Final state of a two-dimensional dynamical system

Calculus Level 4

A two-dimensional dynamical system is characterised by the following state equation:

$x(k+1) = A x(k)$

where

$x(k) = \begin{bmatrix} x_1(k) \\ x_2(k) \end{bmatrix}$

is the state vector, and matrix $A$ is given by

$A = \begin{bmatrix} 0.8 && 0.3 \\ 0.2 && 0.7 \end{bmatrix}$

Given that

$x(0) = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$

Then what is,

$\lim_{k \to \infty} x(k) ?$

\begin{bmatrix} 5 \\ 3 \end{bmatrix}

\begin{bmatrix} 1 \\ 2 \end{bmatrix}

\begin{bmatrix} 3 \\ 2 \end{bmatrix}

\begin{bmatrix} 1.5 \\ 0.5 \end{bmatrix}

1 solution

Otto Bretscher
Apr 11, 2016

Note that $A\begin{bmatrix} 3-k \\ 2+k\end{bmatrix}=\begin{bmatrix} 3 - \frac{k}{2} \\ 2 + \frac{k}{2} \end{bmatrix}$ , so, by induction, $x(k)=\begin{bmatrix} 3 - 2^{1-k}\\ 2 + 2^{1-k}\end{bmatrix}$ and $\lim_{k \to \infty} x(k)=\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ .

Very nice observation. Concise solution and to the point.

Hosam Hajjir - 5 years, 2 months ago

Final state of a two-dimensional dynamical system

1 solution

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