I have a system \begin{align} \dfrac{dx}{dt}&=-x^2 + 4 y^2, \\ \dfrac{dy}{dt}&=-8 - 4 y + 2 x y. \end{align}
There two singular points $A_1(-2;-1), A_2(4,2)$. I need to know the type of these points.
To do it, in case of linear system, I need to find lambdas, which depend on a,b,c,d koefs. Now, in case of nonlinear system, I'm getting problems:
1) To find lamdas I need to make a linearization of the equations. I don't know how to do that. All that I got in mind is the regression, exactly linear regression.
2)Instead of linear system, there are 2 singular points, hence I think, lambda equation should be changed respectively to this fact. How to find lambdas for two or more singular points?
) You need to compute the Jacobi matrix in the stationary points. The linearization is just the first order Taylor expansion of the right side. You have $$\dot x=f(x,y),\\\dot y=g(x,y).$$ In the linearization you replace them with $$\dot x=f(x_0,y_0)+∂_xf(x_0,y_0)(x-x_0)+∂_yf(x_0,y_0)(y-y_0),\\ \dot y=g(x_0,y_0)+∂_xg(x_0,y_0)(x-x_0)+∂_yg(x_0,y_0)(y-y_0).$$
) You compute the eigenvalues separately. The linearization gives insight into the local behavior very close to the stationary points, as long as the stationary points are isolated, there is no need to consider any possible interaction (like connecting trajectories) between them.