Bifurcation of two-dimensional maps

Processing math: 100%

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

mu = 0.1

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.