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Bifurcation: change in the qualitative behaviour of a system at a critical value of parameter

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💃 Example: Buckling of a beam, the beam is subject to a compression force from the top

• At a critical value of the load the beam buckles (either to the right or left)
• We quantify this phenomenon by examining the deflection (degree of freedom) versus the load (parameter)

Saddle-Node Bifurcations

💼 Case: consider the following first order system: $\dot x=r+x^2$

• FP1: $x^*=-\sqrt{r}$ stable

• FP2: $x^* = \sqrt{r}$ unstable

• FP: $x^*=0$ half-stable

• no FP

🗒️ Note: bifurcation occurs at $r=0$ since the vector fields for $r<0$ and $r>0$ are different

We can show the positions of the fixed points and their stability as a function of $r$ in a Bifurcation diagram

• Solid lines are stable solutions
• Dashed lines are unstable solutions
• Vectors represent the direction of velocity
• Bifurcation point: $r$ where there is only 1 FP
• $\rm y$-axis is $x$ and $\rm x$-axis is $r$

🗒️ Note: for this example it follows an $r=-x^2$ curve which we derived from $\pm \sqrt{r}$

• 🗒️ Note: most systems bifurcation systems look like $r\pm x^2$ close to the bifurcation point

Transcritical Bifurcations

💼 Case: The normal form is $\dot x=rx-x^2$

• FP1: $x^*=r$ unstable

• FP2: $x^*=0$ stable

• FP: $x^*=0$ half-stable

• FP1: $x^*=0$ unstable

• FP2: $x^*=r$ stable

We get the following bifurcation diagram

💎 Conclusion: In transcritical bifurcations the two fixed points don’t disappear after the bifurcation, they just switch their stability

Pitchfork Bifurcation

Supercritical Pitchfork Bifurcation

💼 Case: the normal form is $\dot x=rx-x^3$

• FP: $x^*=0$ stable

• FP: $x^*=0$ stable

• FP1: $x^*=-\sqrt{r}$ stable

• FP2: $x^*=0$ unstable

• FP3: $x^*=\sqrt r$ stable