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 #m/def/dynamics/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.

Properties and further terminology

Phenomena

Homoclinic orbit

Techniques

Linear stability analysis

Particular types of flow

Flow in 2D
- Reversible 2D flow

#state/tidy | #lang/en | #SemBr

2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §1.2, pp. 6ff. ↩
2017. Elementary differential equations and boundary value problems, p. 282 ↩

Differential system

Properties and further terminology

Phenomena

Techniques

Particular types of flow

Footnotes