You may not have tried it, but if b is set to less than zero, then the fixed point where the pendulum hangs straight down is actually unstable. As b is increased to be greater than zero the stability of the fixed point changes from unstable to stable. The point where the fixed point changes stability is called a bifurcation point. When b is increased past zero, the system becomes very different in terms of its long term behavior. Bifurcations are not always associated with fixed points.
Cycles can undergo bifurcations as well.
Bifurcations can be classified into a few types. In this section we will study the different types of bifurcations and learn to recognize, and analyze each type. Note that these classifications apply to discrete systems as well, although they will not be dealt with here. Phase Space Phase space is a very nice way of seeing all the dynamics of a system at once. Very simply it is a plot that has every aspect of the dynamics as an axis.