A transcritical bifurcation is one in which the fixed points never disappear (as was the case with saddle-node) but instead they change their stability. For example, if xdot=rx-x^2, then for r<0>0, the fixed point at the origin is unstable and the one at x*=r is now stable. An exchange of stabilities has occured.
Challenges
I had no trouble understanding the idea of a transcritical bifurcations but example 3.2.1 lost me when it was showing that the system undergoes a transcritical bifurcation.
Reflections
I noticed that the bifurcation diagram for the normal form of a transcritical bifurcation is linear, and I was wondering if all diagrams of transcritical bifurcations were linear. Just a thought. Thanks for the help during office hours today.
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