Example 6.2 Consider the map \[ x_{n+1} = \mu y_n +x_n-x_n^2,\qquad y_{n+1} = x_n, \] where \(|\mu|\) is small. The fixed point \((x^*,y^*)\) satisfies the equations \(x=\mu x + x - x^2, y=x\). That is, there are two fixed points \[ (x_1^*,y_1^*) = (0,0),\qquad (x_2^*,y_2^*) = (\mu,\mu). \] From the Jacobian matrix \[ J(x,y) = \begin{pmatrix} \frac{\partial }{\partial x} (\mu y+x-x^2) & \frac{\partial }{\partial y} (\mu y+x-x^2) \cr \frac{\partial }{\partial x} x & \frac{\partial }{\partial y} x \end{pmatrix} =\begin{pmatrix} 1-2x & \mu \cr 1 & 0 \end{pmatrix}, \] we get \[ J(x_1^*,y_1^*) = \begin{pmatrix} 1 & \mu \cr 1 & 0 \end{pmatrix},\qquad J(x_2^*,y_2^*) = \begin{pmatrix} 1-2\mu & \mu \cr 1 & 0 \end{pmatrix}. \] At the fixed point \((x_1^*,x_2^*)\), the two eigenvalues are governed by \[ \det \big(\lambda I - J(x_1^*,y_1^*)\big) =\det \begin{pmatrix} \lambda -1 & -\mu \cr -1 & \lambda \end{pmatrix} =\lambda^2 - \lambda - \mu=0. \] That is \[ \lambda_1^\pm = \frac{1\pm \sqrt{1+4\mu}}{2}. \] As \(\mu\) passes zero, \(\lambda_1^+\) pass 1 and this fixed point \((x_1^*,y_1^*)\) becomes unstable. At the fixed point \((x_2^*,y_2^*)\), the two eigenvalues are governed by \[ \det \big(\lambda I - J(x_2^*,y_2^*)\big) =\det \begin{pmatrix} \lambda -1+2\mu & -\mu \cr -1 & \lambda \end{pmatrix} =\lambda^2 - (1-2\mu)\lambda - \mu=0. \] That is, \[ \lambda_2^\pm = \frac{1-2\mu\pm \sqrt{(1-2\mu)^2+4\mu}}{2} =\frac{1-2\mu\pm \sqrt{1+4\mu^2}}{2}. \] In this case, \(\lambda_2^-\) is close to zero (\(|\lambda_2^+|\) is far away from unit) and can not trigger any instability. If \(\mu\) is small and negative, \[ \sqrt{1+4\mu^2} >\sqrt{1+4\mu+4\mu^2}=1+2\mu \] and \[ \lambda_2^+ = \frac{1-2\mu+\sqrt{1+4\mu^2}}{2} > \frac{1-2\mu+(1+2\mu)}{2} = 1. \] As a result, the fixed point \((x_2^*,y_2^*)\) is unstable. On the other hand, if \(\mu\) becomes positive (and small), \( \sqrt{1+4\mu^2} < \sqrt{1+4\mu+4\mu^2}=1 +2\mu \) and \[ \lambda_2^+= \frac{1-2\mu+\sqrt{1+4\mu^2}}{2} < \frac{1-2\mu+(1+2\mu)}{2}=1 . \] Therefore, the stability of the two fixed points \((x_1^*,y_1^*)\) and \((x_2^*,y_2^*)\) are exchanged, indicating the transcritical bifurcation at \(\mu=0\).

The bifurcation is also clear from Figure 6.8. For \(\mu \in (-1/4,0)\), \(|\lambda_1^\pm| <1\) and the fixed point \((x_1^*,y_1^*)=(0,0)\) is stable. The other fixed point \((x_2^*,y_2^*)=(\mu,\mu)\) is stable for \(\mu>0\), but becomes unstable again when \(\lambda_2^-=-1\), or \(\mu=2/3\). A period-doubling bifurcation occurs her (associated with eigenvalue \(-1\)).