Solve discrete dynamical system I have to resolve the following discrete dynamical system: 
$$x_{n+1}=0.5x_n+0.2y_n+0.5z_n$$
$$y_{n+1}=0.1x_n+0.8y_n+0.1z_n$$
$$z_{n+1}=0.4x_n+0.6z_n$$
with the starting conditions:
$$x_0=12$$
$$y_0=7$$
$$z_0=1$$
solve the system for n=5 and determine the solution when $n \rightarrow \infty$
I already searched in the internet but didn't find anything useful and we never really saw it in class, so I honestly have no clue of how to solve this problem.
I'll appreciate any help.
 A: These are recurrence equations.  The solution is to think that you recurrently substitute each solution to solve n steps:
STEP 1:
$x_1=0.5(12)+0.2(7)+0.5(1)$
$y_1=0.1(12)+0.8(7)+0.1(1)$
$z_1=0.4(12)+0.6(1)$
so 
$x_1=6+1.4+0.5=7.9$
$y_1=1.2+5.6+0.1=6.9$
$z_1=4.8+0.6=5.4$
These values are STEP 1.
To get STEP 2, you increment $n$, you use same equations and just make $x_0=x_1$, $y_0=y_1$, $z_0=z_1$ you will get the new $x_2, y_2$ and $z_2$.  This is a recurrence.
You can use a simple spreadsheet to solve for n=5 and also for a large n to see what happens when $n$ approaches infinity.
A: General procedure: your system can be written in matrix form:
$$v_{n+1} = M v_n$$
with
$$v_n = \pmatrix{x_n\cr y_n\cr z_n},\qquad M=\cdots$$
So
$$v_n = M^n v_0.$$
Diagonalizing/finding the Jordan form of $M$ you can find easily the required limit.
A: For the solution when $n \to \infty$, if there is a stable one you can find it by setting $x_{n+1}=x_n$ and so on.  That gives you three equations in three unknowns which you should be able to solve.  If the sum of the coefficients of the $x_{n+1}$ equation were $1$, the total of the three variables would be a constant and you would know there was a stable solution.  In this case the solution is constantly growing.
