Differential system

A $𝑛$ -dimensional differential system or flow has the form

˙ 𝑥 1 = 𝑓 1 (𝑥 1, \dots, 𝑥 𝑛) = ⋮ ˙ 𝑥 𝑛 = 𝑓 𝑛 (𝑥 1, \dots, 𝑥 𝑛)

or more concisely flow

˙ 𝐱 = 𝑓 (𝐱)

Note that many ODEs may be converted to this form, including those with time-dependent terms.¹² The vector $⃗ 𝐱 = [𝑥 1 \dots 𝑥 𝑛] 𝖳$ is a point in the abstract Phase space, and every solution to the system thus corresponds to a path within this phase space, called a Trajectory.

Examples

The damped harmonic oscillator equation
$𝑚 ¨ 𝑥 + 𝑏 ˙ 𝑥 + 𝑘 𝑥 = 0$
may be rewritten by setting $𝑥 1 = 𝑥$ and $𝑥 2 = ˙ 𝑥$ , thus
$˙ 𝑥 1 = 𝑥 2 ˙ 𝑥 2 = - 𝑘 𝑚 𝑥 1 - 𝑏 𝑚 𝑥 2$
which is a linear differential system since it is a linear combination of $𝑥 𝑖$ .

The swinging pendulum equation
$¨ 𝑥 + 𝑔 𝐿 s i n 𝑥 = 0$
may be rewritten as
$˙ 𝑥 1 = 𝑥 2 ¨ 𝑥 2 = - 𝑔 𝐿 s i n 𝑥 1$
which is a nonlinear differential system.

A non-autonomous system such as a forced harmonic oscillator
$𝑑 ¨ 𝑥 + 𝑏 ˙ 𝑥 + 𝑘 𝑥 = 𝐹 c o s Ω 𝑡$
may be rewritten as
$˙ 𝑥 1 = 𝑥 2 ˙ 𝑥 2 = 1 𝑚 (- 𝑘 𝑥 1 - 𝑏 𝑥 2 + 𝐹 c o s 𝑥 3) ˙ 𝑥 3 = Ω$
hence an $𝑛$ th-order non-autonomous equation is an $(𝑛 + 1)$ -dimensional differential system. This makes sense because the state of such a problem really does depend on time.