Consider the system \begin{align*} \dot{x} &= 2x+y\\ \dot{y} &= x-y. \end{align*} Prove that there exists an infinite number of invariant sets.
I do not know how to proceed. My first attempt was calculate the equilibrium point and analyzed its stability. But I do nos know how to conclude that the exist infinite invariant sets. Another approach I tried is prove that the line $y=kx$ with $k \in \mathbb{R}$ is invariant, but I did not get anything.
Any help?
Clearly the characteristic equation is $$ \lambda^2-\lambda-3 =0 $$ which has two roots $\lambda_{1,2}=\frac{1\pm\sqrt{13}}{2}$. Let $$ y=c_1e^{\lambda_1t}+c_2e^{\lambda_2t} $$ and then $$ x=\dot y+y=c_1(1+\lambda_1)e^{\lambda_1t}+c_2(1+\lambda_2)e^{\lambda_2t}. $$ Now for $k$ which will be determined soon, one has $$ kx-y=c_1(k+k\lambda_1-1)e^{\lambda_1t}+c_2(k+k\lambda_2-1)e^{\lambda_2t}. $$ Let $k+k\lambda_1-1=0$ give $$ k=\frac{1}{1+\lambda_1} $$ and hence $$ kx-y=c_2(k+k\lambda_2-1)e^{\lambda_2t}. $$ Noting that $e^{\lambda_2t}\to 0$ as $t\to\infty$, one can have that $y=kx$ is the invariant (the red line in the graph) and hence conclude an infinite number of invariant sets.