thisPop = rate * lastPop

thisPop = rate * lastPop (1 - lastPop)

 BigDecimal setSmallestScale(BigDecimal value)
  {
     for(int scale = 0; ; scale++)
     {
        try
        {
            value = value.setScale(scale);
        }
        catch(ArithmeticException e)
        {
            continue;
        }
        return value;
     }
  }

BigDecimal d1 = new BigDecimal(2.456);
BigDecimal d2 = d1.setScale(1, BigDecimal.ROUND_DOWN);

BigDecimal dividend = new BigDecimal(5.5);
BigDecimal divisor = new BigDecimal(2.000);
BigDecimal quotient = dividend.divide(divisor,
BigDecimal.ROUND_DOWN);

gives a quotient with a value of 2.7 and a scale of 1. To specify the scale of the quotient use divide(BigDecimal val, int scale, int roundingMode).


Figure 1.
The Chaos user interface. Controls are in the lower panel, results are plotted in the upper panel.

Other methods to watch are hashCode() and equals(). A BigDecimal's hash code is dependent not just on the number's value but also on its scale. This means that 1.5 and 1.50 have different hash codes, and because of the duality between hashCode() and equals(), a test using equals() will return false. However, compareTo() will return 0, because it ignores the scale. In the following code:

BigDecimal d1 = new BigDecimal(2.4);
BigDecimal d2 = new BigDecimal(2.40);
boolean areEqual = d2.equals(d1);
int comparison = d2.compareTo(d1);

areEqual is false because the numbers have different scales but comparison is 0, because they have the same value.


Figure 2.
Class diagram for the Chaos application.

THE CHAOS APPLICATION
User Interface
The Chaos application allows the logistic equation to be plotted and explored. Different initial populations can be chosen as can different growth rates or ratios.

To understand the code, it's best to start with a look at the application
(Figure 1).

The lower panel contains the controls, the upper panel is used to plot the results. From left to right we can set the initial population (0 to 1), the growth rate or ratio (typically 0 to 4), and the number of decimal places (i.e., the scale) to which the calculations will be restricted (a value of 0 will remove any limit, but you won't be able to calculate more than the first few points).

“Zoom” is not really a zoom factor, but the number of pixels (plus one) between successive points. A value of one plots one pixel per point, greater values will join up successive points. If you set the number of decimal places to zero, i.e., unlimited, then set the zoom to something like 80 to 100. This will ensure that only the first few points are plotted.

Plot color is used to set the color of the next plot.

Having set the required values, pressing the Plot button will cause plotting to commence. Multiple plots can be made with the Clear button being used to clear the panel as required.

Experiment with different values and see what happens. Later, I list some interesting values to use and think about.

Program Structure
Figure 2 is a class diagram for the Chaos application. It consists of three classes: GraphPanel, Chaos, and ColorChoice. In the interests of simplicity I have not included any input checking, nor provided axes markers. The complete program is given in Listing 1.

GraphPanel Class
GraphPanel extends Panel and is the class responsible for creating a plot and presenting it.

The plotting is carried out in a for loop (lines 35 to 54), within the makePlot() method. This loops once for each point plotted. The plotting is done into an off-screen image buffer, image. The width of the buffer, and hence the number of points plotted, is determined by the size of the window. g is the buffer's graphics context.

The actual calculation for the logistic equation is carried out on line 36:

newPop = ratio.multiply(oldPop.multiply
(one.subtract(oldPop)));

newPop = ratio *  oldPop * (one - oldPop);

It's worth noting that, because BigDecimals are immutable, three new BigDecimals are created for every point.

Next is a try-catch block that limits newPop's scale (i.e., the number of digits after the decimal point) if decimalPlaces is non zero.

Line 45 converts the new population (a number between 0 and 1) to a y coordinate that can be plotted. The height of the plot is scaled to the size of the window.


Figure 3.
The effect of increasing the ratio or rate. Black = 1.5, green = 2.0, red = 2.9, Zoom = 5.

The actual plotting is done in lines 47 to 50: this will plot a single point if the zoom (the point-to-point step size) is 1 or will draw a line from the old point to the new point if the zoom is greater than 1.

At the end of the loop the old values for the population and its plotted value are set to the new values.


Figure 4.
At a ratio of 3.5, the population cycles among four distinct values. Single plot, zoom = 1.

repaint() is called at the end of makePlot(). repaint() schedules a call to paint(), which in turn simply copies the image buffer into the Panel. If an image buffer wasn't used then every time the application was re-sized, or partially obscured, the last graph would be re-plotted, and the others lost. The downside of double buffering like this is that a plot does not become visible until sometime after it has been plotted.


Figure 5.
Chaos! The ratio is 3.9.

clear() is called when the Clear button is pressed. Its main task is to create a new image buffer that matches the size of the GraphPanel.

update() is overridden to prevent the Panel being cleared to its background color, to save a little time.

Chaos Class
The Chaos class is the application's main class. The constructor (lines 108 to 139) is responsible for creating the control panel (controlPanel), which contains all the UI widgets and an instance of our GraphPanel (gp).

The control panel uses a grid layout to get some control over the layout of the UI components. It's added to the south of the Frame. The GraphPanel (gp) is added to the center.

Table 3. The effect of different ratios, or rates.
Values	Consequence
ratio <>	Population dies out.
1 < ratio=""><>	Population tends to a fixed final value, without any overshoot, irrespective of the initial value.
2 < ratio=""><>	Population tends to a fixed final value, with overshoot.
3 < ratio=""><>	Population oscillates between 2, 4, 8, or 16 different values as the ratio is increased. This is known as bifurcation, with the number of possible values doubling ever more rapidly as the ratio is increased. At a ratio of about 3.57 the number of possible values is infinite. Beyond this is chaos.
3.57 < ratio=""><>	Chaotic behavior but with islands of stability, e.g., try a value of 3.83.

All the action is handled by actionPeformed(). If the Clear button is pressed, then the GraphPanel is cleared (lines 155, 156). If the enter key is pressed when any of the text fields have focus, or the Plot button is pressed, then the set up values are copied into our instance of GraphPanel, and the GraphPanel's makePlot() method is called (lines 142 to 153).

ColorChoice
ColorChoice (lines 77 to 93) is simply a Choice component that presents a list of colors to the user. It provides an additional method, getColor(), that returns the Color object represented by the color chosen by the user.


Figure 6.
Sensitivity to initial values. Red, initial value = 0.10000; green, initial value = 0.10001. The ratio is 3.9. Complete divergence occurs after about 14 cycles.



Figure 7.
The effect of increased precision: The green plot was plotted to 30 decimal places, the red plot to 20. The two plots diverge after about 100 points.

EXPLORING THE LOGISTIC EQUATION

Now it's time to experiment with the logistic equation! Use Table 3 as a guide: some of the results are shown in Figures 3, 4, and 5.

Ratios above 3 start to produce some interesting results. Figure 6 shows how sensitive the results are at high ratios to a small change in the initial value.

So, did I need the extra precision that BigDecimal provided? Yes and no. Figure 7 shows that, in a chaotic situation, increased precision gives different results. However, the critical values (e.g., the values at which bifurcation occurs) appear to be independent of precision.

RECOMMENDED READING

1. Gleick, J., Chaos,Sphere Books, London, 1988. A very readable, non-mathematical, introduction to chaos. It reviews everything from the Mandelbrot set to Jupiter's great red spot.
2. Williams, G. P. Chaos Theory Tamed,Taylor & Francis, London, 1996. A more mathematical look at chaos. It reviews the basic math needed prior to a more detailed analysis.