Definition:
Invariant setA set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$
-if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq S,\forall t$
-if $\forall x\in S,q\left ( t,x \right )\subseteq S,\forall t$
Definition: Positively invariant
A set $S\subseteq \mathbb{M}$ is positively invariant if $\Psi_{t}\left ( S \right )\subseteq S,t \geq 0$
Definition: Negatively invariant
A set $S\subseteq \mathbb{M}$ is negatively invariant if $\Psi_{t}\left ( S \right ) \subseteq S,t\leq 0$
The first definition seems to be saying that for any elements of a set S, the map of elements of a set S under a trajectory results in the image being in the set S-an onto map?
A couple of definitions came up on my notes and I hope to gain a geometric intuition for it. Could someone kindly assists me?
Thanks in advance.
Best Answer
In my opinion, the most intuitive definition of an invariant set is that it does not change when the flow $Ψ$ is applied (pointwise), i.e., $S$ is invariant, iff $Ψ_t(S) = S~ ∀ t$. With other words, trajectories may not leave or enter the set, i.e., for any point $x∈S$, the trajectory passing through that point is entirely in $S$ – for the entire past and future. It suffices to only require $Ψ_t(S)⊆S$ in the definition as this directly yields that $Ψ_t(S)⊆Ψ_{-t}(Ψ_t(S))=S$.
A classical example is are the state-space trajectories of a frictionless pendulum. All states corresponding to a given total energy of the pendulum form an invariant set (and so do all unions of such sets). Trajectories leading into or out of this set would correspond to the pendulum losing or gaining energy, which is impossible. If we add friction to the scenario and this friction is such that the pendulum, the only possible invariant set contains only the rest state of the pendulum. However, this point is only an invariant set if the friction is such that a pendulum in motion never reaches total standstill, but only moves slower and slower.
If the set is only positively invariant, trajectories may enter it but not leave it. For the pendulum with friction, all states with less than a given energy form a positively invariant set: Trajectories leaving the set would correspond to the pendulum gaining energy, which is impossible.
Finally, a negatively invariant set may not be entered by trajectories. In the pendulum with friction, all states with an energy higher than a given energy would comprise a negatively invariant set.