Bifurcation of two-dimensional maps

\[ x_{n+1} = \mu y_n + x_n-x_n^2,\qquad y_{n+1}=x_n \]

Move the slider to choose the value of \(\mu\) between \(-0.2\) and \(0.2\), and click your mouse to select \( (x_0,y_0)\). You will see how \( (x_n,y_n)\) changes (click your mouse, and empty circles will pop up).

There is a transcritical bifurcation at \(\mu=0\): for \(\mu<0\), the fixed point \( (0,0)\) is stable, while \((\mu,\mu)\) (the red moving dot) is unstable; the stability of these two fixed points are exchanged, when \(\mu\) passes zero.