Invariant set

In a dynamical system an invariant set $𝐴$ is a subset of the phase space closed under time evolution, i.e. $Φ 𝑡 𝐱 0 \in 𝐴$ for all $𝐱 0 \in 𝐴$ and all $𝑡$ ¹. #m/def/dynamics

A positively invariant set is invariant under forward evolution
A negatively invariant set is invariant under reverse evolution

Examples

Fixed point
Every trajectory $Φ ℝ 𝐱 0$ is an invariant set
Separatrix
Basin of attraction
Constant manifold for a Conserved quantity
$𝛼$ - and $𝜔$ -limit sets

#state/develop | #lang/en | #SemBr

with $𝑡 \in ℝ$ in the continuous case and $𝑡 \in ℤ$ or $ℕ$ in the discrete case. ↩