We now consider a general non-linear plane autonomous system given by the matrix form X prime is equal to g(x), where g(x) has 2 components, right? A, P(x,y) and the Q(x, y) okay? I'm considering this, the generator of non-linear plan autonomous system. I'm assuming that it has isolated the critical point at x0 and y0. Which has an isolated critical point x0, y0. For some technical reason, I'm assuming that both the p and q, assume, P(x,y) and the Q(x, y) are belonging in say C^2(D). Where the D is the region in IR^2 containing the critical point x0, y0. I will denote this one by the X0. There is the certain region D and the C^2 means what? Both functions of P and Q are 2 times continuously differentiable in the region D. I'm assuming it. 2 times a continuously differentiable. That's a simpler way. In other words, p and the support of the partials dp/dx, and the dp/dy, it's a second-order partial derivatives. We have 3, such a thing as a way d^2p/dx^2, d^2p/ dx^2, d^2p/dxy^2 and d^2p/dy^2. They're all continuous in D. That's what I mean. P belongs to C^2 (D).They are all continuous in the region D. The same for the Q, and the same for Q, right? By which I mean dq/dx, dq/dy, d^2q/dxdy, d^2q/dy^2. They are all continuous in the region D. I'm assuming in. Let us consider the linearization of both the P and Q at point X0 and Y0. We have g(x )is equal to P and Q. I'm considering as a linear regression or linear approximation of both P and Q at point Xo and Yo. They are forced to linearization of p, at point (x0,y0) there is a p(x0,y0) plus dp/dx(x0,y0)(x-x0) plus dp/dy(x0,y0)(y-y0). Same thing for Q, first the Q of x_0 y_0 plus dQ/dx, x_0 y_0, and x minus x_0 plus dQ/dy, x_0 y_0, and y minus y_0. That's the linearization of both P and Q. Because I'm assuming that the point x_0, y_0 is a critical point of this non-linear system, by which I mean both P and Q vanish at point x_0, and y_0, so that we do not have this one. Both are 0. This is the linearization, linearization of both the P and Q at pointy x_0 and y_0. It's rather convenient to rely to this vector into the matrix form. Look at this one. This vector, you can write it as, this is dP/dx, x_0 y_0, dP/dy, x_0 y_0, and dQ/dx, x_0 y_0, and dQ/dy x_0 and y_0, times this 2 by 2 matrix times the vector x minus x_0, and y minus y_0. We get the same thing. This 2 by 2 matrix in front. I will denote it by g prime at x_0, times x minus x_0, and the y minus the y_0. This 2 by 2 matrix I denoted by g prime x_0. This is a very important one. I will give a name on it. This we call the Jacobian or Jacobian matrix of the vector P and Q. At point x_0, y_0. Then what? Starting from this original non-linear plane autonomous system, we consider linearization or the linear approximation of both P, and Q at the critical point, x_0, and y_0. We get the following linear play in autonomous system, x prime is equal to g prime x_0, and times x minus x_0, and y minus y_0 line through the linearization. We may think that they are quite similar. For x, y close to the x_0, y_0. This original problem is possibly non-linear. This new problem we have dual system, we have down there is a linear problem. Not just a linear, but we obtain this one through the linear approximation. About this point. Linearization of original problem, x prime equal to g of x at point x_0, and y_0. It's a quite a natural, kicks back to the fall ramp. Because, this line, that's a good linear approximation of a P about the point x_0, y_0. This second component, this is a good approximation, good linear approximation of the original function Q due at the point x_0, y_0. This linear system or the linearization of the original problem could be a good approximation of this original non-linear system about the point x0, y0. It's natural to expect that the behavior of solutions for this original problem, original nonlinear system, is similar to the behavior of the solution for linearization about the critical point x0, y0. They may behave similar. In fact that this is true and we have the following very useful and interesting result, due to the Lyapunov. Then we stated the following the Lyapunov theorem. Consider the general, the plane on autonomous system, x prime is equal to g of x having two components, capital P and Q, where we continue to assume that can P and Q are C2 in a suitable reason containing a separated, isolated, critical point, the x0, y0 of the system. Consider this way. With the same hypothesis as we make before. The critical point x0, this is asymptotically stable. If all eigenvalues of the Jacobian matrix g prime x0 at the critical point are negative. The all eigenvalues of this Jacobian matrix have negatively real parts. Define the following. If all eigenvalues of g prime x0, have negative real parts, then the critical point x0, y0 for this linearization is asymptotically stable. That property is transported to the original non-linear problem, original non-linear system. That's the conclusion A. The critical point x0 is asymptotically stable for this problem if all eigenvalues of this Jacobian matrix have negative real parts. Same for the unstable case. This critical point x0 is unstable if this Jacobian matrix g prime x0, has an eigenvalue with positive real part, has an eigenvalue and not all of them, because this is a two-by-two matrix, it may have at most two distinct eigenvalues. If any one of them has a positive real part then we can conclude that this critical point is unstable for this non-linear system. That's the conclusion of the theorem.