I am looking at a discrete dynamical system and I wish to show that it is bounded. I know that the displacement after $n$ iterations is given by the product
$$\Delta_n=\prod_{k=0}^n \left(1+\frac{2\cos(k(\alpha+\pi))}{\lambda_1+\lambda_2\sin(k(\alpha+\pi))-\cos(k(\alpha+\pi))}\right)$$
where $\alpha$, $\lambda_1$, and $\lambda_2$ are positive constants based on the system and initial conditions. Numerically it is apparent that (even though the product does not converge) it is well behaved and bounded, but I cannot show this analytically.
The rotational constant $\alpha$ is not a rational multiple of $\pi$, and $\lambda_1$ may be made arbitrarily large, if that leads to an easier proof, but it appears to be bounded for most values that arise. A typical set of values for the constants are $\alpha \approx 1.23$, $\lambda_1=10$, and $\lambda_2=\sqrt{2}$, with the image below showing the displacement graph in this case over 10,000 iterations.
One approach I tried was to look at the corresponding series (making sure the constants gave the needed bounds) with the idea that for large $\lambda_1$ is would be essentially something like $$\sum f(k)\cos(k(\alpha+\pi)),$$ $M-\epsilon<f(k)<M+\epsilon$, yielding something akin to the Lagrange trigonometric identity. However, though I do not believe it happens here, due to possible resonance merely showing $f(k)$ to be nearly constant does not ensure the sum behaves like the Lagrange sum.
