We saw in the previous lecture that the familiar bifurcations from one-dimensional systems can take place in higher dimensions as well. In this lecture we show that in two (and more) dimensions we have the possibility for another type of bifurcation: the (Andronov-) Hopf bifurcation. We show that a Hopf bifurcation takes place when a complex conjugate pair of eigenvalues of the Jacobian about a fixed point cross the imaginary axis. The result of the bifurcation is the emergence of a limit cycle, which comes in both the supercritical and subcritical varieties. We motivate and illustrate these kinds of bifurcations with examples throughout while also providing the general theory.
This course is taught by Jason Bramburger for Concordia University.
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