Dynamic systems may frequently be modeled by systems of ordinary diﬀerential equations (or ODEs) of the form
x˙(t) = f(x, t) x(t0) = x0 (5.1)
where x(t) ∈ Rn is a time-varying function that depends on the initial condition x0. Such problems are often referred to as “initial value problems.” A system of nonlinear diﬀerential equations cannot typically be solved analytically. In other words, a closed form expression for x(t) cannot be found directly from equation (5.1), but rather must be solved numerically.