8.2 Saddle-Node and Transcritical Bifurcations

Saddle-node and transcritical bifurcations are key concepts in understanding system behavior changes. These phenomena occur when fixed points collide or exchange stability, leading to dramatic shifts in system dynamics.

Real-world applications range from power grid failures to ecosystem collapses. By studying these bifurcations, we can predict and potentially prevent catastrophic changes in various systems, from climate to population dynamics.

Saddle-Node and Transcritical Bifurcations

Mechanisms of bifurcation types

• Saddle-node bifurcation occurs when a stable fixed point and an unstable fixed point collide and annihilate each other as a parameter varies causing the system to have a single half-stable fixed point at the bifurcation point and no fixed points after the bifurcation
• Transcritical bifurcation happens when a stable fixed point and an unstable fixed point exchange their stability as a parameter varies allowing the fixed points to "pass through" each other without disappearing resulting in the system having two distinct fixed points with opposite stability before and after the bifurcation

Normal forms for bifurcations

• Saddle-node bifurcation normal form:
  • $\frac{dx}{dt} = r + x^2$ where $r$ is the bifurcation parameter
  • Fixed points: $x_{1,2} = \pm \sqrt{-r}$ for $r < 0$ and no fixed points for $r > 0$
• Transcritical bifurcation normal form:
  • $\frac{dx}{dt} = rx - x^2$ where $r$ is the bifurcation parameter
  • Fixed points: $x_1 = 0$ and $x_2 = r$ for all values of $r$

Stability in bifurcation scenarios

• Saddle-node bifurcation stability:
  • For $r < 0$: $x_1 = -\sqrt{-r}$ is stable and $x_2 = \sqrt{-r}$ is unstable
  • At $r = 0$: The single fixed point is half-stable
  • For $r > 0$: No fixed points exist
• Transcritical bifurcation stability:
  • For $r < 0$: $x_1 = 0$ is stable and $x_2 = r$ is unstable
  • At $r = 0$: The fixed points coincide at $x = 0$ and the stability switches
  • For $r > 0$: $x_1 = 0$ is unstable and $x_2 = r$ is stable

Real-world bifurcation applications

• Saddle-node bifurcation real-world examples:
  • Overloading a power grid leads to a blackout when demand exceeds supply
  • Tipping points in climate systems like the collapse of the Atlantic Meridional Overturning Circulation (AMOC) due to increased freshwater input from melting ice
  • Catastrophic shifts in ecosystems such as the sudden collapse of a fishery population due to overfishing
• Transcritical bifurcation real-world examples:
  • Population dynamics with a carrying capacity where the extinction and survival equilibria exchange stability as the growth rate changes (logistic equation)
  • Buckling of a beam under compression where the straight and buckled states exchange stability as the load increases past a critical value (Euler buckling)
  • Onset of convection in a fluid layer heated from below where the conductive and convective states exchange stability as the temperature gradient increases (Rayleigh-Bénard convection)