This change is called bifurcation . It is accompanied by a change of the stability of an attractor. In a bifurcation point, at least one eigenvalue of the Jacobian gets a zero real part. In a stationary bifurcation , a single real eigenvalue crosses the boundary of stability. In the Pendulum Lab , stationary bifurcations of limit cycles occur quit often.
Hopf bifurcations occur when a conjugated complex pair crosses the boundary of stability.
In the time-continuous case, a limit cycle bifurcates. It has an angular frequency which is given by the imaginary part of the crossing pair. In the discrete case, the bifurcating orbit is generally quasiperiodic, except that the argument of the crossing pair times an integer gives just 2 . In the Pendulum Lab , Hopf bifurcations seem to be absent. The period doubling bifurcation .