The topology of chaotic iterations
aa r X i v : . [ n li n . C D ] O c t The topology of chaotic iterations
Jacques M. Bahi, Christophe GuyeuxLaboratoire d’Informatique de l’universit´e de Franche-Comt´e,90000 Belfort cedexT´el: 03 84 58 77 94; fax: 03 84 58 77 32e-mail: [email protected], [email protected]
Abstract
Chaotic iterations have been introduced on the one hand by Chazan, Mi-ranker [6] and Miellou [10] in a numerical analysis context, and on the otherhand by Robert [12] and Pellegrin [11] in the discrete dynamical systems frame-work. In both cases, the objective was to derive conditions of convergence ofsuch iterations to a fixed state. In this paper, a new point of view is presented,the goal here is to derive conditions under which chaotic iterations admit achaotic behaviour in a rigorous mathematical sense. Contrary to what has beenstudied in the literature, convergence is not desired.More precisely, we establish in this paper a link between the concept ofchaotic iterations on a finite set and the notion of topological chaos [9], [7], [8].We are motivated by concrete applications of our approach, such as the use ofchaotic boolean iterations in the computer security field. Indeed, the conceptof chaos is used in many areas of data security without real rigorous theoreticalfoundations, and without using the fundamental properties that allow chaos.The wish of this paper is to bring a bit more mathematical rigour in this field.This paper is an extension of[3], and a work in progress.
Let us consider the system B = {
0; 1 } , in which each of the two cells c i iscaracterized by a boolean state e i . An evolution rule is, for example, f : B −→ B ( e , e ) ( e + e , e )These cells can be updated in a serial mode (the elements are iterated in asequential mode, at each time only one element is iterated), in a parallel mode(at each time, all the elements are iterated), or by following a sequence ( S n ) n ∈ N :the n th term S n is constituted by the block components to be updated at the n th iteration. This is the chaotic iterations , and S is called the strategy . Let usnotice that serial and parallel modes are particular cases of chaotic iterations.Until now, only the conditions of convergence have been studied.1 priori, the chaotic adjective means “in a disorder way”, and has nothing todo with the mathematical theory of chaos, studied by Li-Yorke [9], Devaney [7],Knudsen [8], etc. We asked ourselves what it really was.In this paper we study the topological evolution of a system during chaoticiterations. To do so, chaotic iterations have been written in the field of discretedynamical system: (cid:26) x ∈ X x n +1 = f ( x n )where ( X , d ) is a metric space (for a distance to be defined), and f is continuous.Thus, it becomes possible to study the topology of chaotic iterations. Moreexactly, the question: “Are the chaotic iterations a topological chaos ?” hasbeen raised.This study is the first of a series we intend to carry out. We think thatthe mathematical framework in which we are placed offers interesting new toolsallowing the conception, the comparison and the evaluation of new algorithmswhere disorder, hazard or unpredictability are to be considered.The rest of the paper is organised as follows.The first next section is devoted to some recalls on the domain of topologicalchaos and the domain of discrete chaotic iterations. In third section is definedthe framework of our study. Fourth section presents the first results concerningthe topology (compacity) of the chaotic iterations. Fifth and sixth sectionsconstitute the study of the chaotic behaviour of such iterations. In section 7,the computer and so the finite set of machine numbers is considered. The paperends with some discussions and future work. This section is devoted to basic notations and terminologies in the fields oftopological chaos and chaotic iterations.
In the sequel S n denotes the n th term of a sequence S , V i denotes the i th component of a vector V , and f k = f ◦ ... ◦ f denotes the k th composition of afunction f . Finally, the following notation is used: J N K = { , , . . . , N } .Let us consider a system of a finite number N of cells , so that each cell hasa boolean state . Then a sequence of length N of boolean states of the cellscorresponds to a particular state of the system .A strategy corresponds to a sequence S of J N K . The set of all strategies isdenoted by S . Definition 1
Let S ∈ S . The shift function is defined by σ : S −→ S ( S n ) n ∈ N ( S n +1 ) n ∈ N initial function is the map which associates to a sequence, its first term i : S −→ J N K ( S n ) n ∈ N S . The set B denoting { , } , let f : B N −→ B N be a function, and S ∈ S be astrategy. Then, the so called chaotic iterations are defined by x ∈ B N , ∀ n ∈ N ∗ , ∀ i ∈ J N K , x ni = (cid:26) x n − i if S n = i ( f ( x n )) S n if S n = i. (1)In other words, at the n th iteration, only the S n − th cell is “iterated”. Notethat in a more general formulation, S n can be a subset of components, and f ( x n ) S n can be replaced by f ( x k ) S n , where k n , modelizing for example delaytransmission (see e.g. [2]). For the general definition of such chaotic iterations,see, e.g. [12]. Consider a metric space ( X , d ), and a continuous function f : X −→ X . Definition 2 f is said to be topologically transitive if, for any pair of open sets U, V ⊂ X , there exists k > f k ( U ) ∩ V = ∅ . Definition 3 ( X , f ) is said to be regular if the set of periodic points is densein X . Definition 4 f has sensitive dependence on initial conditions if there exists δ > x ∈ X and any neighbourhood V of x , there exists y ∈ V and n > | f n ( x ) − f n ( y ) | > δ . δ is called the constant of sensitivity of f . Definition 5 f is said to have the property of expansivity if ∃ ε > , ∀ x = y, ∃ n ∈ N , d ( f n ( x ) , f n ( y )) > ε. Then, ε is the constant of expansivity of f. We also say f is ε -expansive. Remark 1
A function f has a constant of expansivity equals to ε if an arbitrarysmall error on any initial condition is amplified till ε . In this section we will put our study in a topological context by defining asuitable metric set. 3 .1 The iteration function and the phase space
Let us denote by δ the discrete boolean metric , δ ( x, y ) = 0 ⇔ x = y, and definethe function F f : J N K × B N −→ B N ( k, E ) (cid:16) E j .δ ( k, j ) + f ( E ) k .δ ( k, j ) (cid:17) j ∈ J N K , where + and . are boolean operations.Consider the phase space X = J N K N × B N , and the map G f ( S, E ) = ( σ ( S ) , F f ( i ( S ) , E )) (2)Then one can remark that the chaotic iterations defined in (1) can be describedby the following iterations (cid:26) X ∈ X X k +1 = G f ( X k ) . The following result can be easily proven, by comparing S and R , Theorem 1
The phase space X has the cardinality of the continuum. Note that this result is independent on the number of cells.
We define a new distance between two points (
S, E ) , ( ˇ S, ˇ E ) ∈ X by d (( S, E ); ( ˇ S, ˇ E )) = d e ( E, ˇ E ) + d s ( S, ˇ S ) , where d e ( E, ˇ E ) = N X k =1 δ ( E k , ˇ E k ) ,d s ( S, ˇ S ) = 9 N ∞ X k =1 | S k − ˇ S k | k . It should be noticed that if the floor function ⌊ d ( X, Y ) ⌋ = n , then thestrategies X and Y differs in n cells and that d ( X, Y ) − ⌊ d ( X, Y ) ⌋ gives ameasure on how the strategies S and ˇS diverge. More precisely, • This floating part is less than 10 − k if and only if the first k th terms of thetwo strategies are equal. • If the k th digit is nonzero, then the k th terms of the two strategies arediferent. 4 .3 The topological framework It can be proved that,
Theorem 2 G f is continuous on ( X , d ) . Proof
We use the sequential continuity (we are in a metric space).Let ( S n , E n ) n ∈ N be a sequence of the phase space X , which converges to( S, E ). We will prove that ( G f ( S n , E n )) n ∈ N converges to G f ( S, E ). Let usrecall that for all n , S n is a strategy, thus, we consider a sequence of strategy( i.e. a sequence of sequences).As d (( S n , E n ); ( S, E )) converges to 0, each distance d e ( E n , E ) and d s ( S n , S )converges to 0. But d e ( E n , E ) is an integer, so ∃ n ∈ N , d e ( E n , E ) = 0 for any n > n .In other words, there exists threshold n ∈ N after which no cell will change itsstate: ∃ n ∈ N , n > n = ⇒ E n = E. In addition, d s ( S n , S ) −→ , so ∃ n ∈ N , d s ( S n , S ) < − for all indices greaterthan or equal to n . This means that for n > n , all the S n have the same firstterm, which is S : ∀ n > n , S n = S . Thus, after the max ( n , n ) − th term, states of E n and E are the same, andstrategies S n and S start with the same first term.Consequently, states of G f ( S n , E n ) and G f ( S, E ) are equal, then distance d between this two points is strictly less than 1 (after the rank max ( n , n )).We now prove that the distance between ( G f ( S n , E n )) and ( G f ( S, E )) is con-vergent to 0. Let ε > • If ε >
1, then we have seen that the distance between ( G f ( S n , E n )) and( G f ( S, E )) is strictly less than 1 after the max ( n , n ) th term (same state). • If ε <
1, then ∃ k ∈ N , − k > ε > − ( k +1) . But d s ( S n , S ) converges to0, so ∃ n ∈ N , ∀ n > n , d s ( S n , S ) < − ( k +2) , after n , the k + 2 first terms of S n and S are equal.As a consequence, the k + 1 first entries of the strategies of G f ( S n , E n ) and G f ( S, E ) are the same (because G f is a shift of strategies), and due to thedefinition of d s , the floating part of the distance between ( S n , E n ) and ( S, E )is strictly less than 10 − ( k +1) ε .In conclusion, G f is continuous, ∀ ε > , ∃ N = max ( n , n , n ) ∈ N , ∀ n > N , d ( G f ( S n , E n ); G f ( S, E )) ε. Then chaotic iterations can be seen as a dynamical system in a topologicalspace. In the next section, we will study the compacity of such a topologicalspace, with a view to prove the expansive chaos in section 6.3.5
Compacity
To prove that ( X , G f ) is a compact topological space, we have to check whetherit is separate or not. Then, the sequential characterisation of the compacity forthe metric spaces will be used to obtain the result. This section starts with some basic recalls...
Definition 6
A topological space (
X, τ ) is said to be a separated space if forany points x = y ∈ X , there exist two open sets ω x , ω y such that x ∈ ω x , y ∈ ω y and ω x ∩ ω y = ∅ . Theorem 3 ( X , G f ) is a separated space. Proof
Let (
E, S ) = (ˆE , ˆS).1. If E = ˆE, then the intersection between the two balls B (cid:0) ( E, S ) , (cid:1) and B (cid:16) (ˆE , ˆS) , (cid:17) in empty.2. Else, it exists k ∈ N such that S k = ˆS k , then the balls B (cid:0) ( E, S ) , − ( k +1) (cid:1) and B (cid:16) (ˆE , ˆS) , − ( k +1) (cid:17) can be chosen. Definition 7
A topological space (
X, τ ) is said to be compact if it is a separatedspace, and if each of its open covers has a finite subcover.
Definition 8
Let (
X, τ ) be a topological space, and A a subset of X . a ∈ A isan accumulation point if ∀ V ∈ V a , V ∩ A = ∅ , and V ∩ A = { a } .Let us now recall the sequential characterisation of the compacity for themetric spaces: Theorem 4
Let ( E, d ) be a metric space, and K ⊂ E . The following propertiesare equivalents:1. K is a compact space.2. For any sequence of K , it can be possible to extract another sequence whichconverge in K .3. Any sequence of K has an adherence value in K .4. Any infinite subset of K has an accumulation point in K . .3 Compacity result Theorem 5 ( X , d ) is a compact metric space. Proof
First, ( X , d ) is a separate space.Let ( E n , S n ) n ∈ N be a sequence of X .1. A state E ˜n which appears an infinite number of time in this sequence canbe found. Let I = { ( E n , S n ) | E n = E ˜n } . For all (
E, S ) ∈ I , S n [0] ∈ J , N K , and I is an infinite set. Then it can befound ˜k ∈ J , N K such that an infinite number of strategies of I start with˜k.Let n be the smallest integer such that E n = E ˜n and S n [0] = ˜k.2. The set I ′ = { ( E n , S n ) | E n = E n et S n [0] = S n [0] } is infinite, then one of the element of J , N K will appear an infinite numberof times in the S n [1] of I ′ : let us call it ˜l.Let n be the smallest n such that ( E n , S n ) ∈ I ′ and S n [1] = ˜l.3. The set I ′′ = { ( E n , S n ) | E n = E n , S n [0] = S n [0] , S n [1] = S n [1] } is infinite, etc. Let l = (cid:0) E n , ( Sn k [ k ]) k ∈ N (cid:1) , then the subsequence ( E n k , S n k ) converge to l . To prove that we are in the framework of topological chaos, we have to checksome topological conditions.
Theorem 6
Periodic points of G f are dense in X . Proof
Let (
S, E ) ∈ X , and ε >
0. We are looking for a periodic point ( S ′ , E ′ )satisfying d (( S, E ); ( S ′ , E ′ )) < ε .We choose E ′ = E , and we reproduce enough entries from S to S ′ so thatthe distance between ( S ′ , E ) and ( S, E ) is strictly less than ε : a number k = ⌊ log ( ε ) ⌋ + 1 of terms is sufficient.After this k th iterations, the new common state is E , and strategy S ′ is shiftedof k positions: σ k ( S ′ ).Then we have to complete strategy S ′ in order to make ( E ′ , S ′ ) periodic (atleast for sufficiently large indices). To do so, we put an infinite number of 1 tothe strategy S ′ .Then, either the first state is conserved after one iteration, so E is unchangedand we obtain a fixed point. Or the first state is not conserved, then: if the first7tate is not conserved after a second iteration, then we will be again in the firstcase above (due to the fact that a state is a boolean). Otherwise the first stateis conserved, and we have indeed a fixed (periodic) point.Thus, there exists a periodic point into every neighbourhood of any point,so ( X , G f ) is regular, for any map f . Contrary to the regularity, the topological transitivity condition is not automat-ically satisfied by any function( f = Identity is not topologically transitive). Let us denote by T the set ofmaps f such that ( X , G f ) is topologically transitive. Theorem 7 T is a nonempty set. Proof
We will prove that the vectorial logical negation function f f : B N −→ B N ( x , . . . , x N ) ( x , . . . , x N ) (3)is topologically transitive.Let B A = B ( X A , r A ) and B B = B ( X B , r B ) be two open balls of X , where X A = ( S A , E A ), and X B = ( S B , E B ). Our goal is to start from a point of B A and to arrive, after some iterations of G f , in B B .We have to be close to X A , then the starting state E must be E A ; it remainsto construct the strategy S . Let S n = S nA , ∀ n n , where n is chosen in sucha way that ( S, E A ) ∈ B A , and E ′ be the state of G n f ( S A , E A ). E ′ difers from E B by a finite number of cells c , . . . , c n . Let S n + n = c n , ∀ n n . Then the state of G n + n f ( S, E ) is E B .Last, let S n + n + n = S nB , ∀ n n , where n is chosen in such a way that G n + n f ( S, E ) is at a distance less than r B from ( S B , E B ). Then, starting froma point ( S, E ) close to X A , we are close to X B after n + n iterations: ( X , G f )is transitive. Remark 1
If, in the preceeding proof, the strategy were completed using S B ,then it can be proved that there exist a point X close to X A , and k ∈ N , suchthat G k f ( X ) = X B : this property is called strong transitivity . Remark 2
The question of the characterisation of T will be discussed in an-other paper. Theorem 8 ( X , G f ) has sensitive dependence on initial conditions, and itsconstant of sensitiveness is equal to N . Proof
Let (
S, E ) ∈ X , and δ >
0. A new point ( S ′ , E ′ ) is defined by: E ′ = E , S ′ n = S n , ∀ n n , where n is chosen in such a way that d (( S, E ); ( S ′ , E ′ )) < δ ,and S ′ n + k = k, ∀ k ∈ J N K .Then the point ( S ′ , E ′ ) is as close as we want than ( S, E ), and systems of G k + N f ( S, E ) and G k + N f ( S ′ , E ′ ) have no cell presenting the same state: distancebetween this two points is greater or equal than N .8 emark 2 This sensitive dependence could be stated as a consequence of reg-ularity and transitivity (by using the theorem of Banks [4]). However, we havepreferred proving this result independently of regularity, because the notion ofregularity must be redefined in the context of the finite set of machine numbers(see section 7.2).
Theorem 9 ( X , G f ) is an expansive chaotic system. Its constant of expansiv-ity is equal to 1. Proof
If (
S, E ) = ( ˇ S ; ˇ E ), then: • Either E = ˇ E , and then at least one cell is not in the same state in E andˇ E . Then the distance between ( S, E ) and ( ˇ S ; ˇ E ) is greater or equal to 1. • Or E = ˇ E . Then the strategies S and ˇ S are not equal. Let n be the firstindex in which the terms S and ˇ S difer. Then ∀ k < n , G n f ( S, E ) = G kf ( ˇ S, ˇ E ) , and G n f ( S, E ) = G n f ( ˇ S, ˇ E ), then as E = ˇ E, the cell which has changedin E at the n -th iterate is not the same than the cell which has changedin ˇ E , so the distance between G n f ( S, E ) and G n f ( ˇ S, ˇ E ) is greater or equalto 2. Remark 3
It can be easily proved that ( X , G f ) is not A -expansive, for any A >
Definition 9
A discrete dynamical system is said to be K -chaotic if:1. it possesses a dense orbit,2. it has sensitive dependence on initial conditions. Theorem 10 If X is a compact space, then being regular and transitive impliesbeing K -chaotic. Theorem 11 ( X , G f ) is chaotic in the sense of Knudsen, ∀ f ∈ T . Proof X is a compact space, and ( X , G f ) is regular and transitive, then( X , G f ) is K -chaotic. 9 .2 Devaney’s chaos Let us recall the definition of a chaotic topological system, in the sense of De-vaney [7]:
Definition 10 f : X −→ X is said to be D − chaotic on X if ( X , f ) is regular,topologically transitive, and has sensitive dependence on initial conditions.If f ∈ T , then ( X , G f ) is topologically transitive, regular and has sensitivedependence on initial conditions. Then we have the result. Theorem 12 ∀ f ∈ T 6 = ∅ , G f is a chaotic map on ( X , d ) in the sense ofDevaney. Definition 11
A discrete dynamical system is said to be E -chaotic if it hastransitive, regular and expansive properties. Theorem 13 ∀ f ∈ T , ( X , G f ) is E -chaotic. Proof ( X , G f ) is D -chaotic, and has the expansive property, then ( X , G f ) is E -chaotic.We have proven that under the transitivity condition of f , chaotic iterationsgenerated by f can be described by a chaotic map on a topological space indiferent senses.We have considered a finite set of states B N and a set S of strategies composedby an infinite number of infinite sequences. In the following section we willdiscuss the impact of these assumptions in the context of the finite set of machinenumbers. Let (
X, d ) be a compact metric space and f : XX be a continuous map. Foreach natural number n , a new metric d n is defined on X by d n ( x, y ) = max { d ( f i ( x ) , f i ( y )) : 0 ≤ i < n } . Given any ε > n >
1, two points of X are ε -close with respect to thismetric if their first n iterates are ε -close.This metric allows one to distinguish in a neighborhood of an orbit the pointsthat move away from each other during the iteration from the points that traveltogether. A subset E of X is said to be ( n, ε )-separated if each pair of distinctpoints of E is at least ε apart in the metric d n . Denote by H ( n, ε ) the maximumcardinality of an ( n, ε )-separated set. Definition 12
The topological entropy of the map f is defined by (see e.g. [1]or [5]) h ( f ) = lim ǫ → (cid:18) lim sup n →∞ n log H ( n, ε ) (cid:19) . .4.2 ResultTheorem 14 Entropy of ( X , G f ) is infinite. Proof
Let E , ˇE ∈ B N such that ∃ i ∈ J , N K , E i = ˇE i . Then, ∀ S , ˇS ∈ S , d ((E , S); (ˇE , ˇS)) > c of S is infinite, then ∀ n ∈ N , c > e n .Then for all n ∈ N , the maximal number H ( n,
1) of ( n, − separated pointsis greater than or equal to e n , so h top ( G f ,
1) = lim n log ( H ( n, > lim n log (cid:16) e n (cid:17) = lim ( n ) = + ∞ But h top ( G f , ε ) is an increasing function when ε is decreasing, then h top ( G f ) = lim h → h top ( G f , ε ) > h top ( G f ,
1) = + ∞ We have proven that it is possible to find f , such that chaotic iterationsgenerated by f can be described by a chaotic and entropic map on a topologicalspace in the sense of Devaney. We have considered a finite set of states B N anda set S of strategies composed by an infinite number of infinite sequences. In thefollowing section we will discuss the impact of these assumptions in the contextof the finite set of machine numbers. In the computer science framework, we also have to deal with a finite set ofstates of the form B N and the set S of sequences of J N K is infinite (countable),so in practice the set X is also infinite. The only diference with respect to thetheoretical study comes from the fact that the sequences of S are of finite butnot fixed length in the practice.The proof of the continuity, the transitivity and the sensitivity conditions areindependent of the finitude of the length of strategies (sequences of S ), so evenin the case of finite machine numbers, we have the two fundamental propertiesof chaos: sensitivity and transitivity, which respectively implies unpredictabilityand indecomposability (see [7], p.50). The regularity property has no meaningin the case of finite systems because of the notion of periodicity.We propose a new definition in order to bypass the notion of periodicity inpractice. Definition 13
A strategy S = ( S , ..., S L ) is said cyclic if a subset of successiveterms is repeated from a given rank, until the end of S . A point of X that admitsa cyclic strategy is called a cyclic point .For example, • (1 , , , , , , ,
2) and (1 , , , , , , ,
2) are cyclic,11 but (1 , , , , ,
2) and (1 , , , ,
3) are not cyclic.This definition can be interpreted as the analogous of periodicity on finitesets.Then, following the proof of regularity (section 5.1), it can be proved that theset of cyclic points is dense on X , hence obtaining a desired element of regularityin finite sets, as quoted by Devaney ([7], p.50): two points arbitrary close toeach other could have diferent behaviours, the one could have a cyclic behaviouras long as the system iterates while the trajectory of the second could ”visit”the whole phase space.It should be recalled that the regularity was introduced by Devaney in orderto counteract the transitivity and to obtain such a property: two points closeto each other can have fundamental diferent behaviours. It is worthwhile to notice that even if the set ofmachine numbers is finite, we deal with strategiesthat have a finite but unbounded length. Indeed, itis not necessary to store all the terms of the strat-egy in the memory, only the n th term (an integerless than or equal to N ) of the strategy has to bestored at the n th step, as it is illustrated in the fol-lowing example.Let us suppose that a given text is input fromthe outside world in the computer character by char-acter, and that the current term of the strategy isgiven by the ASCII code of the current stored char-acter. Then, as the set of all possible texts of theoutside world is infinite and the number of theircharacters is unbounded, we have to deal with aninfinite set of finite but unbounded strategies.Of course, the preceding example is a simplistic illustrating example. Achaotic procedure should to be introduced to generate the terms of the strategyfrom the stream of characters.In conclusion, even in the computer science framework our previous theoryapplies. We proved that discrete chaotic iterations are a particular case of topologicalchaos, in sense of Devaney, Knudsen and expansivity, if the iteration functionis topologically transitive, and that the set of topologically transitive functionsis non void. 12his theory has a lot of applications, because of the high number of situationsthat can be described with the chaotic iterations: neural networks, cellularautomata, multi-processor computing, , and so on. If this system is requested toevolve in an apparently disorderly manner, e.g. for security reasons (encryption,watermarking, pseudo-random number generation, hash functions, etc. ), ourresults could be useful.More concretely, for example, any medium (text, image, video, etc. ) canbe considered to be an agregation of elementary cells (respectively: character,pixel, image). Thus a digital watermarking of this medium can be describe as aninsertion of cells of a watermark into some cells of a carrier image (the state ofthe system), in a deterministic but unpredictable manner carried by a strategy.Moreover, the theory brings another way to compare two given algorithmsconcerned by disorder (evaluation of theirs constants of sensitivity, expansivity, etc. ), which can be seen as a complement of existing statistical evaluations.In future work, other forms of chaos (such as Li-York chaos [9]) will be stud-ied, other quantitative and qualitative tools such as entropy (see e.g. [1] or [5])will be explored, and the domain of applications of our theoretical concepts willbe enlarged.
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