a system of finite difference equations

Question

Let $a,b>0$ such that $ab<1$ consider the system$$x_{t+1}=x_ty_t+ay_t$$ $$y_{t+1}=x_ty_t+bx_t$$

I would like you to help me answer the following:

1. find values $a$ and $b$ ​​for which the solution $\left(\frac{1-ab}{1+b},\frac{1-ab}{1+a}\right)$ is of saddle type
2. can one obtain the general solution explicitly?

any suggestion is welcome!

Answer 1

Find the Jacobian matrix and put the solution (fixed point) to obtain

$$J = \begin{bmatrix}y & x+a \\ y+b & x\end{bmatrix} \vert_{\left(\frac{1-ab}{1+b},\frac{1-ab}{1+a}\right)} = \begin{bmatrix}\frac{1-ab}{1+a} & \frac{1+a}{1+b} \\ \frac{1+b}{1+a} & \frac{1-ab}{1+b}\end{bmatrix}$$

Then the fixed point is a saddle point if and only if $\text{det} J < 0$ (for 2nd order systems of course).

Finding an explicit solution might be difficult. However, the system equations itself is a recursive solution.