1. 2D chaotic maps

In this section, a topological aspect of coupled chaotic maps is explained with comparison between 2D invertible and non-invertible chaotic maps. The difference between them is concerned with geometry of an attractor and a basin of attraction.

As is well known, a stretch-and-fold operation is essential for a system to exhibit chaotic dynamics. The simplest form of 1D chaotic map is represented as: x(t+1)=f(x(t))=1-ax²(t). In the logistic map, the line segment (1D phase space) is stretched and then folded at the center of the segment. Therefore, the domain is divided into two intervals, Z₂ where a point has two preimages and Z₀ where a point has no preimages. Hence, 1D chaotic map is non-invertible [13].

Extensions to such a 1D chaotic map yield two types of 2D chaotic maps. One is invertible and the other is non-invertible. A topological difference between them is schematically illustrated in the following.

(1) 2D invertible chaotic map

If a contracting map is added to the logistic map, then the total system corresponds to the sequential operation as shown in Fig. 1. The stretch-and-fold operation is qualitatively equivalent to the Smale's horseshue map. The phase plane is divided into two regions, Z₁ where a point X has a preimage X^-1 and Z₀ where a point has no preimages. Since the phase plane is expanding in the horizontal direction and contracting in the vertical direction, the 2D map has one positive and one negative Lyapunov exponents. The resulting chaotic attractor has a cantor-set-like property as illustrated in Fig. 2.

Fig.1: Stretch-and-fold operation in the Hénon map given as:
x(t+1) = f(x(t))+y(t), y(t+1) = bx(t).

Fig.2: A typical attractor in the Hénon map where a=-1.4 and b=0.3.

(2) 2D non-invertible chaotic map [13]

As opposed to the 2D invertible chaotic map, two coupled logistic maps are typically non-invertible. We consider a system of two logistic maps with average coupling. This is the 2-dimensional case of globally coupled logistic maps which have been intensively studied in the two decades [8]. The phase plane is stretched two-dimensionally, and folded horizontally and vertically as shown in Fig. 3. The phase plane is divided into two regions, Z₄ where a point X has four preimages X_k^-1 (k=1, 2, 3, 4) and Z₀ where a point has no preimages. It should be noted that a system of coupled non-invertible maps is also non-invertible in general. Due to the stretch-and-fold operation, an attractor can be hyperchaotic as illustrated in Fig. 4.

Fig.3: Stretch-and-fold operation in two coupled logistic maps given as:
x(t+1) = f(x(t))+b(f(x(t))+f(y(t)))/2, y(t+1) = f(y(t))+b(f(x(t))+f(y(t)))/2.

Fig.4: A typical attractor in two coupled logistic maps where a=1.68 and b=0.06.