Context: I'm trying to write code that generates 2D Iterated Function System (IFS) fractals based on some affine transformations. I want to generate the fractal till convergence when possible.
The system of two recurrence relations I'm looking at are affine transformations of the form:
$x_{n+1} = ax_n + by_n + e$
$y_{n+1} = cx_n + dy_n + f$
where a, b, c, d, e, f are real numbers and $x_i, y_i$ are integers.
I'm trying to figure out the conditions for which $x_i$ and $y_i$ would eventually converge but I have not made much progress yet. The only thing I can think of is that it will only converge if a, b, c and d are within the range (-1, 1). But I'm not sure if this is enough to ensure convergence.
You need the eigenvalues of the matrix $A=\begin {pmatrix} a&b\\c&d \end {pmatrix}$ to be less than $1$ in absolute value. Unless there is something special about your problem, the matrix will have two eigenvalues, each with a corresponding eigenvector. The requirement is the same as for a geometric series, which is the one dimensional equivalent of what you have here. You can write your iteration in terms of $X=\begin {pmatrix} x\\y \end {pmatrix}$ and $E=\begin {pmatrix} e\\f \end {pmatrix}$ as $X_{n+1}=AX_n+E$ with the limit found by $X=AX+E, X=(1-A)^{-1}E$