C.2. Limit Cycles
For non-conservative non-linear systems, it
is possible to have stable periodic solutions called limit cycles. Although such
a solution is not a steady state, all states in its neighborhood will decay towards
it eventually. There are also unstable
periodic solutions away from which all states in its neighborhood will
Show that the set of equations
admits a limit cycle.
It seems natural to change to the polar
coordinates defined by
Integrating (1c) gives
. Solving for
Coupled with (1d), we see that all motions
converges to a clockwise rotation with radius 1 about the origin.