I have the following system of equations:
$$\begin{align} \frac{dX}{dt} &= 2Y-2\\ \frac{dY}{dt} &= 9X-X^3 \end{align}$$
I would like to study the property of solutions to this function about the point $(3,1)$. Namely, I'd like to find a linear approximation for small oscillations about the equilibrium point, and I'd like to estimate the period of oscillation.
I don't exactly know what is meant by "find a linear approximation" for a coupled system, though. I've taken the Jacobian of the system, yielding eigenvalues $6i,-6i$ and corresponding eigenvectors $[1,3i]$ and $[1,-3i]$.
But from here, I don't exactly know where to go.
First write: $$X=x+3,\ \ \ Y=y+1$$ so that the equilibrium point moves to $(0,0)$. The equations become: $$\begin{array}{l}\dot{x}=2y\\\dot{y}=9x+27-(x+3)^3=-x^3-9x^2-18x\end{array}$$ Now linearize in both $x$ and $y$ (i.e.: take the Taylor expansion up to first order). You get: $$\begin{array}{l}\dot{x}=2y\\\dot{y}=-18x\end{array}$$ This equation determinates the dominant behavior of your system near the equilibrium point. I think you can take it from here. Good luck.