CDS - Supercritical Pitchfork Bifurcation vs Subcritical Pitchfork Bifurcation

Remember:

If we analyze the supercritical pitchfork bifurcation, with respect to the subcritical pitchfork bifurcation, we can see that, for the SUPERcritical case, the system is a lot more unstable, since for the supercrictical case, the system will never diverge:

• While for the SUBcritical case the system converges only for specific values of $r$ and $x_0$.

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If we analyze the supercritical pithfork bifurcation, we can see that:

• For whichever value $r$ assumes, the system will converge, except for $x^{\ast }=0,r=0$ for which the system will remain at $0$ (unstable steady state)

While for the subcritical case:

• The system is a lot more unstable.

NOTE: A bifurcation only appers when there is a non-hyperbolic steady state, this is true also for higher dimensions systems.