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Entropy 2013, 15(6), 2363-2383; doi:10.3390/e15062363
Abstract: The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox.
The time-switched dynamics of two one-dimensional, dissipative affine maps,
The situation envisaged in  is special in several regards.
The state space is .
Both the forward and backward dynamics are amenable to detailed analysis.
The non-autonomous pullback attractor consists of singleton component sets.
In this follow-up paper we continue studying this question in an extended setting, namely, we consider this time switching between difference equations (called constituent maps) that have pullback attractors with nontrivial component sets. Our scope is to separate the particular results from the general ones, and so better understand the intricacies of switching and non-autonomous dynamics [2,3,4]. Specifically, in this paper:
The state space is or a proper subset of it.
The constituent maps are not specified, except for the fact that they are supposed to have attractors.
The component sets of the pullback attractor of the systems under switching are supposed to be uniformly bounded.
Both from an instrumental and a conceptual point of view, the main challenge is introduced by the generalization (iii’). To begin with, working with attractors with nontrivial component sets instead of point-valued components sets requires using the Hausdorff distance. Contrarily to what happens in the latter case, the Hausdorff distance between component sets of a set-valued attractor is not continuous, in general, but only upper-continuous when a control sequence converges to another one . This technical shortcoming is behind Assumption 1 in Section 3 (“Results using measurability”), and Assumption 3 in Section 5 (“Results using continuity”). Also the generality of (ii’) will be limited in different ways (injectivity of a certain map Φ in Section 3, or Assumption 2 in Section 4) in order to derive sharper results.
On the conceptual side, new features related to the extended geometry of set-valued attractors manifest themselves. Indeed, the observable complexity is going to be the result of two contributions. One, the “macroscopic” complexity, comes from the trajectory segments connecting different component sets of the pullback attractor (think that each component set has been lumped to a point). The other one, the “microscopic” complexity, comes from trajectory segments within the component sets. No geometrical meaning should be attached to this denomination since the (in general, uncountably many) component sets might be packed in a state space region of a size comparable to the attractor itself. The rationale for distinguishing two sorts of complexity is rather that our analytical tools will only be able to resolve the trajectories up to the precision set by the component sets. An offshoot of this picture is that the entropy of the coarse-grained dynamics (or “macroscopic entropy”) is a lower bound of the entropy of the switched dynamics. This fact will again allow us to relate the entropy of trajectories and control sequences, once we have shown that the latter coincide with the macroscopic entropy.
Thus, the main purpose of the present paper is the study of the macroscopic complexity in switched dynamics, as measured by the macroscopic topological entropy. In Section 3 and 5 we prove that, under certain provisos, the macroscopic topological entropy coincides with the topological entropy of the switching sequence generator. In this case, the complexity of the switched dynamics (a mixture of macroscopic and microscopic complexities) is certainly greater than the complexity of the switching sequence. This result can be brought in connection with Parrondo’s paradox, i.e., the emergence of new properties via switching [5,6,7,8]. Indeed, it suffices that the entropy of the switching sequence generator is higher than the entropies of the constituent maps for the switched dynamics to be more complex than the constituent dynamics.
This paper is organized as follows. Section 2 sets the mathematical framework (switching systems, Hausdorff distance and pullback attractors) of the paper. Additional materials on the Hausdorff distance have been collected in the Appendix for the reader’s convenience. A few results for further reference, but also interesting on their own, are proved as well in this section and in the Appendix. The main theoretical results of the paper are derived in Section 3 (Theorem 3) and Section 5 (Theorem 4) depending on whether the switched dynamics complies with Assumption 1 (Section 3), or with Assumption 3 (Section 5). In between, Section 4 scrutinizes a property needed in Theorem 3 and reformulates it as Assumption 2 to be used in the proof of Theorem 4. Some numerical simulations illustrating our theoretical results make up Section 6. The highlights of all these sections are summarized in the Conclusion, followed by the references and the Appendix.
This section introduces the background on switching and non-autonomous dynamical systems needed to make this paper self-contained. The interested readers are referred to the books [2,3], the review , and the papers [9,10,11,12,13].
2.1. Switching Systems
Let = ∈ be the collection of all switching controls, which is a compact metric space with the metric
The metric space is compact; see, e.g.,  (p. 207).
Consider the left shift operator σ defined on ,
We consider discrete time dynamical systems generated by switching between two continuous mappings : → , where the autonomous dynamical systems generated by each mapping are dissipative and have global attractors . In other words, the (time) switched dynamics (or switching system) we are going to study in the following is described by the non-autonomous difference equation
As an important example, let be affine maps,
Likewise if the matrices are invertible, then from
As usual, let denote the norm of and a compatible norm of the matrix M.
From Equation (6), we have
Linear systems with are globally dissipative. The origin is a fixed point and, according to Equation (10) with , all other orbits converge to it for each .
2.2. Hausdorff Metric and Sequences of Compact Subsets
Let be the space of nonempty compact subsets of , which is a complete metric space with the Hausdorff metric
Let and be the open and closed balls of radius and center , respectively. Define = , i.e., the family of nonempty compact subsets of that are contained in . Then, is a compact metric space.
Let be continuous, and . Then
In order to apply Proposition 2 to our constituent maps , whose definition domain is in principle all , it suffices to consider their restrictions to .
Further, we define a metric on the space of bi-infinite sequences of nonempty compact subsets of in by
2.3. Skew Product Flows and Pullback Attractors
Define φ: → by = and
An entire solution of a discrete time skew product flow is a mapping such that
A pullback attractor is a family of nonempty compact subsets, = ⊂ which is φ-invariant, i.e.,
We will assume that the switched dynamics (4) has a pullback attractor = such that the are nonempty, uniformly bounded compact subsets of . This means that there is such that for every . If the constituent maps are both affine, then it is inferred from Proposition 1 that one may choose . A sufficient condition ensuring the existence of such a pullback attractor is that the two unswitched systems have a common bounded, positively invariant absorbing set.
(Theorem 3.34) The set-valued mapping ↦ is upper semi-continuous in , i.e.,
Counterexamples show that, in general, we cannot replace the Hausdorff semi-distance here by the Hausdorff metric, but in special cases we can do that, e.g., when the pullback attractor consists of singleton sets as in  (see Example 1 below), since then obviously
A general, sufficient condition for the continuity of the map ↦ in is provided by the following result.
Suppose that uniformly in for some nonempty, bounded set . Then the map ↦ is continuous in , i.e.,
By the property (18) of pullback attractors, for all there exists an N such that
It follows from Proposition 3 that the pullback attractor is also a forward attractor, i.e., with
3. Results Using Measurability
Remember that is endowed with the metric (13). Let Φ: be the map defined by . We show first that Φ is Borel measurable.
By definition the Borel sigma-algebras of the product spaces and are generated by the corresponding cylinder sets or just cylinders. If , , the cylinders of have the form
If, as before, σ is the (left) shift on , and Σ is the shift on , then and, similarly, for all .
is Borel measurable.
Since the cylinder sets build a semi-algebra of the product sigma-algebras they generate, it suffices to prove that is Borel measurable for every , , and every (see Theorem 1.1 in ). Note that unless [see Equation (17)] for some , and for . In any case,
Suppose now that Φ is one-to-one and let us explore when is also Borel measurable, resulting in a Borel bimeasurable mapping. For , ,
The sets and are Borel measurable for every .
Continuity or closedness of the mapping are obvious sufficient conditions for Assumption 1 to hold. (As a matter of fact, by the closed map lemma, if is continuous then it is closed because is compact and is a Hausdorff topological space.) For example, in the affine, one-dimensional case studied in  is a singleton, i.e., , for all . That all the sets are compact (hence Borel measurable) follows in this case from the continuity of . In Section 5 we will study with more detail the consequences of assuming the mapping continuous.
If is one-to-one and Assumption 1 holds, then Φ is Borel bimeasurable.
We only need to prove that Φ transforms cylinders into Borel measurable sets. In view of Equation (21) this boils down to showing that is Borel measurable for every , .
From Equation (22) and Assumption 1, it follows that is a finite intersection of Borel measurable sets, hence it is Borel measurable. □
Consider the diagram
Let be a bijection and suppose that Assumption 1 holds. Then .
Other conditions leading also to will be discussed in Section 5.
Notice that corresponds to what we called macroscopic entropy in the Introduction. Indeed, since (i) is a trajectory in , a set whose “points” are the component sets of ; and (ii) implies for all , it holds that measures the complexity of the trajectories up to the precision set by the distinct component sets . To relate the macroscopic entropy, , to the entropy of the switched dynamics (referred to as microscopic entropy in the Introduction), , let be the set of entire orbits, and consider the commutative diagram
Under the hypotheses of Theorem 2, .
This theorem provides sufficient conditions for a complexity version of Parrondo’s paradox. Indeed, it suffices that (along with the assumptions of Theorem 2), where is the topological entropy of , for the complexity of the switched dynamics to exceed the complexity of the constituent maps.
4. Conditions for the Injectivity of Φ
According to Theorem 2, the injectivity of Φ (along with Assumption 1) is instrumental for and to coincide. We explore next sufficient conditions for the injectivity of Φ.
If Φ is not injective, then there are control sequences such that , i.e., for all . Then
There is no such that or, equivalently, for all .
This condition can be reworded as follows: and distinguish all pullback attractors of the switching dynamics.
Let be the attractors of , i.e., , and . If , then Assumption 2 does not hold since , and . Think, for instance, of two linear dynamics having only 0 as a fixed point. Therefore, in this case the topological entropy of the switched dynamics may be different from the topological entropy of the control sequence.
Now suppose that at least one of the maps is invertible and define if is invertible, or otherwise; if both maps are invertible, then either choice is the inverse of the other one. Then
In case that one of the maps is invertible, there is no g-invariant .
Suppose that are two -matrices, with being invertible, and , so that with . Thus, when applying Assumption 2 for invertible mappings, we are asking about the existence of invariant sets of known linear maps. Linear maps can be decomposed as actions of scalings (expansion/contraction, projection, reflection) and/or rotations. Therefore, the existence of G-invariant pullback attractors boils down to well-known geometrical facts.
The only invariant set of an (in general anisotropic) expansion/contraction is .
If a linear map is a projection onto a subspace V, then any set contained in V is invariant.
If a linear map is a reflection, then any set symmetric under reflections (like a star-like shaped object) is invariant.
If a linear map is a rotation by an angle α, then any set symmetric under that rotation is invariant, in particular any ball centered at the origin of radius and dimension n or if contained in the orthogonal complement of the rotation axis.
Since the origin is invariant in any case, it might happen that for some (for example, for or , since , where are the attractors of ). In this case, the version of Assumption 2 for invertible mappings would not hold.
From the preceding discussion we conclude the following result.
Suppose that the maps are such that Assumption 2 holds. Then the map Φ: → defined by is one-to-one.
5. Results Using Continuity
The main technical difference between the setting of  and the present general setting is that the map is not necessarily continuous (Proposition 3). It is, however, worth following the same path as in  to see how far we can go and under which conditions. The first condition is obvious.
The set-valued mapping is continuous in , i.e.,
Proposition 4 provides a sufficient condition for Assumption 3 to hold.
We are going to show in this section that if Assumptions 2 (Section 4) and 3 are fulfilled, then is a homeomorphism.
Under Assumption 3, the map Φ: defined by is uniformly continuous.
We have to show that for any there exists such that
Consider first the second term in the decomposition
The above proof exploits the special structure of the model and provides insight into it. The result, in fact, follows from the continuity of the mapping Φ and the compactness of the metric space .
If the correspondence is invertible, then given a sequence there exists a unique control sequence such that for every n. Therefore, can be determined from the knowledge of and .
Suppose that Assumption 2 holds so that the mapping Φ is one-to-one. Then, : is continuous.
Given , we have to prove that there exists such that
Now suppose that, contrarily to what we need to prove, there is an n, , such that . For brevity assume , i.e.
First of all, by Proposition 2 and Equation (32)
Notice that we did not assume Φ to be continuous in Lemma 5. With Lemmas 4 and 5 we derive a different version of Theorem 2.
Suppose that Assumptions 2 and 3 hold. Then the mapping Φ: defined as is a homeomorphism, hence , i.e., .
6. Numerical Simulations
In this section we illustrate the previous theoretical results with a few numerical simulations. As in  we resort to two-state Markov processes with transition probability matrices
6.1. Case 1
The topological entropy of these two maps in bits per iteration (i.e., taking logarithms to base 2) are
It turns out that, except for the periodic switching , the complexity of the switched dynamics is higher than the complexity of the control switching sequence, and also higher than the entropies of the constituent maps, see Equation (41). This provides an instance of Parrondo’s paradox with regard to the complexity as measured by the topological entropy.
6.2. Case 2
Consider the constituent affine maps
A special feature of the matrices is that they are simultaneous diagonalizable. Indeed, in the common eigenvector base
Consider the time-switched dynamics associated with two maps having global attractors . In this paper we studied the relation between the topological entropy of the control sequence generator and the topological entropy of the switched dynamics. Compared with the previous paper , we found some new technical obstacles.
First of all, we had to circumvent the shortcoming that the map is not necessarily continuous in ; here is a control sequence and is the corresponding component set of the pullback attractor, . To this end we assumed in Section 3 (“Results using measurability”), (i) the measurability of certain unions of component sets (Assumption 1); and (ii) the injectivity of . It follows then, Theorem 2, that Φ is a topological conjugacy between the shift on the control sequences, , and the shift on the trajectories of the switched dynamics up to the precision set by the component sets, . Being the entropy of coarse-grained trajectories, is a lower bound of , the topological entropy of the switched dynamics. We call the macroscopic entropy and write . Likewise, is the entropy of the control sequence generator, thus we write . In sum, [Equation (25) and Equation (26)] under the hypotheses (i) and (ii).
In Section 4 we surveyed sufficient conditions for the injectivity of Φ postulated in Section 3. We formulated our conclusions as Assumption 2, both in a general version, and in a special version when one of the maps is invertible.
In Section 5 (“Results using continuity”) we hypothesized from the outset that the mapping is continuous (Assumption 3) and paralleled the approach of . In Theorem 4 we proved that Φ is a homeomorphism (or topological conjugacy), Assumptions 2 and 3 granted. Once again we concluded that .
The main results of this paper, Theorems 3 and 4, are in line with what we call the complexity version of Parrondo’s paradox, namely, that the complexity of a switched dynamics may be higher than the complexity of its constituent maps. Moreover, they provide sufficient conditions as well, to wit: .
Finally, both possibilities (Case 1) and (Case 2) have been numerically illustrated in Section 6.
J.M.A. and A.G. were financially supported by the Spanish Ministerio de Economía y Competitividad, project MTM2012-31698. P.E.K. was financially supported by the Spanish Ministerio de Ciencia e Innovación, project MTM2011-22411, and Conserjería de Innovación, Ciencia y Empresa (Junta de Andalucía) under the Ayuda 2009/FQM314 and the Proyecto de Excelencia P07-FQM-02468.
Conflict of Interest
The authors declare no conflict of interest.
Recall that the Hausdorff semi-distance and Hausdorff distance between two nonempty subsets X and Y of a metric space are denoted by and , respectively, while
Let X and Y be compact subsets of . Then
If , then and . This amounts to
It follows from Proposition A1 that
Among the different properties of the Hausdorff distance, we mention here only two of them. Let be the set of all nonempty compact subsets of M.
If is complete, then is also a complete.
If is compact, then is also a compact.
In Section 2.2 we consider the metric space endowed with the Euclidean distance , as well as the complete, compact metric space , where .
Let be a compact metric space and a complete metric space. If is a continuous map, then there exists a real-valued function with such that, for any ,
The function ω here is called a modulus of continuity. Moduli of continuity are used to express in a convenient way both the continuity at a point and the uniform continuity of maps between metric spaces as, for instance, in Equation (49).
Let X and Y be compact subsets of such that . By Equation (48),
Let be the modulus of continuity of the uniformly continuous map f of into , i.e.,
All in all, given and compact such that , we can choose δ sufficiently small so that for , and . From Equation (50) we obtain
This, in fact, proves the uniform continuity (since is compact) of the set-valued map ∋ ∈ . Equation (49) is just an equivalent formulation of this fact. □
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