TranscriticalBifurcation - crowlogic/arb4j GitHub Wiki

A transcritical bifurcation is a type of local bifurcation that occurs in a dynamical system when a stable equilibrium (fixed) point and an unstable equilibrium point exchange their stability as a control parameter is varied. Bifurcations are critical points in a dynamical system where small changes in the system's parameters lead to qualitative changes in the system's behavior.

In a transcritical bifurcation, there are two equilibria that collide as the control parameter passes through a critical value. As a result, one of the equilibria becomes unstable, while the other becomes stable. This exchange of stability is the key feature of a transcritical bifurcation.

Mathematically, a transcritical bifurcation can be described by a normal form, which is a simplified version of the dynamical system that captures the essential behavior of the bifurcation. The normal form for a transcritical bifurcation is given by the following equation:

$$\frac{dx}{dt} = \mu x - x^2$$

where $x$ is the state variable, $t$ is time, and $\mu$ is the control parameter. When $\mu < 0$, there is a stable fixed point at $x = 0$ and an unstable fixed point at $x = \mu$. When $\mu > 0$, the stability of these fixed points is reversed, with the fixed point at $x = 0$ becoming unstable and the fixed point at $x = \mu$ becoming stable.

Transcritical bifurcations can be observed in various physical, biological, and engineering systems. They are essential for understanding the transitions between different dynamical regimes and the onset of instability in these systems.