Simulation of Linear_dynamical_system

Linear time invariant systems

To simulate the solution of a continuous-time linear dynamical system given the system matrix $A$ and initial condition $x_0$:

$\dot{x} = Ax, x(0) = x_0, $

We will use the matrix exponential formula to compute the state transition matrix and then use it to evolve the system over time.

Unstable

The system matrix $A$ is given by:

$ A = \begin{bmatrix} 1 & 2 \ -2 & 1 \ \end{bmatrix}. $

The system of equations can be written as:

\begin{align*} \frac{dx_1}{dt} &= 1 \cdot x_1 + 2 \cdot x_2 \ \frac{dx_2}{dt} &= -2 \cdot x_1 + 1 \cdot x_2 \ \end{align*}

where $x_1$ and $x_2$ are the state variables, and the coefficients of the system matrix $A$ determine the dynamics of the system.