Why nonlinear?

Linear dynamical system

A linear dynamical system is defined as

\[\fracx}t}=Ax(t),\]

where $x(t)$ is a state of this system at time $t$, and $A$ determines the relation from $x(t)$ to $x(t+{\delta}t)$. Suppose the state $x(t)$ is characterised by $d$ numeric variables, i.e., $x(t)\in\mathbb{R}^{d}$. We denote $x(t)$ as a column vector, and then $A$ is a square matrix of size $d$.

Importantly, we can solve this system analytically. A particular solution is found as

\[x(t)=e^{At},\]

so the general solution is

\[x(t)=e^{At}c\]

where $c$ is an arbitrary constant vector. Then, clearly

\[x(t)=e^{At}x(0).\]

Furthermore, discretising the time by $t=k{\Delta}t$ yields

\[\begin{aligned} x(k{\Delta}t)&=e^{Ak{\Delta}t}x(0) \\ &=\left(e^{A{\Delta}t}\right)^{k}x(0), \end{aligned}\]

and by setting $B=e^{A{\Delta}t}$,

\[x(k)=B^{k}x(0).\]