set of periodic points on are when there exists a fixed point of such that crosses the line at . In other words, we obtain
But, is a residual set in therefore from Remark 6, we conclude that is of first category in . Equivalently, does not have a dense set of periodic points, for most functions , which ends the proof.
Finally based on the following result,
Theorem 29 () For most functions , the set of all points where is sensitive is dense in the set of all periodic points of .
we show an analogous result for the sensitivity property, as follows.
Theorem 30 For most functions , the extension is not sensitive.
Proof. This is a direct consequence of Theorem 28 and Theorem 29.
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 on a Riemannian manifold with chaotic sets of the shift flow induced by on . 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.
Let be a connected dimensional Lie group with Lie algebra . A linear control system on is an affine system determined by
where is linear, that is, its flow is a one-parameter group of -automorphism, the control vectors are invariant vector fields, as elements of . The restricted class of admissible control is the same as before.
Certainly, the drift vector field is complete and the same is true for every invariant vector field . As usual, we assume that satisfy the Lie algebra rank condition, i.e.
The system is said to be controllable if is
The class of systems is huge and contains many relevant algebraic systems as the classical linear and bilinear systems on Euclidean spaces , and the class of invariant systems on Lie groups, . Furthermore, according to the Jouan Equivalence Theorem , is also relevant in applications. It approaches globally any affine non-linear control system on a Riemannian manifold when the Lie algebra of the dynamics of is finite dimensional.
One can associate to a derivation of defined by Indeed, the Jacobi identity shows is in fact a derivation. The relation between and is given by the formula
Consider the generalised eigenspaces of defined by
where . Then, when is an eigenvalue of and zero otherwise. Therefore, it is possible to decompose as , where
Actually, are Lie algebras and , are nilpotent. Denote by , and the connected and closed Lie subgroups of with Lie algebras , and 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  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  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 class. We begin with a fundamental result.
Theorem 31 Assume the system satisfy the Lie algebra rank condition. Therefore, there exists a control set
which contains the identity element in its interior. Here, is the set of states of that can be sent by to in positive time.
For a proof in a more general set up, see .
Recently, we were able to establish some algebraic, topological, and dynamical conditions on 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 , as in  we assume here that has finite semisimple centre, i.e. all semisimple Lie subgroups of 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 is open. This is true if for example, the system satisfy the -rank condition. About the uniqueness and boundness of control sets of a linear systems, we know few things .
Theorem 32 Let a linear control system on the Lie group
1. If is decomposable, is the only control set with non-empty interior. In particular, this is true for any solvable Lie group.
2. Suppose that is semisimple or nilpotent, it turns out that
3. If is a nilpotent simply connected Lie group, it follows that
Furthermore, it is possible to determine algebraic sufficient conditions to decide when is bounded. Actually, in a forthcoming paper we show that
Theorem 33 Let be a linear control system on the Lie group Assume that is decomposable and is a normal subgroup of . Hence, is compact.
A analogous result is obtained for assuming that is normal. Of course, is a normal subgroup of if and only if is an ideal of . On the other hand,
We start with an explicitly relationship between chaotic subsets of and the -control sets.
Theorem 34 Let and the canonical projection Hence,
is compact and its non-void interior consists of locally accessible points. Then,
is a maximal topologically mixing set if and only if there exists a control such that
In this case, is unique and .
The periodic points of are dense in .
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 is a bounded control set with non-empty interior of and define where
The Lie group is finite dimensional and is a closed subset of the compact class of admissible control with the weak* topology. Since the projection is a continuous map, it turns out that is compact and are uniquely defined.
On the other hand, we are assuming that 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 , first, for some classes of controllable linear systems, as follows.
Theorem 35 Let be a linear control system on a Lie group . Any condition.
is compact, or
is Abelian, or
has the finite semisimple centre property and the Lyapunov spectrum of is implies that the skew flow is chaotic in .
Proof. Under the hypothesis in , any control set is bounded. Furthermore, if is compact, the Lie algebra rank condition assures that the linear control system is controllable on see . Hence, is topologically mixing, topologically transitive and the periodic points of are dense in , which give us the desired conclusion.
It is well known that any Abelian Lie group is a product between the Euclidean space and the torus (times), for some In this case, is also controllable . Indeed, since the automorphism group of is discrete, any linear vector field on the torus is trivial. But, we are assuming the Lie algebra condition on which coincides with the Kalman rank condition in And, on the compact part, we apply Hence, the skew flow is chaotic in . In fact, and the hypothesis of the compacity on the projection in Theorem is not necessary for the lifting, see Proposition 4.3.3 in . The same is true for 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 to translate these properties to its associated chaotic set as follows.
Theorem 36 Let be a linear control system on a Lie group It holds.
If there exists one and only one chaotic set in given by
If is nilpotent and has only eigenvalues with non-positive real parts, then the only chaotic set in is closed
If is nilpotent and has only eigenvalues with non-negative real parts then the only chaotic set in is open
Proof. If is decomposable, we know that there exists just one control set: the one which contains the identity element. Hence, is the only chaotic set of on which proves To prove and , we observe that the Lyapunov spectrum condition on the derivation associated to the drift vector field is equivalent to the control set be closed or open, respectively. Since the projection is a continuous map with the weak* topology on the lifting is both closed and open, respectively.
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 and the special linear group . We finish by computing an example on .
1. The nilpotent Lie algebra has the basis with Here, denotes the real matrix of order with zero everywhere except in the position The associated Heisenberg Lie group has the matrix representation
As invariant vector fields, the basis elements of has the following description
The canonical form of any -derivation is given by
Any linear vector field reads as
2. The vector space of all real matrices of order three and trace zero is the Lie algebra of the Lie group . Let us consider the following generators of : and The Lie group G is semisimple, then any derivation is inner which means that there exists an invariant vector field such that represents Thus, a general form of a derivation reads as
Example 7.1 On the Heisenberg Lie group, consider the system
where is determined by the derivation Since the group is nilpotent, it has the semisimple finite centre property. The Lyapunov spectrum of reduces to zero. Finally, the reachable set from the identity is open. In fact, the -rank condition is obviously true because . It turns out that the skew flow is chaotic in
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