CDS - Supercritical Pitchfork Bifurcation

Remember:

Supercritical Pitchfork Bifurcation:

• $\mu <{\mu }^{\ast }$ : $x_1^{\ast }$ is stable.
• $\mu >{\mu }^{\ast }$ : $x_1^{\ast }$ is unstable, $x_2^{\ast }$ and $x_3^{\ast }$ appear and are stable
• $\mu ={\mu }^{\ast }$ : $x_1^{\ast }$, $x_2^{\ast }$ and $x_3^{\ast }$ coincide.

~Ex.:

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• The blue curve has a $0$-derivative for $x=0$

Let’s analyze the steady steates. For $r<0$:
And in case of the blue curve, $r=0$:
While for $r>0$:

• From the red curve, to the green curve, we have that:
  1. $x_1^{\ast }$ looses stability.
  2. $x_2^{\ast },x_3^{\ast }$ are born, and are stable.
• We say that for $r=0$ is when this change happens.

• This is called the “Supercritical Pitchfork Bifurcation”
• We can say that globally this system is stable.