I have a data set of a time series, and determined that the data fits this equation, where $y_{n}$ is the dependent variable, $a$ and $b$ are independent variables and $c$ is a constant $y_{i+1} = 0.5(a_{i+1}/b_{i+1})y_{i} + 0.5(y_{i} + (b_{i+1}-a_{i+1})c)$
$a_{n} / b_{n}$ is never greater than 1 and $a_{n} <= b_{n}$
[Edited] For certain values of a and b it seems like the series oscilates around some value - determined by c, but past some threshold of a/b the series becomes very small.
What is the stability of this equation, specifically what range must a,b,c be in for the equation to stay close to its fixed points. Are there fixed points beyond when
Example values are below:
$c$ may be $500$, $a$ may be $80000$ * $10^{-6}$ and $b$ may range from $80000$ * $10^{-6}$ to $120000$ * $10^{-6}$
The linear recurrence
$$ y_{i+1} = 0.5(a_{i+1}/b_{i+1}+1)y_{i} + 0.5(b_{i+1}-a_{i+1})c $$
can be easily solved. Calling $u_i = 0.5(a_{i}/b_{i}+1)$ and $v_i = 0.5(b_{i}-a_{i})c$ we follow with
$$ y_i = u_i y_{i-1}+v_i $$
with solution
$$ y_i = \left(c_0 +\left(\sum _{k=1}^{i} \frac{v_k}{\prod _{j=1}^k u_j}\right)\right)\prod _{j=1}^{i} u_j $$