I'm trying to do the following exercise from Devaney's Introduction to Chaotic Dynamical Systems, exercise 2.6.1. The problem is this:

Consider the diffeomorphism $Q_\lambda$ of the plane given by \begin{align*} & x_1 = e^x - \lambda \\ & y_1 = - \frac{\lambda}{2}\arctan{y} \end{align*} where $\lambda$ is a parameter. Find all fixed points and periodic points of period $2$ for $Q_\lambda$. Classify each of these points as sinks, sources or saddles. If the point is a saddle, identify and sketch the stable and unstable manifolds.

Now here's as far as I've gotten: starting with the fixed points, the equation $x = e^x - \lambda$ only has a solution when $\lambda \geq 1$, in which case the solution to $y = - \frac{\lambda}{2}\arctan{y}$ is just $y=0$. So the fixed points, when they exist, are of the form $(a,0)$ where $a$ is a solution to the equation in $x$. But when it comes to classifying them, I don't know how to proceed.

To classify a point as a sink, source or saddle, I need to look at the Jacobian matrix $DQ_\lambda$ and its eigenvalues. It is given by $$DQ_\lambda = \begin{pmatrix} e^x & 0 \\ 0 & -\frac{\lambda}{2(y^2+1)} \end{pmatrix}$$ so its nature clearly depends on the value of $\lambda$. If $\lambda$ is small, both eigenvalues have absolute value less than $1$ so that the point is attracting; if $\lambda$ is large, they are both greater than $1$ so the point is repelling; and for certain values in-between, they're mixed, so the point is a saddle. In fact, there are even certain values (like $\lambda = 2$) for which we get non-hyperbolic eigenvalues!

I feel like I'm not supposed to solve this problem on such a general level. Surely I must have missed some really important detail? How am I supposed to sketch the stable and unstable manifolds? And don't even get me started on the period 2 points...!


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