CDS - Subcritical Hopf Bifurcation

Remember:

CDS - Hopf Bifurcation General Formula

The basic example of the Hopf Bifurcation:$\{\begin{array}{l}\dot{x}=\beta x-y+\sigma x(x^2+y^2)\\ \dot{y}=x+\beta y+\sigma y(x^2+y^2)\end{array}$Where $\beta$ and $\sigma$ are parameters and we have that:

• $\sigma =-1$, we will have the SUPERcritical hopf bifurcation.
• $\sigma =+1$, we will have the SUBcritical hopf bifurcation.

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To understand better this bifurcation also refer to its counterpart, the supercritical hopf bifurcation.

Subcritical Hopf Bifurcation $(\sigma =+1)$: For the linear approximation, the steady state $(0,0)$ is:

• stable if $\beta <0$.
• unstable if $\beta >0$.
• marginally stable if $\beta =0$.

NOTE: this is exactlty like in the supercritical case, however the bifurcation diagram will be very different.

The bifurcation diagram:

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• The same eigenvalues as for $\sigma =-1$

• In this case we have a divergence.

• If you choose inital condition outside of the unstable limit cycle (or better a “repuslive limit cycle”), then you have a divergence.
• We say that for positive value we have “full instability”, and for negative value we have “local stability” (inside the limit cycle).