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My textbook states that for a nonlinear autonomous system $$x^\prime = F(x,y)\qquad y^\prime = G(x,y)$$

The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever the functions $F$ and $G$ have continuous partial derivatives up to order two. To show this , we use Taylor expansions about the point $(x_0,y_0)$ to write $F(x,y)$ and $G(x,y)$ in the form $$F(x,y)=F(x_0,y_0)+F_x(x_0,y_0)(x-x_0)+F_y(x_0,y_0)(y-y_0)+\eta_1(x,y)\\ G(x,y)=G(x_0,y_0)+G_x(x_0,y_0)(x-x_0)+G_y(x_0,y_0)(y-y_0)+\eta_2(x,y)$$ where $\eta_1(x,y)/[(x-x_0)^2+(y-y_0)^2]^{1/2}\to 0$ as $(x,y)\to(x_0,y_0)$, and similarly for $\eta_2$.

By locally linear, the book means that the nonlinear system of differential equations around the critical point can be approximated by the linear system $$\mathbf{u}^\prime=\left( \begin{array}{@{}cc@{}} F_x(x_0,y_0)&F_y(x_0,y_0)\\ G_x(x_0,y_0)&G_y(x_0,y_0) \end{array} \right)\mathbf{u}$$ where $\mathbf{u}=\mathbf{x}-\mathbf{x}^0$. The critical point $\mathbf{x}^0=(x_0,y_0)$ is an isolated critical point where $$F(x_0,y_0)=0\quad G(x_0,y_0)=0\\ \det\left(\begin{array}{@{}cc@{}} F_x(x_0,y_0)&F_y(x_0,y_0)\\ G_x(x_0,y_0)&G_y(x_0,y_0) \end{array}\right)\neq0$$

Why does $F$ and $G$ need to have second order partial derivatives if the first order partial derivatives only show up in this expression? And how does that guarantee that $\mathbf{\eta} /||\mathbf{x}-\mathbf{x}^0||$ will go to $0$ as $\mathbf{x}$ approaches the critical point?

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  • $\begingroup$ The fact that you are asking is standard fact about truncations of Taylor series and could be found in any quite rigorous book on calculus. Just out of curiosity. It seems that your question lacks a lot of information: what do they exactly mean by locally linear and are there any other additional conditions on Jacobi matrix. $\endgroup$ – Evgeny Dec 24 '15 at 9:19
  • $\begingroup$ By locally linear, the book means that the nonlinear system behaves similar to a linear one around a critical point. This is useful because analyzing the linear system's critical point can give insight into whether the critical point of the nonlinear system behaves the same way. I've updated the original post to reflect this. Also I've never encountered Taylor series in the context of multivariable functions. Can you explain this to me or direct me to an appropriate resource? $\endgroup$ – cheesyfluff Dec 24 '15 at 13:31
  • $\begingroup$ The problem is in statement "The system is locally linear in the neighborhood of a critical point $(x_0,y_0)$ whenever the functions $F$ and $G$ have continuous partial derivatives up to order two.". If Jacobi matrix is non-hyperbolic, then it might be simply wrong. See this discussion for example: math.stackexchange.com/questions/1380801/… $\endgroup$ – Evgeny Dec 24 '15 at 13:36
  • $\begingroup$ And this is just one of examples when absence of non-hyperbolicity condition leads to the fact that linearized behaviour is not the same as local non-linear behaviour. $\endgroup$ – Evgeny Dec 24 '15 at 13:51
  • $\begingroup$ I see what you're saying. If the Jacobi matrix has pure imaginary eigenvalues, the book concedes that the associated linear system gives no information about whether the critical point can be a spiral point or a center. It also gives no information about the stability of the critical point. Also I forgot to mention that $\mathbf{x}^0$ is an isolated critical point, so $\det J=0$. $\endgroup$ – cheesyfluff Dec 24 '15 at 13:57

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