Tues Nov 18

MAIN POINTS
A limit cycle is a closed trajectory surrounded by non-closed trajectories. It can be stable (attracting) or unstable (or even half-stable....?). They cannot exist in linear systems because they cannot be isolated (the trajectory of a linear system can be multiplied inwards and outwards). 7.1 gives examples. The first example shows all trajectories attracted to a unit circle trajectory. The second example shows a trajectory that is not a circle. 7.2 shows that closed orbits are impossible in gradient systems. This is used to show that a given system has no closed orbit. One way this is done is to give an integral for the energy of some closed periodic solution, and show that it is nonzero, reaching a contradiction. Also, Liapunov functions (energy-like function that decreases along trajectoris) can be used to show there is no periodic solution. Otherwise, Dulac's criterion (finding a function that, multiplied by a gradient, has one sign), can be used.

CHALLENGES
The first two chapters were quite short and low on math. Thus, I consider the third:
In example 7.2.2, they use an energy function... when will we not be able to consider a function like this? When it's impossible to show that the integral is nonzero? Is there a typical case when we would use Liapunov functions? Is there a rule to see which method we should use to show that a system has no isolated, closed orbit?

REFLECTIONS
He says that constructing Liapunov functions usually require "divine inspiration" to construct. If we are not prone to divine inspiration, should we still remember their use in this respect?