Suppose that we have a two-dimensional nonlinear ODE of the form \begin{align*} \frac{dx}{dt} = f(x,y) \\[5pt] \frac{dy}{dt} = g(x,y) \end{align*}
where $f:\mathbb{R}^2 \to \mathbb{R}$ and $g: \mathbb{R}^2 \to \mathbb{R}$ are $C^1$ functions. Suppose the nullclines of the system are as shown below.
The intersection of the nullclines, call it $(x^{*},y^{*})$, is a fixed point of the system. Suppose that
- $(x^{*},y^{*})$ is unstable
- $(x^{*},y^{*})$ is the only fixed point of the system
- The system has a limit cycle about the fixed point.
The presence of a limit cycle enclosing the fixed point suggests that $(x^{*},y^{*})$ should be an unstable spiral or unstable node. So with $J(x^{*},y^{*})$ denoting the Jacobian matrix of the system evaluated at the fixed point, this would tell us that $\text{tr}(J(x^{*},y^{*})) < 0$ and $\det J(x^{*},y^{*}) > 0$. Is there anything else we can say about the Jacobian matrix at the fixed point, based on the (local) configuration of the nullclines near the fixed point? Can anything be said about the signs of the individual entries?