What is chaos? Colloquially, it is a state of disorder or confusion.
But in physics it has a more specific (albeit in some sense related)
definition: A system is said to be *chaotic* if a small change in the
initial conditions results in an exponentially divergent final state.

So far in this course we have mostly been integrating the equations
of motion for systems that give deterministic solutions; *i.e.*,
for a given set of initial conditions, we can use these equations to
determine the state of the system for all time, and small changes in
those initial conditions would result in small changes in the final state.
For instance, for orbits Earth around the Sun, one could imagine changing
the orbital parameters of the Earth slightly, and the result would be
that Earth would settle into a slightly different orbit; it would not,
for example, rapidly get shot out beyond the orbit of Pluto. Hence this
is a non-chaotic system.

Even for the orbital problem, however, the behavior can quickly become quite complicated. Consider setting up a problem with three bodies, all mutually interacting, with a particular set of initial positions and velocities. Let's say two bodies are orbiting each other calmly, when the third body falls into the system. The near-collision of third body and one of first two will result in a large deflection of their directions into new orbits. Eventually (if all three objects are bound gravitationally) they will come together again resulting in another set of large deflections into another set of new orbits, and so on. Now, each deflection varies strongly with the distance of closest approach. A small difference in distance will change the deflection angle appreciably, which will then change the time and manner of subsequent interactions, and so on. The resulting motion is very complex; it is in fact chaotic. In particular, if we sent in the third particle on a slightly different orbit, that particle could end up very far from where it did previously, because of the strong perturbations inherent in the close approaches.

Rather than examining chaotic orbital dynamics in two dimensions, it is easier to get a flavor of what is going on by examining systems in one dimension. As an example, let us consider a situation that arises in simple models of ecology and economics: the logistic equation.

Consider a population of organisms whose rate of reproduction
depends somehow upon the number density of individuals (similar to
the Lotka-Volterra equations we had earlier, but only for a single
species). Clearly, the rate of reproduction depends upon the number
of individuals already present as these are the ultimate means of
reproduction. The rate at which these individuals reproduce is also
frequently dependent on the availability of resources, which in turn is
proportional to the number of individuals using up those resources. We
can represent this behavior schematically as

(11) |

(12) |

If we adopt a non-zero , however, the equation exhibits more
interesting behavior. To cast this equation into the form usually
employed, let
and let :

It is easy to show that if we restrict
, then the
population is bounded by (assuming the initial population
is within these bounds). This equation is therefore a mapping of
a point in the interval back onto another point in the same
interval which is known as a *logistic map*. For a given
initial condition , we can determine the state of the system at all
future times. The set is called the trajectory, or orbit, of the
system that we generate by iteration, by applying equation (13)
recursively for many generations.

The results of solving equation (13) are shown in
Figure 3. Here the logistic equation has been solve for a
range of values, and it is clear that varying can give more or
less complex behavior. For some values, the solution oscillates at
first, and then settles down to a constant value. This value is known
as an *attractor* for the system; no matter what the starting value,
the solution settles down to, or is attracted to a given value. The set of
initial points which are attracted to this value is called the *basin*
of attraction. Think of a dropping a marble into a basin; wherever you
drop it, it will fall toward the bottom as if it were attracted to it.

For other values of , the solution oscillates between several values in
what is known as a *limit cycle*. For the *limit cycle*
has an *attractor* of period 2. As we increase , the behavior
becomes even more complex, so that the attractors have a period of 4
or larger.

We can describe the behavior of this system as we increase by using
a *bifurcation diagram*. A bifurcation diagram shows all the
attractor values at any given value of .

The top curve in Figure 4 shows a bifurcation diagrams for our logistic equation. For each value of , we have iterated the logistic map for some fairly large number of iterations to allow the initial transient effects to damp out, and then plot the points attained for some additional number of iterations. By varying the value of , you will then have a plot of the values of visited by the dynamical system (the orbits) as a function of .

For values of , there is an attractor of period 1, as you can see from Figure 3. When is increased, however, the line of bifurcates into an attractor of period 2: the solution oscillates between two values of . For slightly greater than 0.862, the line bifurcates again to an attractor of period 4. This period doubling continues as we further increase , until the attractor has a very large period indeed. But look in detail at on , as shown in the expanded view in Figure 5: at we are back down to a period of six! Hence there can be ``islands'' of simpler behavior within the chaotic motion.

How does this relate to our original definition of chaos? Well, imagine that two populations are started with slightly different values of . In general, the value of attained after a long period of time will be any one of the attractor values accesible to it in the bifurcation diagram. If is small (i.e. less than 0.75), then the small difference in initial populations will damp out and they will both converge to the single attractor value. However, as becomes large, the late-time value of will be (effectively) randomly selected from all the attractor values accesible to each system. For large , since there are a wide range of attractor values, even very small initial differences can result in large deviations in the final value. This is the essence of the physical definition of chaos.

Returning to the example discussed earlier, the population of an organism after many generations does not depend on initial population so much as on the parameter which describes the interplay between population growth and limits on the population. Thus, for any initial population between 0 and 1 and a value of , the population will converge to the same steady state value. For , any value of converges to a solution with period 2. This means if the time step for each iteration is 1 year then the population of the organism is the value of the upper curve one year, the lower curve the next year, the upper again the next, etc. Thus, the population has a 2 year cycle. For the population has a 4 year cycle. corresponding to the shaded areas of the logistic map indicate the population from year to year is effectively chaotic.

By now you should be convinced that even a fairly simple dynamical system can exhibit extremely complex behavior. To get an idea of the exponential divergence of orbits discussed above, try the following experiment. Choose a value of large enough to be in the ``chaotic'' regime. Now pick an initial condition for one orbit (e.g. ) and another, similar, initial condition for another (say ). How long does it take for the two orbits to diverge significantly as a function of ?

We can quantify how fast chaotic trajectories diverge using the *Lyapunov exponent* , defined by

(14) |

For our simple logistic map, unfortunately, the magnitude of the separation
ceases to increase once it reaches of order one half (why?), so we need a
better method to characterize the divergence. Take the logarithm of the
previous equation to write

(15) |

(16) |

(17) |

(18) |

(19) |

A plot of the Lyapunov exponent as a function of is shown as the lower curves in Figures 4 and 5. Note the behavior of in relation to the behavior of the logistic map. increases from negative values to zero at a bifurcation, and then falls to more negative values. In those regions where chaotic behavior occurs, it rises to a positive value, indicating exponential divergence.