On new chaotic mappings in symbol space

Transcription

1 Acta Mech Sin DOI /s RESEARCH PAPER On new chaotic mappings in symbol space Inese Bula Jānis Buls Irita Rumbeniece Received: 24 August 2010 / Revised: 15 September 2010 / Accepted: 15 September 2010 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011 Abstract A well known chaotic mapping in symbol space is a shift mapping. However, other chaotic mappings in symbol space exist too. The basic change is to consider the process not only at a set of times which are equally spaced, say at unit time apart (a shift mapping), but at a set of times which are not equally spaced, say if the unit time can not be fixed. The increasing mapping as a generalization of the shift mapping and the k-switch mapping are introduced. The increasing and k-switch mappings are chaotic. Keywords Symbol space Chaotic mapping Increasing mapping k-switch mapping The project was supported by Latvian Scientific ( ) and ESF Project (2009/0223/1DP/ /09APIA/VIAA/008). I. Bula ( fi ) University of Latvia, Department of Mathematics, Zellu 8, Rīga, LV 1586, Latvia, and Institute of Mathematics and Computer Science of University of Latvia, Raina bulv. 29, Rīga, LV 1048, Latvia ibula@lanet.lv J. Buls I. Rumbeniece University of Latvia, Department of Mathematics, Zellu 8, Rīga, LV 1586, Latvia 1 Introduction The technique of characterizing the orbit structure of a dynamical system via infinite sequences of symbols is known as symbolic dynamics. If the process is a discrete one such as the iteration of a function f, then the theory hopes to understand the eventual behavior of the points (the orbit of x by f ) x, f (x), f 2 (x),, f n (x), as n becomes large. That is, dynamical systems ask two somewhat nonmathematical sounding questions: where do points go and what do they do when they get there? A well known chaotic mapping in symbol space is a shift mapping [1 4]. However, other chaotic mappings in symbol space exist too. The basic change is to consider the process (physical or social phenomenon) not only at a set of times which are equally spaced, say at unit time apart (a shift mapping), but at a set of times which are not equally spaced, say if the unit time (an increasing mapping) can not be fixed. There is a philosophy of modelling in which the idealized systems that have properties and can be closely approximated by physical systems are studied. The experimentalist takes the view that only quantities that can be measured have meaning. This is a mathematical reality that underlies what the experimentalist can see. The article is structured as follows. It starts with preliminaries concerning notations and terminology that is used in the paper followed by a definition of the chaotic mapping. The increasing mapping is considered in Sect. 3. The k- switch mapping is considered in Sect. 4. Much of what many researchers consider dynamical systems has been deliberately left out of this text. For example, the continuous

2 2 I. Bula, et al. systems or differential equations are not be treated at all. For this reason, the Sect. 5 is devoted to some interpretations. 2 Preliminaries This section presents the notation and terminology used in this paper. Terminology comes from combinatorics on words (for example, Ref. [5] or Ref. [6]). Some notations at first are given Z + = {x x Z & x > 0}, N = Z + {0}, k, n = {k, k + 1,, n}, k n and k, n N. where Z is the set of integers. From now on A will denote a finite alphabet, i.e., a finite nonempty set {a 0, a 1,, a n } and the elements are called letters. Assume that A contains at least two symbols. By A the set of all finite sequences of letters will be denoted, or finite words, this set contains empty word (or sequence) λ too. A + = A /{λ}. A word ω A + can be written uniquely as a sequence of letters as ω = ω 1 ω 2 ω 3 ω i ω l, with ω A, 1 i l. The integer l is called the length of ω and denoted ω. The length of λ is 0. An extension of the concept of finite word is obtained by considering infinite sequences of symbols over a finite set. One-sided (from left to right) infinite sequence or word, or simply infinite word, over A is any total map ω : N A. The set A ω contains all infinite words. A = A A ω.ifthe word u = u 0 u 1 u 2 A ω,whereu 0, u 1, u 2, A, then finite word u 0 u 1 u 2 u n is called the prefix ofu of length n + 1. The empty word λ is assumed to be the prefix ofu of length 0. Pref(u) = {λ, u 0, u 0 u 1, u 0 u 1 u 2,, u 0 u 1 u 2 u n, } is the set of all prefixes of word u. Secondly in A ametricd is introduced as follows. Definition 1 ([6]). Let u, v A. The mapping d : A A R is called a metric ( prefix metric) in the set A if 2 m, u v, d(u, v) = 0, u = v, where m = max{ ω ω Pref(u) Pref(v)}. The term chaos in reference to functions was first used in Li and Yorke s paper [7]. The following definition of Ref. [8] is used. Let (X,ρ) be metric space and A X and F A. Itsays that the set F is dense in A [1,4,8] if for each point x A and each ε>0, there exists y F such that d(x, y) <ε. It says that the function f is topologically transitive on A [1,4,8] if for any two points x and y in A and any ε>0, there is z A such that d(z, x) <εand d( f n (z), y) <εfor some n. It says that the function f : A A exhibits sensitive dependence on initial conditions [1,4,8] if there exists a δ>0such that for any x A and any ε > 0, there is a y A and natural number n such that d(x, y) < ε and d( f n (x), f n (y)) >δ. Definition 2 ([8]). The function f : A A is chaotic if (a) the set of periodic points of f is dense in X,(b) f is topologically transitive and (c) f exhibits sensitive dependence on initial conditions. Devaney s definition is not the only classification of a chaotic map. For example, another definition can be found in Ref. [4]. Also mappings with only one property sensitive dependence on initial conditions frequently are considered as chaotic (see Ref. [9]). Banks, Brooks, Cairns, Davis and Stacey [10] have demonstrated that for continuous functions, the defining characteristics of chaos are topological transitivity and the density of the set of periodic points. Theorem 1 ([10]). Let A be an infinite subset of metric space X and f : A A to be continuous. If f is topologically transitive on A and the set of periodic points of f is dense in A, then f is chaotic on A. But if the set of periodic points of f is dense in A and there is a point whose orbit under iteration of f is dense in the set A,then f is topologically transitive on A ([1]). Therefore in this case (A is infinite set) if f : A A to be continuous, then it is chaotic mapping. 3 Increasing mapping The notion of increasing mapping in Ref. [11] had been introduced. Let f ω (x) = x f (0) x f (1) x f (2) x f (i), i N, x A ω. In this case the function f is called the generator function of mapping f ω. Definition 3 ([11]). A function f : N N is called a positively increasing function if 0 < f (0) and i j [i < j f (i) < f ( j)]. The mapping f ω : A ω A ω is called the increasing mapping if its generator function f : N N is positively increasing. The well known shift map is increasing mapping in one-sided infinite symbol space A ω, in this case the generator function is a positively increasing function f : N N, where f (x) = x + 1. Theorem 2 ([11]). The increasing mapping f ω : A ω A ω is continuous in the set A ω. For increasing mapping f ω : A ω A ω exists a dense orbit in the set A ω.thesetofperiodic points of increasing mapping f ω : A ω A ω is dense in the set A ω. Theorem 3 ([11]). The increasing mapping f ω : A ω A ω is chaotic in the set A ω.

3 On new chaotic mappings in symbol space 3 Lastly two conclusions ([11]) about mappings in symbol space are recalled which are not chaotic: (1) If the generator function f : N N of mapping f ω : A ω A ω is such that f (0) = 0, then the generated mapping f ω is not chaotic in the set A ω ; (2) If the generator function f : N N of mapping f ω : A ω A ω is not one-to-one function, then the generated mapping f ω is not chaotic in the set A ω. 4 k-switch mapping In this section, another class of mappings such that are not increasing mappings are considered. The results of this section at first are represented in Ref. [12]. Definition 4. The mapping : A ω A ω is called k-switch mapping if s = s 0 s 1 s 2 s 3 s k s k+1 A ω (s) = s 1 s 2 s 3 s k s k+1 s k+2, where s i, i = 1, k, is the same symbol (letter) as s i or a A: s i = a. In other words, at first, this mapping is shift and secondly, this mapping switches some symbols (not more as k symbols). For example, let A = {0, 1} and f 1,3 (s 0 s 1 s 2 s 3 s 4 ) = s 1 s 2 s 3 s 4 s 5 It is a 3-switch mapping that switchesfirst and third symbols. Indices by mapping f show which symbols switches to another by the defined rule. For example, if s = , then f 1,3 (s) = If consider situation with A that contains at least three symbols A = {a 0, a 1, a 2,, a n }, then a rule which a i switches to a j and for every a j A exist only one a i A with this rule is defined. For example, if A = {a, b, c} and a switches to b, b to c and c to a,then f 1,3 (aaaaaaaaaa ) = babaaaaaa or f 1,3 (ababababab ) = cacababab. More formally, set a bijection : A A, andfix indices 1 i 1 < i 2 < < i n = k. Then (s 0 s 1 s 2 ) = σ 0 σ 1 σ 2 σ t where s t+1, if η i η = t + 1, σ t = s t+1, otherwise. Theorem 4. The k-switch mapping :A ω A ω is continuous in the set A ω. Proof. Fix word u A ω and ε > 0. It needs prove that there is δ>0such that whenever d(u, v) <δ, v A ω,then d( (u), (v)) <ε. Choose m such that 2 m <ε.let0<δ<2 (m+1).if d(u, v) <δ, then by definition of prefix metric follows that u i = v i for all i = 0, 1,, m. From definition of k-switch mapping: (1) if k < m, thenu i = v i, i = 1, 2,, k, andu i = v i, i = k + 1, k + 2,, m or (2) if k m,thenu i = v i, i = 1, 2,, m. Therefore d( (u), (v)) 2 m <ε. Since A ω is an infinite metric space and k-switch mapping : A ω A ω is continuous then by Theorem 1 for chaotic mapping is sufficient to prove that this mapping is topologically transitive on A ω and the periodic points of mapping are dense in A ω. Theorem 5. The k-switch mapping :A ω A ω is topologically transitive in the set A ω. Proof. Fixwordsu, v A ω and ε>0. It needs prove that there is word z A ω such that d(z, u) <εand that there exists iteration n such that d( f n (z), v) <ε. Choose m such that 2 m <ε. Then there exist a word z A ω such that u i = z i, i = 0, 1,, m 1, and therefore d(z, u) 2 m <ε. Choose n = m. Byn iterations f n (z) = z mz m+1 z m+k 1 z m+k z m+k+1, where the word z m z m+1 z m+k 1 is obtained from word z m z m+1 z m+k 1 switching some symbols several times but some symbols are not switched. Equivalent words are needed to find (1) if k < m, thenz m z m+1 z m+k 1 z m+k z m+k+1 z 2m 1 = v 0 v 1 v 2 v m 1 or (2) if k m,thenz m z m+1 z 2m 1 = v 0 v 1 v 2 v m 1. Symbol switch by rule so that it is possible to find z m, z m+1, z m+2,, z 2m 1 such that (1) or (2) follows has been defined. 4, z = u 0 u 1 u 2 u 3 z 4 z 5 z 6 z 7 A ω and f 1,3 : A ω A ω.then (the switch of symbol we denote with overline) f 1,3 (z) = u 1 u 2 u 3 z 4 z 5 z 6 z 7, f1,3 2 (z) = u 2u 3 z 4 z 5 z 6 z 7, f 3 1,3 (z) = u 3z 4 z 5 z 6 z 7 z 8, f 4 1,3 (z) = z 4z 5 z 6 z 7 z 8. If A = {0, 1}, then0 = 1, 1 = 0, 0 = 0and1 = 1. If v = v 0 v 1 v 2 v 3,thenz 4 = z 4 = v 0, z 5 = v 1, z 6 = v 3, z 7 = v 4. If A = {a, b, c} and a = b, b = c, c = a, andv = aabcaabc, then z 4 = a, z 5 = a, z 6 = b, z 7 = c, therefore z 4 = b, z 5 = c, z 6 = a, z 7 = c. It has proved that there exist z such that d(z, u) <εand d( f n (z), v) <ε.

4 4 I. Bula, et al. Theorem 6. The set of periodic points of k-switch mapping : A ω A ω is dense set in the set A ω. Proof. Let ε>0. Then there exists m such that 2 m <ε. Assume that u A ω and v is a prefix ofwordu of length m. Then there exist a point (infinite word) x A ω with the same prefixasvand this x is a periodic point for k-switch mapping with period m. In this case d(u, x) 2 m <ε. Let x = u 0 u 1 u 2 u m 1 x 0 x 1 x 2. Consider two cases: (1) if k < m, then the m-th iteration of x by : A ω A ω is in form f m(x) = y 0y 1 y k 1 x k x m 1. In this case from equality y 0 y 1 y k 1 x k x m 1 = u 0 u 1 u 2 u m 1 it is possible to determine symbols x 0, x 1, x 2,, x m 1 and then x is in the form x=u 0 u 1 u 2 u m 1 x 0 x 1 x m 1 x 0 x 1 x m 1 x 0 x 1 x m 1, and this point is periodic point with period m. 4, A = {a, b, c}, a = b, b = c, c = a, v = baca and f 1,3 : A ω A ω.then f 4 1,3 (x) = x 0x 1 x 2 x 3 x 0 x 1 x 2 x 3 x 0 x 1 x 2 x 3 and x 0 = b, x 1 = a, x 2 = c, x 3 = a therefore x 0 = c, x 1 = c, x 2 = b, x 3 = a. Theinfinite word x = baca ccba ccba ccba is a periodic point for f 1,3 with period 4 and the distance between u = baca and x is less than 2 4 <ε. (2) if k m,then f m (x) = y 0y 1 y m 1 y m y k 1 x k x k+1. In this case from equality y 0 y 1 y m 1 y m y k 1 = u 0 u 1 u 2 u m 1 x 0 x 1 x k m 1 it is possible to determine symbols x 0, x 1, x 2,, x k 1 and then x is in the form x = u 0 u 1 u 2 u m 1 x 0 x 1 x k m 1 x k m x k m+1 x k 1 x k m x k m+1 x k 1 x k m x k m+1 and this point is a periodic point with period m. 3, A = {a, b, c}, a = b, b = c, c = a, v = bac and f 1,2,5 is 5-switch mapping. Let x = bacx 0 x 1 x 2 x 3 x 4 x 2 x 3 x 4 x 2 x 3 x 4. Then f 3 1,2,5 (x) = x 0x 1 x 2 x 3 x 4 x 2 x 3 x 4 x 2 x 3 x 4 and equalities could be obtained x 0 = b, x 1 = a, x 2 = c, x 3 = x 0, x 4 = x 1, therefore x 0 = x 1 = c, x 2 = x 3 = x 4 = b and the point x = bac cc bbb bbb bbb bbb is a periodic point with period 3 for mapping f 1,2,5. Finally the following could be concluded. Theorem 7. The k-switch mapping : A ω A ω is chaotic in the set A ω. Two different classes of chaotic mappings have been demonstrated. It is possible for increasing mapping (from two symbols 0 and 1 space) to construct corresponding mapping in unit segment that is chaotic [13]. Similarly the chaotic map in the interval [0, 1] from every chaotic mapping of two symbols 0 and 1 space is obtained. Models with chaotic mappings are not predictable in long-term. 5 Interpretations Let x(t 0 ), x(t 1 ),, x(t n ), (1) be the flow of discrete signals. Suppose that the experimentally observed subsequence is as follow x(t 0 ), x(t 1 ),, x(t n ), (2) If T 0 = t 1, T 1 = t 2,, T n = t n+1,, the shift map could be obtained. Notice if the infinite word x = x 0 x 1 x n instead of sequence (1) is got, then the infinite word y = y 0 y 1 y n instead of Eq. (2) could be achieved. Here t y t = x t 1. Hence, the shift map f (t) = t + 1 is obtained, namely, y = f ω (x) = x f (0) x f (1) x f (2) x f (n) It did not claim that the function f (t) = t + 1 is chaotic on the real line R but this function as a generator creates the chaotic map f ω in the symbol space A ω had proved. It had proved [11] something more, namely, every positively increasing function g as a generator creates the chaotic map g ω in the symbol space A ω. In other words, if in our experiment only subsequence had been detected x(t 1 ), x(t 3 ),, x(t 2n 1 ), even then chaotic behavior can be revealed. Now it has proved that every k-switch mapping : A ω A ω is chaotic in the symbol space A ω.inotherwords, if in our experiment only subsequence detected x(t 1 ), x(t 2 ),, x(t n ), with some kind of regular distortion even then we can reveal chaotic behavior. References 1 Holmgren, R.A.: A First Course in Discrete Dynamical Systems. Universitext, 2nd edn., Springer-Verlag (1996)

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