RANDOM MAPPINGS WITH A SINGLE ABSORBING CENTER AND COMBINATORICS OF DISCRETIZATIONS OF THE LOGISTIC MAPPING

Distributions of basic characteristics of random mappings with a single absorbing center are calculated. Results explain some phenomena occurring 
in computer simulations of the logistic mapping.


Introduction
Analysis of combinatorical properties of discretizations of dynamical systems constitutes a new challenging and important area.In this area a special spot is occupied by analysis of discretizations of the logistic mapping f(x) 4x(1-x), xE[0,1].
(1.1) 1This research has been supported by the Australian Research Council Grant A 8913 2609.
2permanent address: Institute of Information Transmission Problems Russian Academy of Science, Moscow.This paper was initially written when Pokrovskii was working at Deakin University and was supported by School of Computing and Math- ematics.

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A. KLEMM and A. POKROVSKII The reason is that although this mapping is the very simplest example of continuous mapping with quasi-chaotic behavior [9,11], nevertheless, its discretizations demon- strate unexpected behavior in many respects [3].In particular, the methods suggested in [10] and refined in [8] are not adequate in investigating the mapping (1.1).In this paper we will show how some properties of discretizations of the logistic mapping can be explained on the basis of properties of a special family of random mappings.This paper extends an approach suggested in [6].
Other related studies include research on period lengths of one-dimensional discre- tized systems carried out in detail by Beck [1, 2], similar questions with different maps by Percival and Vivaldi  [13], and some general questions concerning noisy orbits by Nusse and Yorke [12].

Auxiliary Notations
In this paper [N denotes the N-dimensional coordinate Euclidean space; elements from R g will be denoted by x-(xl,...,xg).Let   QN {(xl,...,xN).0_ x n _ 1,n 1,...,N} be the unit cube in RN.For each N denote by #N the product measure over QN with identical absolutely continuous coordinate probability measures #.We will be interested in limit behavior as N--,cx of measures #N(sN)of some spe- cial sets S N which are described in this paragraph.Denote r0(x1) 1 and define in- ductively sequences of functions by N rN(xl"'"xN)-H (1-Xi), N-1,2,...
Define the functions D k(z;#)-#N xe 0<Yk(x)_<z RN(c; #) #N({x e QN'O <_ rN(x where inequality is understood to be coordinate-wise. Lemma 1" There exists the continuous uniform limit 0 <_ c _< 1, (2.4) F/c(z;#)lim DN(z;#) z e QI, (2.5) Prfi By the definition, the set yN(x) is a subset of yM(x) for N < M, x QM; therefore YkN(x) < YkM(x), N < M, x e QM (2.7) and further On the other hand, YN(x) depends only on the first N coordinates of x E QM and the measure #M is a coordinate probability measure, which yields The last two displayed inequalities imply DkN(z; #) >_ DM(z; #), z e Qk for each positive integer M > N (note that the inequality sign has been reversed while converting (2.7)into (2.8)).
Denote by ^N Yk (x) the k-th largest element of the set Yg(x), that is the last coordinate of YV(x).The inequality AN N(X) < (x) implies r(x) Yf(x), x e Q for all M _> N because all elements of the set yM(x)\yN(x) are not greater than r N(x).In particular, for each z Qk the following inclusion holds C{ --l<_i<_kmin {zi}}.
On the other hand, YkN(x) depends only on the first N coordinates of x E QM and the measure #M is a coordinate probability measure, which yields 0_< <_ 0_<
Denote by ^N 9k (x) the k-th largest element of the set yN(x), that is the last coordinate of YkN(x).The inequality implies ^N YN(x)-YkM(x), x e QM (1.9) for all M >_ N because all elements of the set yM(x)\yN(x) are not greater than r N(X).In particular, for each z E Qk the following inclusion hold {xQM'O<rN(x)< min {zi}) and, further, k DkM(z; #) >_ DN(z; #) RN(   Combining (1.8) and (1.10) yields min {z/t;#), z e Qk. (1.10 Because of continuity of the measure # for each positive e (1.11) N lim sup RN(ct; #) 0 e>0 andlim sup D k(z;#)-O.
Let Z be the set of all finite sequences with the sum 1.Let g(z) be the nonnegative scalar function on Z with the following properties: (a) g is symmetric, that is, the value g(z) does not change under permutations of coordinates of z Z; (b) the value g(z) does not change if we add to z some zero coordinates; (c) if z 1 is longer than z2 and they coincide for all coordinates but one then a( l) -< Examples are given by the maximal coordinate, the second maximal coordinate, the sum of squares of coordinates g,(z) E (zi)2 and many others.
Proof: This follows the same arguments as the proof of the previous lemma and so is omitted.
This lemma is effective as a tool for numerical computation of the corresponding limit functions.Consider the case when the measure # is given by the distribution function #([0, x]) 1 V/1 x (1.17) which will be important in the next section.Here usually the gap between upper and lower estimates is of the magnitude 10-3 for N-3 and decreases very fast in N. ..:. Figure 1.The limit functions FI(Z;#), Fg,(Z;)and the distribution #([0,z]) 1 V/1 z (bold).
Define the set (A,K) of all mappings : X(A,K)--,X(A,K) satisfying (i)-0 for _ 0. This collection is called a random mapping, with an absorbing center.The set { e X(A,K)'x _ 0} is the absorbing center; once a trajectory of enters this set it cannot leave.If S is a subset of (A,K), associated with some given property A, then the proportion of elements of which belong to S will be called the probability of the event A and is denoted by P(A; (A, g)).
Random mappings with an absorbing center are similar to mappings with a single attracting center [4, 5, 15].

Distributions of Basins of Attractions
For each E (A,K) the set X(A,K) is partitioned naturally into a disjoint union of basins of attractions of different cycles of the mapping .Denote by %() the set of cardinalities of basins of attractions and (here the i-th element of Ordk(%()) is defined as zero if is greater than the total number of different basin of attractions).Recall that the distribution function of the finite set S C Qk is defined as (; s) ({ s: < }) R(S) (2.1) for z E Qk (here and below IR(S) denotes the cardinality of the set S). Denote DN, k(z;A'K)-*;L+ " e and c)-e 0 ord ( , ,}.
The proof is relegated to the next section.By this proposition, only the value c a/v/-influences the limit behavior of distributions D%,k(z;[an],[bx]) as n--+oo.
This value c is called the absorbing coefficient.
It is instructional that even rather small values of the absorbing coefficient c-a/v/ influence significantly and in nonevident manner the behavior of the corresponding limit distributions.Figure 2 graphs the case c 0, that is the function from the previous section against the cases c-0.25, c--1 and c-2.Clearly the weak absorbing center c--0.25 increases significantly the corresponding distribution function whereas the strong absorbing center c 2 decreases it.Figure 2. Limit distributions F%, 1(; a/x/) of maximal basin distributions for a/v/--0 (bold)and for a/V-0.25, 1.0, 2.0 (from above).
A. KLEMM and A. POKROVSKII Consider for each the sum of squares This characteristic is especially important because this estimates the probability that two random points from X(A,K) generate the same cycle.Analogous to the previous corollary the following can be established.
Let us mention on simple explicit formula in this direction.Denote by %(i, ) the cardinality of the basin of attraction which contains a particular element E X(A,K) and introduce the function D%(c;)-) c; A+ We emphasize that this characteristic is a scalar function on [0, 1].Denote finally 1 E D%(c; ).

M%(a;A,K)-R(#(A+ K)) e
That is the mean value of funct'ions D%(a; ) over E (A,K).
A. KLEMM and A. POKROVSKII 2.2 Distribution of the Cycle Lengths Denote by () the set to cycle lengths of the mapping and denote

Ok(C) Ordk(C()).
Introduce the distribution function De, k(z; A, K) z; V/A + K where the operation is defined as in (2.1) with the difference that .belongs to the set of k-dimensional vectors with nonnegative components.In line with Proposition 1, it can be proved.Proposition 3: There exist the uniform limits where the equality with In particular, consider the case k-1.By the Stepanov formula 1+ioo 1 f E(ap) + p2 Fe, 1(6; 0, 1 &-S(a) e /2dp 1 -ioo with E(x) / e- see [14], Formula (16), item 9, p. 919 (note that "i" in front of the integral in this formula is a misprint).So Proposition 3 above implies .Limit distributions of normed maximal cycle length b-1 and for a/V'b --O, 0.25, 1.0, 2.0 (from below in this order).
Denote by (i; ), E X(A, K) the length of the unique cycle which is generated by a particular element E X(A, K) and DC(; the mean value of functions De(a; b) over E (A, K).It is convenient to define (0, K) as a completely random mapping on the set X(0, K) { 1,..., K}, that is, as the totality of all possible mappings X(O,K)X(O,K).Denote by fl(1,) the cardinality of a basin of attraction which contains the element 1 and r(1,)= K-fl(1,).
Denote by /3(0,) the cardinality of the points which are eventually absorbed by zero.
A. KLEMM and A. POKROVSKII See [6], p. 562-564 for justification and discussions of this principle.The key parameter c-a/x/ was identified in [7] as approximately 0.9.Making use of Corollaries 1, 2 and Proposition 2 above, we can get from this principle the following.Corollary 6: (a) For typical large N and 1 << n << N, the distribution D%, l(a; N, n) A_ (a; B (N, n)) is close to the function F%, l(a; 0.9/v/In(N)).
The above formulated assertion admits to experimental testing.See, for example Figures 6 and 7. A large number of other experiments have also been carried out.All our experiments support strongly the principle of correspondence within the range of several percent.Figure 7.The same as in Figure 6 for N-107 and n-103.

Cycles
For each u, the set L u is partitioned into a disjoint union of basins of attraction of different cycles.Denote by C(fu) the set of cardinalities of such cycles.For each u and each positive integer k there are defined k-sequences Ctc(fu) ordk(C(fu)).
Analogous to the principle of correspondence for large basins distribution, there is: Principle of Correspondence for Large Cycle Distributions For large N and n the statistical properties of distribution of the set Ck(N,n) {Ck(f N + 1),'",Ck(f N + n)} where is a random element from the set and a, b are the same as in the first principle of correspondence.
The parameter b was identified as approximately 4.45.Therefore, by Corollary 3 and Proposition 4, we can state the following: Corollary 7: (a) For typical large N and 1 << n N, the distribution is close to the function r,l(a; 0.9/V/I(N)).
For typical large N and 1 n N, the function N+n Me(a ;N,n) A_ lp E v--N is close to the function F(x/r-a;O.9/v/ln(N)).
See Figure 8 for numerical testing at n-105, n-104.
periments have been carried out.

Figure 3 Figure 3 .
Figure 3 is analogous to Figure 2 but deals with the limit distribution of sum of squared basins of attraction.

Figure 4
Figure 4 is analogous to Figures 2 and 3; this figure graphs mean distributions for the given parameter values.
Figure5.Limit distributions of normed maximal cycle length b-1 and for a/V'b --O, 0.25, 1.0, 2.0 (from below in this order).