| We have seen that chaos is filled
with cycles, but all of them are unstable. |
| The idea of control of chaos is
very simple: |
| By a small change of a
system parameter,
force the system near the desired cycle. |
| Because the cycle is unstable, eventually the system will wander
away from the cycle. |
| When the system has wandered too far from the cycle, apply another parameter change
to force it back to the cycle. |
|
| How can this idea be implemented, and has it been applied to any real systems? |
| First, we illustrate this method by stabilizing a fixed point of the tent map. | |
| Next, we describe the method of Ott, Grebogi, and Yorke
for implementing control in physical situations where no model equations are known. | |
| Here is an example of applying this method to control an
oscillating magnetic ribbon. | |
| Here is an example of applying this method to control
rabbit heart arrhythmias. | |
| Here is an example of applying this method to control
atrial fibrillation in human hearts. | |
|
| One possible problem is that typically parameter changes
should be small, and so we can only stabilize a cycle if an iterate lands near that
cycle. This may take some time, but sensitivity to initial conditions can be
harnessed, in a method called targeting, to force the iterates near the desired cycle. |
| New applications of this idea appear frequently. Control of chaos may be the
most important impact of chaos on engineering. |