Some of the characteristics of the systems of ODEs selected for this tour are described. These insights suggest interesting explorations.
For the linear system,
produces an unstable spiral,
is a center, and
produces a stable spiral.
For the van der Pol equation,
reduces to a linear center. As
increases, a limit cycle forms.
The critical value of the parameter
for all the bifurcation cases occurs at
, where the number and stability of the fixed points change, causing a qualitatively different dynamical picture.
The saddle node has no fixed points for
and fixed points at
The transcritical fixed points at
switch stability at
The supercritical pitchfork has one stable fixed point at
becomes unstable and a pair of symmetric stable points forms at
The subcritical pitchfork has a stable fixed point at
and a symmetric pair of unstable points at
becomes unstable and all trajectories diverge.
For both the supercritical and subcritical Hopf cases, the origin is a stable spiral when
and bifurcates to an unstable spiral when
, the trajectories approach limit cycles.
S. H. Strogatz, Nonlinear Dynamics and Chaos
, Jackson, TN: Perseus Books Publishing, 1994.