A few students have asked me what they need to know about bifurcation points. Of course we only touched upon this topic superficially (and a more detailed analysis is well beyond the scope of M2AA1).
If an equilibrium point is hyperbolic (no eigenvalues of the Jacobian are on the imaginary axis), then the flow near the equilibrium is determined by its linearized flow (Hartman Grobman theorem) and the flow does not essentially change under sufficiently small perturbations, so hyperbolicity is a counter indicator for (local) bifurcation. If an equilibrium is not hyperbolic, then small perturbations to the vector field may lead to substantial changes of the flow near the equilibrium, which would amount to a (local) bifurcation. We have not really discussed the precise analysis of the flow
at a non-hyperbolic equilibrium point, so this is not something you need to master. However, if we have a parameter in our problem, and at one value of the parameter we have a non-hyperbolic equilibrium, we can often induce from the local behaviour near the hyperbolic equilibrium/equilibria before and/or after this parameter value, what may have happened at the bifurcation point. Nothing beyond this superficial level of analysis will be expected or required at the exam.

## Saturday, 30 April 2016

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