Relationship between input and output coefficients

Question

This is the starting point:

Chop[Simplify[
   RSolve[{x[t] == .45*x[t - 1] - .11*y[t] + b1*c1 + e1*d1 + g1*f1, 
           y[t] == .54*y[t - 1] - .11*x[t] + b2*c2 + e2*d2 + g2*f2, 
           x[1] == A1, y[1] == A2}, {x[t], y[t]}, t]] /. t -> ∞]

What that does is solve the coupled recursive system an then takes the limit as t goes to ∞. The output is

{{x[∞] -> 1.90951 b1 c1 - 0.456621 b2 c2 + 1.90951 d1 e1 - 0.456621 d2 e2 + 1.90951 f1 g1 - 
        0.456621 f2 g2, 
  y[∞] -> -0.456621 b1 c1 + 2.28311 b2 c2 - 0.456621 d1 e1 + 2.28311 d2 e2 - 0.456621 f1 g1 +
        2.28311 f2 g2}}

Here's the question: I start out with 3 constants: .11, .45 and .54. In the end I end up with 3 constants: 1.90951, .456621 and 2.28311. I want to know the algebraic relationship between the starting constants and the ending constants.

The obvious solution is simply to make them arbitrary variables. The problem with that is an intractable formula results. I strongly suspect that there is a simple formula relating these starting 3 and ending 3, but don't know how to do it analytically.

Does anyone have an idea about how to uncover the relationship I want?

Answer 1

It looks like a linear dynamical system which presumably has a single fixed point which is independent of the starting value. Just solve

Solve[{x == .45*x - .11*y + b1*c1 + e1*d1 + g1*f1,   y == .54*y - .11*x + b2*c2 + e2*d2 + g2*f2}, {x, y}]

or

Solve[{x == A x - B y + b1*c1 + e1*d1 + g1*f1,   y == C y - D x + b2*c2 + e2*d2 +  g2*f2}, {x, y}]