Volume 107A, number 2 PHYSICS LETTERS 14 January 1985 STRUCTURE AND CRISES OF FRACTAL BASIN BOUNDARIES Steven W. McDONALD 1, Celso GREBOGI 1, Edward OTT 1,2 and James A. YORKE 3 University of Maryland, College Park, MD 20742, USA Received 21 November 1984 We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon. Basin boundaries for dynamical systems with coex- isting attractors can be smooth or fractal. An under- standing of fractal boundaries may be important for a number of physically significant reasons [1,2] (e.g., they affect the degree to which final states can be pre- dicted in situations where error is present in the spec- ification of initial conditions). Until recently [1-5], however, studies focusing on the properties of fractal basin boundaries have been primarily restricted to mappings of a single complex variable, z ~ F(z), where z - x + iy and F is an analytic function (hence- forth referred to as analytic maps). In this case the formalism of analytic function theory can be applied, and many beautiful results have been obtained [6], dating back to the classic work of Julia and Fatou at the beginning of this century. We note, however, that analytic maps represent a very restricted class of two- dimensional mappings (e.g., due to the Cauchy- Riemann conditions they cannot possess chaotic at- tractors). The work reported in this letter was initial- ly motivated by the desire to see whether the proper- ties of fractal basin boundaries of analytic maps ex- tend to basin boundaries of more general mappings of the plane (such as those that might arise from a sur- face of section for systems of ordinary differential equations). In addition, we shall also describe a new type of boundary crisis [7] (in a boundary crisis, a 1 Laboratory for Plasma and Fusion Energy Studies, and De- partment of Physics and Astronomy. 2 Also, Department of Electrical Engineering. 3 Institute for Physical Science and Technology, and Depart- ment of Mathematics. 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) chaotic attractor is destroyed in a collision with its basin boundary as a parameter is varied). As an example of an analytic mapping, we show in fig. 1 the basin structure for the quadratic analytic map given by F(z) = z 2 + 0.9z exp(i[2) with ~2 = n(1 + x/if); the blank region consists of initial condi- tions which are attracted to the fixed point at the ori- gin while the dark region is the basin of attraction for the point at infinity. The magnification in fig. lb in- dicates the local structure of the basin boundary, and it appears to exhibit the same degree of convolution in the plane on arbitrarily small scale. The capacity dimension of the boundary has been numerically mea- sured [2] to be approximately 1.3. Expressing the analytic map of fig. 1 in terms of its real and imaginary parts, we obtain the map of the plane given by (x, y) I-+ (x 2 - y2 + x cos [2 - y sin [2, 2xy + x sin [2 +y cos f2). We changed the coefficients of the linear terms and added constant terms in search of "nonspecial" quadratic maps (i.e., maps for which the Cauchy-Riemarm conditions do not hold) which possess both a fractal basin boundary and a chaotic attractor. In our survey of about 6000 quadratic maps, we found about 1000 cases which exhibited at least one bounded attractor in addition to the point at in- finity (which was almost always an attractor). From among these, we chose several at random for further study. As an example, consider the family of maps given by (x, y) ~ (x 2 - y2 + x - ~y + 0.048, 2xy + x + 0.6y). The basin structure for the case X = 0.297 is shown in fig. 2. Orbits generated by initial condi- tions in the blank region are attracted to the chaotic 51