The Big Theorem

This goal of the following is to give us a practical feel for our big theorem. In practice this theorem is at the heart of how we can qualitatively the behavior of a system near and equilibrium point. In order to state it we need to recall that a wiggle of a linear system is a small change in the constants that arise when describing the system. The big theorem is...

The Big Theorem: After an arbitrarily small wiggle, every linear system is qualitatively equivalent to one in which the variables decouple into a collection of exponential pieces in the form dx/dt= lambda x and pairs in the form dx/dt= lambda x - omega y, dy/dt = omega x + lambda y.

The nicest part of the big theorem theorem is that if you understand what happens in the two by two case then you essentially understand all the other cases as well. Recall in these two by two cases we explicitly solved these 2 by 2 systems and explored their phases portraits .

The constants showing up in the Big Theorem had for us a very special significance and were even given a name, the set of eigenvalues . More precisely each exponential piece contributed its associated lambda to the system's list of eigenvalues while each pair in the form dx/dt= lambda x - omega y, dy/dt = omega x + lambda y contributed a lambda+ omega I and a lambda - omega I to the system's eigenvalues. Fortunately we can find these eigenvalues with the the detection theorem.