So this problem statement says
Use analytical or graphical techniques to find the positive and the negative limit sets of the orbits through the listed initial points.
I have a system $$x^{'}=y$$ $$y^{'}=x$$ and the points $(1,1), (1,-1), (1,0)$
I have read through my notes and book and am still confused how I determine this.
Clearly, the equilibrium is at $(0,0)$ and then putting this in linear system form, I have $$\textbf{x}^{'}=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\textbf{x}$$
which has characteristic polynomial $P(\lambda)=(\lambda +1)(\lambda -1)$. Thus, $\lambda_1=-1<0<\lambda_2=1$, so the orbits are an unstable saddle point.
Then I find the solution $$\textbf{x}(t)=c_1e^{-t}\begin{bmatrix} -1 \\ 1 \end{bmatrix}+c_2e^t\begin{bmatrix} 1 \\ 1 \end{bmatrix}$$.
Now, the book says:
The set of points an orbit $Γ$ approaches as $t\rightarrow+\infty$ is its $\textit{positive limit set}$ $ω(Γ)$. We define the $\textit{negative limit set}$ $α(Γ)$ similarly, except that $t\rightarrow-\infty$.
So as $t\rightarrow+\infty$, the solution $x\rightarrow+\infty$ and $y\rightarrow+\infty$, and as $t\rightarrow-\infty$, the solution $x\rightarrow-\infty$ and $y\rightarrow+\infty$. So that means the limit sets would be $\emptyset$?
Am I supposed to look at what the graph does at each point given?
e.g. at $(1,-1)$ it approaches $(0,0)$ and $-\infty$, so then $ω(Γ)=\left\{\textbf{0}\right\}$ and $α(Γ)=\emptyset$
At $(1,1)$ it approaches $+\infty$ and $(0,0)$ so then $ω(Γ)=\emptyset$ and $α(Γ)=\left\{\textbf{0}\right\}$.
But at $(1,0)$ the orbit approaches $+\infty$ and $-\infty$ so then $ω(Γ)=\emptyset$ and $α(Γ)=\emptyset$.
And at equilibrium $(0,0)$ the orbit would stay at $(0,0)$? So then $ω(Γ)=\left\{\textbf{0}\right\}$ and $α(Γ)=\left\{\textbf{0}\right\}$.
My teacher briefly went over this the last day of class and I am still so confused, I know it's not a difficult concept I am just not approaching the problem correctly. Any help is appreciated!!