Question: Consider the following autonomous vector field on the plane: $$\dot x = ax+by$$ $$\dot y = cx+dy $$ $$(x,y) \in \mathbb{R}^2 \qquad (a, b, c, d) \in \mathbb{R}$$
$\bullet$ Give a set of conditions on $a, b, c, d$ for which the vector field has no periodic orbits.
$\bullet$ Give a set of conditions on $a, b, c, d$ for which all of the orbits are periodic.
$\bullet$ Using $V(x,y)= \frac{1}{2}(x^2 +y^2)$ as a Lyapunov function, give conditions on $a, b, c, d$ for which $(x,y) = (0,0)$ is asymptotically stable.
$\bullet$ Give conditions on $a, b, c, d$ for which $x = 0$ is the stable manifold of $(x,y) = (0,0)$ and $y = 0$ is the unstable manifold of $(x,y) = (0,0)$.
Answers:
$\bullet$ By the Bendixson criterion I can take $\dot x = f(x,y)$ and $\dot y = g(x,y)$ and find that $\frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y} = a + d$ so for $a +d \neq 0$ the vector field has no periodic orbits.
$\bullet$ Using the Lyapunov function at the point $(x,y)=(0,0)$, I have that $V(0,0) = 0$ and $V(x,y) \gt 0$ for any neighborhood around the origin, thus it is Lyapunov stable.
We have: $\dot V(x,y) = \frac{\partial V}{\partial x}\dot x + \frac{\partial V}{\partial y}\dot y$
$=x(ax+by) + y(cx+dy)$
$=ax^2+dy^2+(b+c)xy \lt 0$ for asymptotical stability.
Hence for the vector field to be asymptotically stable, we must have $a, d \lt 0$ and $b + c = 0$
$\bullet$ We have a linear system: $$\begin{pmatrix} \dot x \\ \dot y \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y\end{pmatrix}$$
So we can analyse the stability through the matrix of the form $\dot x = Ax$ with $A= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is the Jacobian at $(0,0)$
For $a \le 0, x = 0$ is the stable manifold, and for $d \gt 0, y = 0 $ is the unstable manifold. The sign and magnitude of $b,c $ does not affect the stability, thus $(b,c) \in \mathbb R $ is valid.
These are my answers so far, however I am unsure of what conditions I could give for $a,b,c,d$ for all possible orbits to be periodic. I would also very much appreciate if anyone can point out any errors or incorrect statements in my answers.