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

for

.

The transcritical fixed points at

and

switch stability at

.

The supercritical pitchfork has one stable fixed point at

for

. When

,

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

for

. When

,

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

. When

, the trajectories approach limit cycles.

S. H. Strogatz,

*Nonlinear Dynamics and Chaos*, Jackson, TN: Perseus Books Publishing, 1994.