Fixed point

A fixed point $𝑥 *$ , also called an equilibrium solution, of a dynamical system is a phase point which remains constant for all time, i.e. a system in this state will remain this way. #m/def/dynamics

For a differential system this means $𝑓 (𝐱) = 0$ ; whereas
For an iterated map this means $𝑓 (𝐱) = 𝐱$ .

A fixed point corresponds precisely to an Invariant set of cardinality one.

Classification

Stability

A fixed point may be

stable, meaning small perturbations decay back to the point
- Lyapunov stable if for every neighbourhood $𝑈$ of $𝑥 *$ there exists a neighbourhood $𝑉$ of $𝑥 *$ such that $Φ 𝑡 𝑥 \in 𝑈$ for all $𝑡 > 0$ and $𝑥 \in 𝑉$ .
- Asymptotically stable if it Lyapunov stable of in addition to being Lyapunov stable, there exists a neighbourhood $𝑈$ of $𝑥 *$ such that $l i m 𝑡 \to \infty Φ 𝑡 𝑥 = 𝑥 *$ for all $𝑥 \in 𝑈$ .
unstable, meaning small perturbations increase away from the point
- Lyapunov unstable if for every neighbourhood $𝑈$ of $𝑥 *$ there exists a neighbourhood $𝑉$ of $𝑥 *$ such that $Φ 𝑡 𝑥 \in 𝑈$ for all $𝑡 < 0$ and $𝑥 \in 𝑉$ .
- Asymptotically unstable if it Lyapunov stable of in addition to being Lyapunov stable, there exists a neighbourhood $𝑈$ of $𝑥 *$ such that $l i m 𝑡 \to - \infty Φ 𝑡 𝑥 = 𝑥 *$ for all $𝑥 \in 𝑈$ .
A centre.

These can be identified qualitatively in a Phase portrait. Quantitatively it is described by

Contraction

The dynamics at a fixed point is said to be contracting iff an infinitesimal ball of initial conditions around a fixed point shrink over time. This can occur for both unstable and half-stable points. Using Linear stability analysis, contracting dynamics corresponds to a negative Jacobian trace in the continuous case and a Jacobian determinant less than one in the discrete case. A system that is contracting at all points (fixed or not) is called a Dissipative system.

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