PARABLE OF THE PARABOLA
     Chaotic dynamics was made popular by the computer experiments of Robert May and Mitchell Feigenbaum on a mapping known as the logistic map. The remarkable feature of the logistic map is in the simplicity of its form (quadratic) and the complexity of its dynamics. It is the simplest model that shows chaos.

The logistic map is the simplest model in population dynamics that incorporates the effects of both birth and death rates. It is given by the formula

xn+1 º f(xn) = b*xn*(1-xn)
(1)
where the function f is called the logistic mapping and the parameter b models the effective growth rate. The population size, (xn) at the nth year, is defined relative to the maximum population size the ecosystem can sustain and is therefore a number between 0 and 1. The parameter b is also restricted between 0 and 4 to keep the system bounded and therefore the model to make physical sense.

The logistic equation gives the rule for determining the relative population xn+1 at the (n+1)th year in terms of the population in the nth year. To get a physical understanding of the terms in the the logistic equation, we can think of the b*xn term as a positive feedback term in the sense that as xn increases so does the value of b xn. This is same as saying that the population size in the next year (xn+1) is determined by the product of the previous population size xn and the rate (b) at which the population grows. Similarly, the term (1-xn) can be thought of as a negative feedback, since increasing xn will decrease (1-xn) and therefore (1-xn) can be thought of as population decline due to over population and scarce resources.

So what is the big deal about logistic equation. Well, it is the simplest one dimensional, nonlinear (x squared term), single parameter (b in equation (1)) model that shows an amazing variety of dynamical response.

As in the study of any dynamical system, we would like to know, what happens to the long term behavior of the system, for fixed value of the parameter b and any given initial condition, say, x0. In the next two paragraphs, with the help of two different kinds of data representation, the behavior of the logistic equation for values of b less than 3.52 will be summarized. To that end, we will show the time series and what is called the graphical iteration plots for four different values of b.



 FIXED POINTS

It is clear that for values of b between [0,1], if we start iterating the equation with any value of x the value of x will settle down to 0. This can be understood from the fact that the the logistic equation is a product of three numbers, namely, b, xn and (1-xn) which are all between [0,1], the population at the next time step must always be smaller than what is at the current time step. Thus, no matter what initial value we choose for xn, if b <= 1.0, the population is doomed to extinction. The point zero is called the fixed point of the system and is stable for b = [0,1]. The first two figures show the time series and the graphical iteration plot for b = 0.9. It should be noted that 'time' for mappings are discretized and are represented by the iteration number n. Thus, a time series for maps is a plot of xn against n. For the case of graphical iteration plots below, the logistic curve (parabola) is plotted in yelow, the y=x (diagonal line) is plotted in red, and the movement of the iterates is followed by the blue line. The reasons for plotting the curve, shown in green, (an additional curve in pink in the last plot) will be explained later.

TIME SERIES PLOT GRAPHICAL ITERATION PLOT

For values of b between (1,3) the iterates instead of being attracted to zero, get attracted to a different fixed point (as shown in both the plots above for b = 2.8). How did the fixed point change ? What happened to the fixed point located at x = 0. Without going into the details now, we state that the fixed point x = 0 is unstable for b between (1,3) and a new fixed point exists for b >= 1. To determine the fixed point of a map, we do the following. Note, by definition, a fixed point is a point which when fed back into the map gives back the same point. Mathematically, this expressed by the condition xf = x = b*x*(1-x). Solving this equation gives two values for x, xf = 0 and xf = 1 - 1/b. The second fixed point changes its position according to the value of b and doesn't exist for b < 1, because xf becomes negative and that is not allowed domain for the logistic equation. The fixed point can also be determined graphically. All fixed points, regardless of the value of b, can be found by seeing where the parabola (shown in yellow) intersects the identity line (shown in red). For values of b>=1, the parabola will always intersect at two points. The fixed point at zero is unstable for b>=1 and the second fixed point is stable for b between (1,3). The bottom two plots in figure above show the time series and graphical iteration plot for b = 2.8.



 LIMIT CYCLES

This is where all the fun begins. We begin by asking, what happens to the long term behavior of the logistic equation when b>= 3 ? As long as the choice of the initial value of the iterate is not 0 or 1 - 1/b, the logistic equation will never converge to any fixed point. The first set figures below show what happens when b = 3.2. In the jargon of dynamicists, you say, the system settles down to a period 2 limit cycle. That is, the iterates of the logistic equation oscillate between two values. The fixed points 0 and 1 - 1/b still exist but they are unstable.

To understand the creation of this 2 period limit cycle, we will use the graphical method, though it can be done explicitly too. To do this, we note that the 2 period limit cycle of the logistic equation can be thought of as the fixed point of the two composition of the logistic equation. As a matter of fact, an m period limit cycle of the logistic equation can be thought of as the fixed point of the m composition of the logistic equation. Confused ! lost !! Befuddled !!! Ok, let's do this again. The 2 period limit cycle implies that xa = f ( xb ) and xb = f ( xa ). Substituting the second equation in the first gives xa = f ( f ( xa )) = f2( xa ). Thus, to find the 2 period limit cycle, graphically, we look for the intersection of the diagonal line with the graph of the second composition of the logistic equation. Such a curve is shown in green for all the graphical iteration plots. It can be seen that the green curve doesn't intersect the diagonal line for the first two plots and it does for b = 3.2 and 3.52. In fact for any value of b > 3, the f2 curve intersects the diagonal line. Generalizing, the above argument for an m period limit cycle, we say, that the existence of m period limit cycle is established by real solutions to the equation x = fm( x ), provided the solutions lie between 0 and 1.

On further increasing b the period 2 limit cycle becomes unstable and a period 4 limit cycle is created. The figures below show the time series and graphical iteration plot for b = 3.20, and 3.52 where the system settles down to a 2 period and 4 period limit cycle. It should be noted that the transients (the first few iterates) in graphical iteration plot, are not drawn to keep the graph readable. The curve, shown in pink, in the last graphical iteration plot is for y = f4( x ) and graphically shows the existence of 4 period limit cycle.



 STABILITY - INSTABILITY OF FIXED POINTS AND LIMIT CYCLES

For one-dimensional maps like the logistic map, there is a very simple method to determine if a fixed point or limit cycle is stable or unstable. The extension of the method to N dimensional is given here. The idea behind the technique is to examine the local behavior of the map in the vicinity of the fixed point or the limit cycle. The first derivative of the logistic equation is given by f' ( x ) = b (1 - 2*x) which is the slope of the parabola at point x. To find the stability of the fixed point, we evaluate the slope at the fixed point and the following conditions characterize the behavior of the fixed point:

| f' ( x ) |  <     1     attracting and stable
f' ( x )   =     0     super-stable
| f' ( x ) |   >     1     repelling and unstable
| f' ( x ) |   =     1     neutral

The extension of the above analysis to determine the stability of a limit cycle is nearly identical except for the fact that a limit cycle will oscillate between, say, m points in an orbit: x1, x2, x3, ... , xm, where we have labeled the orbit in the order they are visited. As mentioned earlier, an m period limit cycle of the logistic equation is the fixed point of the m composition of the logistic equation, therefore to determine the stability of m period limit cycle, we evaluate the slope of fm( x ) at any one of the m points of the limit cycle and the stability of the limit cycle is determined by checking which one of the above four conditions is satisfied. Actually, we can do a little better than this. Using the chain rule for differentiation, the above condition can be reduced to the evaluation of the expression

| f'( x1 ) × f'( x2 ) × ... , f'( xm ) |
and then checking to see which one of the above four conditions is satisfied, to determine the stability of the limit cycle.

If we increase the value of b even more, we see 4 period limit cycle bifurcating to 8 period limit cycle, then a 16 period limit cycle, and so on. At a very special critical value, the logistic system falls into what is essentially an infinite-period limit cycle. This is chaos. But then that is a story that unfolds in the next page !!