What does it mean to take the Jacobian of a system of Differential Equations?

Question

When solving nonlinear differential equations, we often use the "Jacobian of the system" to determine if fixed points are stable.

As an example, suppose I have a nonlinear system

$$x_{t} = f(x,y)$$ $$y_{t} = g(x,y)$$

And let $(\alpha,\beta)$ be a fixed point. Then we consider the Jacobina defined as

$$\begin{bmatrix} x_{tx} & x_{ty} \\ y_{ty} & y_{ty} \\ \end{bmatrix}$$

Which is the same as $$\begin{bmatrix} f_{x} & f_{y} \\ g_{y} & g_{y} \\ \end{bmatrix}$$

And then the eigenvalues of this approximation at the fixed points give us the information we need about stability.

Now, what I am confused about is what exactly is happening beneath the hood here. How is the Jacobian, normally defined over a single function, acting here as a linear approximator for a system of functions? Put differently, what's the mathematical intuition behind this form of approximator?

My class has been very handwavy with this.

Answer 1

the critical thing to realize here is that there is an isomorphism between a system of equations and a vector valued function. rather than thinking about two equations, think about [x_t, y_t] = h([x,y]) where h returns a two element vector, the first element is the result of f(x,y) etc,

Answer 2

This is rather bad notation at the start. The solutions are functions of $t$, and not of the space coordinates $x,y$.

In the linearization around some stationary point or constant solution $(x^*,y^*)$ you consider the Taylor expansion of the ODE function and explore what the solution of the equation reduced to the linear terms tells about the solution of the non-linear system.

In the first step, the functions $f,g$ are just functions of two variables, not connected to any differential equation or its solutions. Thus you get $$ f(x^*+u,y^*+v)=\partial_xf(x^*,y^*)u+\partial_yf(x*,y^*)v+...\\ g(x^*+u,y^*+v)=\partial_xg(x^*,y^*)u+\partial_yg(x*,y^*)v+...\\ $$ Then consider the linear system $$ \dot u(t)=\partial_xf(x^*,y^*)u(t)+\partial_yf(x*,y^*)v(t)\\ \dot v(t)=\partial_xg(x^*,y^*)u(t)+\partial_yg(x*,y^*)v(t) $$

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One can also explore how the solution depends on the initial values and how small changes in the initial values influence the solution. In that case the derivative of the solution for some space variables makes sense.