Dynamical system

Differential system

A -dimensional differential system or flow has the form

or more concisely flow

Note that many ODEs may be converted to this form, including those with time-dependent terms.¹² The vector 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

may be rewritten by setting and , thus

which is a linear differential system since it is a linear combination of .

The swinging pendulum equation

may be rewritten as

which is a nonlinear differential system.

A non-autonomous system such as a forced harmonic oscillator

may be rewritten as

hence an th-order non-autonomous equation is an -dimensional differential system. This makes sense because the state of such a problem really does depend on time.

Properties and further terminology

• Existence and uniqueness of flows
• Continuity of flows
• Equivalence of flows
• Fundamental theorem of flows
• Fixed point
• Hartman-Grobman theorem
• Lyapunov stability criterion

Phenomena

• Homoclinic orbit

Techniques

• Linear stability analysis

Particular types of flow

• Flow in 2D
  • Reversible 2D flow

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tidy | en | sembr

Footnotes

1. 2024. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering, §1.2, pp. 6ff. ↩

2. 2017. Elementary differential equations and boundary value problems, p. 282 ↩