International Journal of Bifurcation and Chaos, Vol. 11, No. 1 (2001) 19–26 c World Scientific Publishing Company CHAOS AND COMPLEXITY RAY BROWN and ROBERT BEREZDIVIN Raytheon Systems Company, Falls Church, VA, USA LEON O. CHUA Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 94720, USA Received February 20, 2000; Revised June 15, 2000 In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how “near-chaotic” complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The rela- tionship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain. 1. Introduction presently, making a contribution to the study of complexity. They are, roughly: (1) the the- The study of chaos has raised many interesting ory of complexity or computability ala Chaiten, questions about highly complicated nonlinear sys- Kolomogrov, Church and others; (2) information tems generally. This is because, in an effort to an- theory; (3) ergodic theory and dynamical systems; swer questions about chaos, it has been necessary (4) cellular automata and artificial life; (5) large to undertake the study of nonchaotic processes such random physical systems; (6) self-organized critical- as skew translations and infinite dimensional rota- ity; (7) artificial intelligence; and (8) neuroscience. tions which can produce dynamics that can appear Gell-Mann, Crutchfield, and their associates con- related to chaos. The nonchaotic strange attractor tribute to (1), (3)–(7), and there is also a long list is one example. The study of dynamics at the edge of other contributors. In general, transitioning from of chaos that has resulted from investigations into these individual disciplines to rigorous mathemati- complex nonchaotic systems has led to the study of cal theorems about complexity appears to be diffi- complexity as a separate discipline. Thus we arrive cult. It is on this point that we present this paper. at the question: What is the relationship between Specifically, we present a rigorous connection be- chaos and complexity? In this paper we provide a tween complexity, chaos and various forms of com- discussion of one possible bridge between these two plicated dynamics that have been studied exten- important areas of research. sively. Our connection proceeds through area (3) cited above. 1.1. Background 1.2. Notation Phillip Anderson, in [Anderson, 1994], describes eight disciplines that either have been, or are Let T be a transformation on Rn that preserves 19 20 R. Brown et al. n a special finite subset which we will call V .We Then ST is defined for four points in I. They are 0.0, define this subset as follows: 0.25, 0.5, 0.75. ST(0.0) = 0.25, ST(0.25) = 0.75, ST(0.5) = 0.5, ST(0.75) = 0.5. Since ST is not Vn { ∈ n| ∈{ } = x R x =(a1,a2,..., an),ai 0,1 ) defined for the point (1, 0) by T, we extend ST to include the interval (0.75, 1.0) by assigning it to be As we have required that T preserves this set, we the identity on this subinterval. As the dimension Vn ⊆Vn have T( ) . We allow that T, under iteration, of T increases, this subinterval, which for an n- Vn may map to a proper subset of itself. dimensional map is ((2n − 1)/2n, 1.0), goes to zero It is our objective to show that a certain class and so this convention is both harmless and useful. Vn of transformations which preserve can produce It is clear that there are n! representations of a very complex dynamics. As an aid to seeing how given T, since the coordinates of T have n!permuta- complex dynamics can occur in this setting, we tions. Thus there are n! different ways of graphing Vn construct an invertible mapping of into the T on I. However, the dynamics of all representa- unit interval I. This is done as follows: Let X = tions are qualitatively the same. For example, if a ∈Vn ⊂ n (x1,x2,x3,..., xn) R . We define the in- point converges to a given attractor in one represen- Vn → vertible mapping π : Ias tation, then it converges to an attractor of the same size in all representations. Thus, the dynamics of T π(X)=(0.x x x ,..., x ) 1 2 3 n are invariant with regard to the representation. It For example, let (1, 1, 0, 0, 1) ∈Vn,thenπ(1, 1, 0, may happen, however, that one graphical represen- 0, 1) = 0.11001. tation may be more appropriate than another for a For any high-dimensional dynamical system T, given T. We will see an example of this in the next having a finite invariant subset of the type Vn,the section. For convenience, we will refer to π as the mapping π canbeusedtodefineamappingonI standard representation based on a given labeling that relates T to a one-dimensional map of I. In par- of the coordinates of T. −1 In the next section we will show how to define ticular, if we define ST(X) ≡ π(T(π (X))), then n-dimensional maps which preserve Vn and have ST(0.x1x2x3,..., xn)=π(T(x1,x2,x3,..., xn)) nearly chaotic dynamics. Before that, we give some examples of how to construct mappings that just Thus, when T is restricted to Vn, T is conjugate preserve Vn. to a one-dimensional map of a finite subset of I to itself. ST may be conveniently viewed as a mapping Example. Consider the following transformation on a subset of fractions between 0 and (2n − 1)/2n on R4: in order to facilitate graphical representation of the one-dimensional map. This is expressed by trans- T(x, y, z, w) forming the point 0.x1x2x3,..., xn by the formula =(x·y·z, x + y − x · y, 1 − z · x, x + y − 2x · y) Xn x V4 (0.x x x ,..., x )→ i which preserves . We are able to construct such 1 2 3 n 2i i=1 transformations generally by the requirement that each coordinate of the transformation defines a We will use ST to mean either the decimal or binary mapping of Vn to {0, 1}. The following are some mapping so long as it is not ambiguous. examples of such functions: The mapping ST, as we have defined it, is de- termined only on a finite subset of I by the mapping (x, y, z) → x + y + z − 2(x · y + y · z + z · x) T. It can be extended to be a continuous mapping +3x·y·z in any number of ways, all of which are equivalent so long as we are only examining the dynamics of (x, y, z) → x · y + y · z + z · x − 2x · y · z. TonVn. In the first example, if either of x, y, z is one, the Example. Define T as result is one, otherwise it is zero. For example, if ! ! both of x, y are one, the result is zero. In the sec- − · x → x + y x y ond example, at least two coordinates must be 1 for y 1 − x the result to be 1, otherwise it is zero. Chaos and Complexity 21 We note that we may increase the exponent of any factor without losing the invariance of Vn since 1 or 0 to any power is still 1 or 0, respectively. Thus (x, y, z) → x3 · y + y · z + z · x − 2x · y5 · z0.5 also defines a mapping of V to the set {0, 1}. 2. Construction of Complex Dynamics in High-Dimensional Spaces Using the notation of the previous section, we can generate complex orbits in high-dimensional spaces with very long periods by writing down a transfor- mation that is conjugate to 3x mod (1) when re- stricted to Vn.Themap3xmod (1) is chosen as it is able to generate complex orbits from points which have zeros in every coordinate position except 1. Fig. 1. In this figure we show the one-dimensional map, This would not be possible using 2x mod (1). To ST(X) corresponding to Eq. (1). It is equal to the define T to be conjugate to 3x mod (1), T must one-dimensional map x → 3x mod (1) on the domain of carry out the function of a coordinate shift followed definition. by a binary addition operation. The ith coordinate function for the shift-and-add transformation, T, is given by: Some important observations about the shift- and-add transformation are: It can be defined in xi+1 = yi + vi−1 · (1 − 2 · (xi + xi−1)+4·zi) any number of dimensions. When T is restricted yi =xi +xi−1 −2·zi to points whose coordinates are all less than 1, it (1) converges to the origin as a local attractor. For z =x ·x− i i i 1 values greater than 1, it is unbounded. An inter- · vi =zi +yi vi−1 esting feature of T is that, as a dynamical system, it would not be readily recognized that T is conju- The equation for v does not have to be recursive in i gate to a unilateral shift on a finite subset.