CDS - Transcritical Bifurcation

Remember:

Transcitical Bifurcation:

• We have two steady states and after the bifurcation value those steady states invert their stabilities.
  • $\mu <{\mu }^{\ast }$, $x_1^{\ast }$ is stable and $x_2^{\ast }$ is unstable.
  • $\mu >{\mu }^{\ast }$, $x_1^{\ast }$ is unstable and $x_2^{\ast }$ is stable.
  • $\mu ={\mu }^{\ast }$, $x_1^{\ast }$ and $x_2^{\ast }$ is coincide.

~Ex.:

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• This is considered the “simplest case of bifurcation”.
• This system has 2 steady states $x_1^{\ast }=0$ and $x_2^{\ast }=r$.
• Red parabola: $r<0$.
• Green parabola: $r>0$.
• Blue parabola: $r=0$.
• Changin $r$ this time changes the stability of one of the steady states, and differently from before no steady states disappears.

• Analysis of the stability of the steady states.
• Again when the two steady stated conicide for $r=0$ the steady state is stable from on part, and unstable from the other part, like for the saddle-node bifurcation.

• This is called the “Transcritical Bifurcation”