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Transcritical Bifurcation 📂Dynamics

Transcritical Bifurcation

Definition 1 2

Transcritical Bifurcation is a bifurcation in a dynamical system where two fixed points exchange stability as a parameter of the system varies.

Normal Form

The normal form of a transcritical bifurcation is given by: $$ \dot{x} = rx - x^{2} $$

Diagram

The bifurcation diagram of a transcritical bifurcation is as follows:

Explanation

The normal form of a transcritical bifurcation always has two fixed points, $x_{1} = 0$ and $x_{2} = r$, regardless of $r$. The exchange of stability between the two fixed points means that at the critical point $r = 0$, the stability of fixed points $x_{1} = 0$ and $x_{2} = r$ reverses.

  • $r < 0$: $x_{1} = 0$ is stable and $x_{2} = r$ is unstable.
  • $r = 0$: The two fixed points merge and become a saddle node. At $\dot{x} = -x^{2}$, because $x$ is always negative, trajectories converge to $x_{1} = 0$ in $x > 0$ and diverge from $x_{1} = 0$ in the negative direction in $x < 0$.
  • $r > 0$: $x_{1} = 0$ is unstable and $x_{2} = r$ is stable.

  1. Strogatz. (2015). Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering(2nd Edition): p50. ↩︎

  2. Kuznetsov. (1998). Elements of Applied Bifurcation Theory: p75. ↩︎