Int. J. Appl. Math. Comput. Sci., 2012, Vol. 22, No. 2, 259–267 DOI: 10.2478/v10006-012-0019-4 ERGODIC THEORY APPROACH TO CHAOS: REMARKS AND COMPUTATIONAL ASPECTS ∗ ∗∗ PAWEŁ J. MITKOWSKI ,WOJCIECH MITKOWSKI ∗ Faculty of Electrical Engineering, Automatics, Computer Science and Electronics AGH University of Science and Technology, al. Mickiewicza 30/B-1, 30-059 Cracow, Poland e-mail: [email protected] ∗∗Department of Automatics AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland e-mail: [email protected] We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the pre- viously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota’s conjecture concerning nontrivial ergodic properties of the model. Keywords: ergodic theory, chaos, invariant measures, attractors, delay differential equations. 1. Introduction man, 2001). Transformations (or flows) with an invariant measure display three main levels of irregular behaviour, In the literature concerning dynamical systems we can i.e., (ranging from the lowest to the highest) ergodicity, find many definitions of chaos in various approaches mixing and exactness. Between ergodicity and mixing we (Rudnicki, 2004; Devaney, 1987; Bronsztejn et al., 2004). can also distinguish light mixing, mild mixing and weak Our central issue here will be the ergodic theory appro- mixing (Lasota and Mackey, 1994; Silva, 2010) and, on ach. Ergodic theory in general has its origin in physical the level similar to exactness, the type of K-flows (or K- systems of a large number of particles, where microsco- property, K-automorphism) (cf. Rudnicki 1985a; 1985b; pic chaos leads to macroscopic (statistical) regularity. As 2004; Lasota and Mackey,1994). In this article we will the beginning of ergodic theory, the moment when Bolt- consider only ergodicity and mixing. First we formalize zmann formulated his famous ergodic hypothesis, in 1868 these notions and show some simple examples of ergodic (see, e.g., Nadzieja, 1996; Górnicki, 2001) or in 1871 and mixing transformations. Then in Section 3. we analy- (Lebowitz and Penrose, 1973), can probably be conside- ze an infinite dimensional system which additionally has red. For more information about the ergodic hypothesis, interesting medical (hematological) interpretations. consult also the works of Birkhoff and Koopman (1932) By {St}t≥0 we denote a semidynamical system or a as well as Dorfman (2001). semiflow on the metric space X, i.e., 2. Ergodic theory and chaos: Basic facts (i) S0(x)=x for all x ∈ X; + One of the most fundamental notions in ergodic theory (ii) St(St (x)) = St+t (x) for all x ∈ X,andt, t ∈ R ; is that of invariant measure (see Lasota and Mackey, 1994; Fomin et al., 1987; Bronsztejn et al., 2004; Rud- (iii) S : X × R+ → X is a continuous function of (t, x). nicki, 2004; Dawidowicz, 2007), which is a consequence of Liouville’s theorem (see, e.g., Szlenk, 1982; Landau By a measure on X we mean any probability measure de- and Lifszyc, 2007; Arnold, 1989; Nadzieja, 1996; Dorf- fined on the σ-algebra B(X) of Borel subsets of X.A 260 P.J. Mitkowski and W. Mitkowski measure μ is called invariant under a semiflow {St}t≥0, we can observe the result of the action of S on the en- −1 3 if μ(A)=μ(St (A)) for each t ≥ 0 and each A ∈B. semble of 10 points distributed randomly in the area [0, 0.1] × [0, 0.1]. The transformation (2) shifts the initial 2.1. Ergodicity. ABorelsetA is called invariant with area and does not spread the points over the space. When −1 respect to the semiflow {St}t≥0 if St (A)=A for all we measure the Euclidean distance during iterations be- t ≥ 0. We now denote by (S, μ) asemiflow{St}t≥0 with tween two arbitrarily chosen close points, we notice that an invariant measure μ.Thesemiflow(S, μ) is ergodic it is constant (Fig. 2(d)). Thus the popular criterion of cha- (we say also that the measure is ergodic) if the measure os, i.e., sensitivity to initial conditions, is not a property of μ(A) of any invariant set A equals 0 or 1. Let us now ergodic transformations. Their property is the dense tra- consider two simple examples. jectory (we formalize this fact in the last paragraph of this section). Example 1. Let S :[0, 2π) → [0, 2π) be a transformation One of the most important theorems in ergodic the- generating rotation through an angle φ on a circle with ory is the Birkhoff individual ergodic theorem (Birkhoff, unit radius (see Lasota and Mackey, 1994; Bronsztejn et 1931a; 1931b; Birkhoff and Koopman, 1932; Lasota and al., 2004; Devaney, 1987; Dorfman, 2001): Mackey, 1994, Fomin et al., 1987; Szlenk 1982; Dawi- dowicz, 2007; Nadzieja, 1996; Gornicki, 2001; Dorfman, S(x)=x + φ (mod 2π). (1) 2001). Here we cite a popular extension of this theorem (see Lasota and Mackey, 1994, p. 64; Fomin et al., 1987, If φ/2π is rational, we can find invariant sets which have p. 46). Recall that by (S, μ) we denote a semiflow {St}t≥0 S measure different from 0 or 1, and thus is not ergodic. with an invariant measure μ. However, if φ/2π is irrational, then S is ergodic (for a proof, see the work of Lasota and Mackey (1994,√ p. 75) Theorem 1. (Extension of the Birkhoff theorem) Let or Devaney (1987, p. 21)). If we take, e.g., φ = 2 and (S, μ) be ergodic. Then, for each μ-integrable function pick an arbitrary point on the circle, we can observe that f : X → R, the mean of f along the trajectory of S is successive iterations of this point under the action of S equal almost everywhere to the mean of f over the space will densely fill the whole available space (circle) (see Fig. X, that is, 1). 1 T 1 Example 2. To understand better the typical features of lim f(St(x)) dt = f(x) μ(dx), (3) T →∞ T 0 μ(X) X ergodic behaviour, let us consider the following transfor- mation (see Lasota and Mackey, 1994, p. 68): μ √ √ -almost everywhere. S(x, y)=( 2+x, 3+y)(1). mod (2) If we substitute f = 1A in Eqn. (3) (1A is the cha- racteristic function of A) (see Lasota and Mackey, 1994; This is an extension of the rotational transformation (1) Rudnicki, 2004; Dawidowicz, 2007), then the left-hand on the space [0, 1] × [0, 1] → [0, 1] × [0, 1].InFig.2 1 1 1 90 0.2 120 60 0.15 150 0.1 30 0 1 0 1 0 1 0.05 (a) (b) (c) −6 x 10 180 0 1 210 330 0 1 50 100 240 300 iteration 270 (d) Fig. 2. Iterations of the ergodic transformation (2) acting on an Fig. 1. Normalized (to the probability density function) round ensemble of 103 points randomly distributed in [0, 0.1]× histogram (bars inside the circle) showing that a single [0, 0.1]: 1st iteration (a), 2nd iteration (b), 3rd iteration point under√ the action of the ergodic transformation (1) (c), Euclidean metric between two arbitrarily chosen clo- with φ = 2 fills densely the whole circle. se points from the ensemble (d). Ergodic theory approach to chaos: Remarks and computational aspects 261 side of (3) is the mean time of visiting the set A and (see Lasota and Mackey, 1994, p. 57, pp. 65–68) the right-hand side is μ(A), and this corresponds to er- S(x, y)=(x + y,x +2y)(1). godicity in the sense of Boltzmann, which roughly spe- mod (5) aking is the mean time that a particle of a physical sys- This is an example of the Anosov diffeomorphism tem spends in some region and it is proportional to its (Anosov, 1963) (see also Bronsztejn et al., 2004, p. 903). natural probabilistic measure (Dawidowicz, 2007; Dorf- In Fig. 3 we can see the first the fifth and the tenth ite- man, 2001; Nadzieja, 1996; Górnicki, 2001; Birkhoff and ration of the mixing tranformation (5) acting on the en- Koopman, 1932; Lebowitz and Penrose, 1973) semble of 103 points distributed randomly in the area We can see that ergodic behaviour in the “pure” form [0, 0.1] × [0, 0.1]. The points are being spread over the does not need to be very irregular and unpredictable. In space and afterwards that transformation is literally mi- fact, an invariant and ergodic measure should have so- xing these points in the whole space. The Euclidean di- me additional properties to be interesting from the po- stance between close points first grows quickly and then int of view of dynamics. Briefly speaking, it should be fluctuates irregularly (Fig. 3 (d)). The difference between nontrivial—for example, we intuitively understand that to the ergodic transformation (2) (cf. Fig. 2) is noticeable. have interesting dynamics the measure should not be con- Typical for mixing is the sensitivity to initial conditions centrated on a single point. According to our knowled- (we will formalize this fact further on). ge, two approaches to this problem appear in the litera- ture. In the main ideas, both seem to be similar, but in We can say more about the chaoticity of mixing sys- the literature exist separately.