Linear dynamical systems
It is often assumed that the exponential function is the solution to the linear dynamical system.
\[\begin{align*} \dot{x} &= Ax \\ x(0) &= x_0 \end{align*}\]The solution is given by:
\[x(t) = e^{At}x_0\]Where \(e^{At}\) is the matrix exponential, not specifically 2.718… to the power of a matrix (which is not defined), but \(e\) as the exponential function. This is defined as the infinite series:
\[e^{A} = \sum_{n=0}^{\infty} \frac{A^n}{n!}\]Let’s start by taking small steps \(\Delta t\) and approximating the solution to the differential equation as:
\[\begin{align*} x_{\Delta t} &= x_0 + A\Delta t x_0 \\ x_{2\Delta t} &= x_{\Delta t} + A\Delta t x_{\Delta t} \\ &= x_0 + A\Delta t x_0 + A\Delta t (x_0 + A\Delta t x_0) \\ &= x_0 + A\Delta t x_0 + A\Delta t x_0 + A^2\Delta t^2 x_0 \\ &= x_0 + 2A\Delta t x_0 + A^2\Delta t^2 x_0 \\ &= x_0 (I + 2A\Delta t + A^2\Delta t^2) \\ &= x_0 (I + A\Delta t)^2 \end{align*}\]We can see that expanding to the \(n\)th step is:
\[x_{n\Delta t} = x_0 (I + A\Delta t)^n\]We set \(\Delta t = \frac{t}{n}\) and take the limit as \(n \to \infty\) (making the steps infinitely small):
\[\lim_{n \to \infty} \Big[I+\frac{At}{n}\Big]^n\]We can recognise the definition of the exponential function, which is a reformulation of the infinite series:
\[e^{x} = \lim_{n \to \infty} \Big[1+\frac{x}{n}\Big]^n\]Thus, we have shown that:
\[x(t) = e^{At}x_0\]