Ch 9.7: Periodic Solutions and Limit Cycles. In this section we discuss further the possible existence of periodic solutions of second order autonomous systems x \' = f ( x )
x\' = f(x)
has eigenvalues 1 i, and hence the origin is an unstable spiral point.
are r = 0 (the origin) and r = 1, which corresponds to the unit circle in the phase plane.
is therefore equivalent to the system
where t0 is an arbitrary constant.
can be found by separation of variables: For r 0 and r 1,
and after using a partial fraction expansion and some algebra,
where c0 and t0 are arbitrary constants.
is given by
but it also attracts other nonclosed trajectories that spiral toward it as t .
whose eigenvalues are [ (2 – 4)½]/2.
and R given by r1 r r2, where r1 is small and r2 is large.