CDS - Supercritical 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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Supercritical 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 $\beta =0$.

For the Hartman-Grobman Theorem we know that when the system is NOT marginally stable, so for $\beta \ne 0$ (in this case) the behaviour of the non-linear system is qualitatevly similar to the linearized one. While for $\beta =0$, since the system is marginally stable, meaning at least one of the eigenvalues is null or has real part $0$, then the steady state is not hyperbolic, so we are not ablo to reconstruct correctly the phase space near the steady state.

The bifurcation diagram:

Here some example for the flow given different values of $\beta$ and different initial conditions:NOT-IMPORTANT


Here’s what happens more in detail:NOT-IMPORTANT

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• For a negative value of $\beta$: $\beta <0$

• For a positive value of $\beta$: $\beta >0$
• This is a “real” example of a limit cycle.

• For a positive value of $\beta$: $\beta >0$, like before, but this time the initial conditions this time are inside the limit cycle.
• NOTE: the size and shape of the limit cycle is the same (even if it does not look like it), look at the scale of the graphs.

• The amplitude/size of the limit cycle depends on $\beta$.

• The limit cycle is part/called an attracting set.