The Mandelbrot set is the set of points in the complex c-plane that do not go to infinity when iterating z_{n}_{+1} = z_{n}^{2} + c starting with z = 0. One can avoid the use of complex numbers by using z = x + iy and c = a + ib, and computing the orbits in the ab-plane for the 2-D mapping
x_{n}_{+1} = x_{n}^{2}
- y_{n}^{2} + a
y_{n}_{+1} = 2x_{n}y_{n}
+ b
with initial conditions x = y = 0 (or equivalently x = a and y = b). It can be proved that the orbits are unbounded if |z| > 2 (i.e., x^{2} + y^{2} > 4). The boundary of the Mandelbrot set is a very complicated fractal with a Hausdorff dimension of 2. Bounded orbits may attract to a fixed point, a periodic cycle, or they may be chaotic. More details and references for the Mandelbrot set are included in the sci.fractals FAQ.
There are also various ways to express the Mandelbrot set in terms of a single time-delayed scalar variable. One such representation is
y_{n}_{+1} = 2y_{n}[(y_{n} - b)^{2}/4y_{n}_{-1}^{2} - y_{n}_{-1}^{2} + a] + b
with initial conditions y_{0} = b, y_{-1} = 0.
An interesting question is to ask for what values of (a, b) are the orbits chaotic. This question was addressed numerically using a BASIC program MANCHAOS.BAS which has been compiled with PowerBASIC. The program does period-checking for periods up to 16,000 and considers the orbit to be chaotic if it is bounded but not periodic up to the limit checked. The following figure shows the output of that program.
In this figure, white represents unbounded orbits, black represents chaotic orbits, dark blue represents fixed points, and other periodic cycles are plotted modulo the remaining 13 colors. The chaotic regions appear to be restricted to the boundary of the set and to a portion of the real (x) axis toward the left. Along the x-axis, the dynamics are governed by the 1-D map, x_{n}_{+1} = x_{n}^{2} + a, which is known to have chaotic solutions over most of the range -2 < a < -1.4011. The number of apparently chaotic orbits off the real axis shrinks as period-checking is extended to higher periods, seemingly tending to zero.
If chaotic orbits are limited to the boundary of the Mandelbrot set, as appears to be the case, then they occur with a probability less than or equal to the probability that a point falls on the boundary of the set. Apparently it has not been proved whether the boundary of the Mandelbrot set has non-zero area. Similarly, it is not known whether a point in any finite region of the ab-plane has a non-zero probability of exhibiting chaos. It appears that the probability is extremely small and very likely zero.
The few black points off the x-axis may just be examples of transient chaos; they may eventually go to infinity or settle into a periodic cycle, although some of these cases have been followed for over 10^{10} iterations. Some of these aperiodic points can be shown to have a Lyapunov exponent of exactly zero (see note by Jay Hill below). For these points, the orbits are aperiodic and fractal, and the separation of orbits with two nearby initial conditions fluctuates, but it doesn't grow, exponentially or otherwise. These points seemingly fail to satisfy the sensitive dependence on initial conditions that is usually the defining characteristic of chaos. This is in contrast to most orbits in the range -2 < a < -1.4011, b = 0, which are truly chaotic. Thus the Mandelbrot set, with all its complexity, apparently admits a negligibly small number of truly chaotic orbits.
Several interesting responses to the above observations were posted on the newsgroup sci.fractals:
On 26 June 1997, Bob Beland (bg570@FreeNet.Carleton.CA) wrote:
"Points in the Mandelbrot plane come in several types:
Outside set, nowhere near it: Orbit diverges to infinity
Inside set, not on boundary: Orbit converges to a finite attracting point or cycle
Edge of set, tip or branch of filament: Orbit converges on finite repelling cycle
Edge of set, other filament points, with irrational external angle: Chaotic orbit
Edge of set, cusp of a cardioid or bud attachment point (One or more rational external angles): parabolic, orbit goes to weakly attracting cycle or point.
Edge of set component, not on a cusp, perhaps at the limit of an alternating series of buds, with irrational external angle: Orbit chaotic. (Julia set has Siegel disks.)"
On 26 June 1997, Jay R. Hill (Jay.R.Hill@cpmx.saic.com) wrote:
"There are special values of c with convergent behavior at 'rational' points on the boundary of a component. And there are 'irrational' points on the boundary that are chaotic. But even these are at the edge of chaos, with a Lyapunov exponent equal to zero. On 16 Feb 1994 I posted a list of these chaotic points. Four of them are c = 0.33 + 0.06i, 0.37 + 0.16i, -0.23 + 0.64i, -0.47 + 0.54i. As for these being only transiently chaotic, I have followed them for more than 10^{7} iterations. They are still going, forever, in their chaotic path. Their paths are fractal, by the way, and fun to plot."
On 3 July 1997, Jay R. Hill (Jay.R.Hill@cpmx.saic.com) added:
"The value of c [0.33 + 0.06i) is exactly on the edge of the cardioid with theta related to the 3-4-5 triangle. For the Mandelbrot Set the Lyapunov exponent is When lambda = 0, this implies If the cycle ever becomes periodic, the product will go to zero violating [2]. There is a path normal the component for which inside the component lambda is negative and outside positive. On the edge it must be zero and cannot ever be a cycle.
c = 0.5 * exp (i*theta) - 0.25 * exp (2*i*theta).
lambda = lim (1/N) sum{ log2 | 2*z[n] | } [1]
lim prod (2*z[n]) = 1. [2]
The iteration orbit follows a fractal path always finding a new spot between earlier ones. If that means asymptotic to a cycle, the cycle has infinite length. I wouldn't call that a cycle. The lambda=0 orbits are very different from the periodic orbits located at the touch points between components. When lambda=0, we get a fractal path. The others have star patterns which develop as the iterations converge to the limit cycle. The number of star 'spikes', the period of the orbit, is the same as the period of the 'outer' attached component."
On August 5, 1997, Paul Derbyshire (ao950@FreeNet.Carleton.CA) wrote:
"The book The Beauty Of Fractals calls these hypothetical objects "queer components". It is strongly suspected the ordinary z^{n }+ c Mandelbrots do not have these. Other Mandelbrots for more complex (non polynomial) formulas likely can have them. They would be associated, I guess, with Siegel disk and Herman ring attractors. (Say, what is a Herman ring? They are mentioned briefly anywhere Siegel disks are, but Siegel disks are described in detail and Herman rings are not. It is mentioned that Herman rings do not occur in z^{2 }+ c though.)"
On August 6, 1997, Peter T. Wang (peterw@cco.caltech.edu) wrote:
"Yes, the boundary of the m-set contains points which correspond to j-sets of Siegel disk type; in a sense, the orbit of the iteration is quasiperiodic. Instead of being attracted to some n-cycle, an iterated point in the basin of attraction orbits an "invariant circle" about some fixed point in the plane (we're talking about the function space, not the parameter space of the m-set). Anyway, what this basically amounts to is that the "period," if you say it has one, is effectively infinite. It is bounded behavior but no periodicity occurs. However, I wouldn't say the behavior is chaotic."
On September 3, 2008, Adam Majewski wrote:
"After reading your page http://sprott.physics.wisc.edu/chaos/manchaos.htm I have made new image of boundaries of hyperbolic components of Mandelbrot set. http://en.wikibooks.org/wiki/Image:Components1.jpg That algorithm is based on polynomials defining boundaries."