I have just started a dynamical systems course and I am a bit confused as to how to determine if something is positively or negatively invariant, or just invariant.
I know the defintions for invariance are as follows:
Let $B \subset X$.
$B$ is called
Positively Invariant if $S_tB\subset B$ for all $t \geq 0$
B is called negatively invariant if $S_tB\supset B$
B is called invariant if $S_tB=B$ for all $t\geq 0$
Here are some questions I am stuck on:
Considering the continuous time dynamical system given by the ODE $x'=-x$
with solution $x(t)=xe^{-t}$, determine if the following sets are invariant, positively or negatively invariant.
1.) $M \subset \mathbb R : M=(0,1)$
So, I have that $S_t((0,1))=(0,-e^{-t})\not\subset B=(0,1)$ for all $t>0$
Then, for $t=0$, we have $S_t((0,1))=(0,-1)\not\subset(0,1)$ for all $t=0$.
Hence, I cannot see how this is invariant at all? None of the defintions apply here or have I done it all wrong?
2.) $[-1,1]$
I have that this one is positively invariant as $S_tx=xe^{-t} \in [1,1]$ for all $t \geq 0$ and $x\in[-1,1]$
3.) (1,2)
I have this as invariant.
Any help on these much appreciated. Surprisingly few other examples or resources on this on line.
Best Answer
The solution should be written as $x(t) = x_0 e^{-t}$, where $x_0 = x(0)$. $S_t$ is the time evolution operator, so $S_t(x) = x e^{-t}$. Thus for #1, if $x \in (0,1)$, $S_t(x) = x e^{-t} \in (0, e^{-t}) \subset (0,1)$.