Analysing Transcritical Bifurcation Points

transcritical bifurcation points (equilibra that don’t dissapear, but may change stability) are determined by the linear and quadratic terms of the vector field near the equilibrium. Hence simplifying an equation like

$$\dot{x}=x(1-x^2)-a(1-e^{-bx})$$

Can be done by expanding the first term and discarding everything above quadratic power, and doing the same for the taylor expansion of $e^{-bx}$ in the second term. We get

$$\dot{x}=x-a\left(bk-\frac{1}{2}{b}^2{x}^2\right)+O(x^3)$$

With $O(x^3)$ the remainder of all cubic terms and above.

Normal Form of transcritically bifurcating equations

$$\dot{x}=ax-x^2$$