We begin by considering the consequence of using a linear function as a projection function. Mathematics texts usually introduce linear functions using the slope–intercept form, \(f(x)=mx+b\) where \(m\) and \(b\) are model parameters (slope and intercept). However, we will more frequently use the point–slope form, \(f(x)=m(x-a)+b\), which has slope \(m\) and passes through a point \((a,b)\), or more precisely, \(f(a)=b\). As an additional warning, note in advance that we will often use symbols other than \(m\) to represent slope.

The first step in analysis is to identify the fixed points which we use to rewrite our function. A linear projection function \(f(x)=\alpha x + \beta\) for constants \(\alpha\) (alpha) and \(\beta\) (beta) has exactly one fixed point so long as \(\alpha \ne 1\). Recall that a fixed point is a solution \(x^*\) to the equation \(x=f(x)\): \begin{gather*} x^* = \alpha x^* + \beta\\ x^*-\alpha x^* = \beta \\ (1-\alpha) x^* = \beta \\ x^* = \frac{\beta}{1-\alpha} = \frac{-\beta}{\alpha-1}. \end{gather*} Because it is a fixed point, we know \(f(x^*)=x^*\) and the slope is \(\alpha\). Rewriting \(f(x)\) in point–slope form, we find the alternative representation \begin{equation*}f(x) = \alpha(x-x^*) + x^*.\end{equation*}

The new representation allows us to rewrite our recursive equation for the sequence. Suppose that \(x_n\) refers to a sequence generated by the projection function \(x_{n+1} = f(x_n)\). Using the new representation, this becomes \begin{equation*}x_{n+1} = \alpha (x_n-x^*) + x^*.\end{equation*} This is equivalent to the equation \begin{equation*}x_{n+1} - x^* = \alpha (x_n - x^*).\end{equation*} If we define a new sequence \(z_n\) as the difference between \(x_n\) and the fixed point \(x^*\), \begin{equation*}z_n = x_n - x^*,\end{equation*} then our equation is showing that the difference is a geometric sequence, \begin{equation*}z_{n+1} = \alpha z_n.\end{equation*} Geometric sequences have an explicit formula \begin{equation*}z_n = z_0 \cdot \alpha^n,\end{equation*} which leads to an explicit formula for \(x_n\) given by \begin{equation*}x_n = x^* + z_n = x^* + (x_0 - x^*) \cdot \alpha^n.\end{equation*} For this reason, I call such sequences a shifted geometric sequence.

Now that we have an explicit formula for our sequence, we can analyze the stability. The geometric sequence \(z_n = z_0 \alpha^n\) corresponds to repeated multiplication by the same number \(\alpha\). When \(\alpha \lt 0\) (negative), the sings of \(z_n\) alternate between positive and negative. When \(|\alpha| \gt 1\), the sequence \(z_n\) grows in magnitude; when \(|\alpha| \lt 1\), the magnitude of \(z_n\) converges to zero.

Most functions are not linear, so it might seem that the previous results are limited to just a few special cases. However, one of the basic premises of calculus is that many functions can be approximated by linear functions locally. That linear function corresponds to the tangent line, which exists so long as the function is differentiable.

Now, suppose that we are working with a projection function \(f\) with a fixed point at \(x=x^*\). This means that \(f(x^*) = x^*\). The tangent line of the projection function around the fixed point will be \begin{equation*}L_{f,x^*}(x) = f(x^*) + f'(x^*) \cdot (x-x^*) = x^* + f'(x^*) \cdot (x-x^*).\end{equation*} By construction, we know that \(x=x^*\) is a fixed point of the linearized projection function and the stability of the fixed point as an equilibrium is determined by the slope \(f'(x^*)\). The following theorem guarantees that so long as \(|f'(x^*)| \ne 1\), the stability of the fixed point for the projection function is the same as the linearized function.

Sage is a computer algebra system, meaning that it can work with formulas symbolically. When we use an algebra system, we have confidence that the algebra and calculus operations are performed properly. Here we will see how we can use Sage to assist in computing the value of the derivative quickly in order to analyze stability of fixed points.

The first thing we need to do is tell Sage what our independent variable will be. If we want to use \(x\) as our independent variable, then we start with a command var('x'). Technically, this is only needed for variables other than \(x\). But it is good practice for those times we choose another variable. Sometimes we will have other parameters in our model. We include those as variables as well.

The second step is to define our projection function. Recall that you must show every multiplication symbol. Once this is complete, we can solve the fixed point equation. In Sage, an assignment (setting a variable or function to an expression) is performed with a single equal sign =. A comparison of equality is performed with two equal signs ==. There is a possibility of multiple fixed points, so an algebra system reports the solutions to the equation as a list. An example follows.