The implicit function theorem

The implicit function theorem is an important example where the linearized system qualitatively determines the behavior of the nonlinear system. We modify (2.1) by letting f depend on a parameter:  

$$\dot x=f(x,\lambda).$$ (61)

As before x and f take values in ${\rm I\!R}^n$, while $\lambda$ is a real parameter. Suppose that we know an equilibrium solution x_c for $\lambda= \lambda_c$: $f(x_c,\lambda_c)=0$. If we linearize around this point (x=x_c+y, $\lambda=\lambda_c+\mu$), we find the linearized system  

$$\dot y_i=\sum_{j=1}^n {\partial f_i\over\partial x_j}(x_c,\lambda_c)y_j +{\partial f_i\over\partial\lambda}(x_c,\lambda_c)\mu.$$ (62)

This system is of the form  

$$\dot y=Ay+\mu g,$$ (63)

where the matrix A and the vector g are given by

$$A_{ij}={\partial f_i\over\partial x_j}(x_c,\lambda_c),\quad g_i={\partial f_i\over\partial\lambda}(x_c,\lambda_c).$$ (64)

If A is nonsingular, equilibrium solutions of (2.7) are given by $y=-\mu A^{-1}g$, i.e. we can solve for y as a function of $\mu$.

Intuitively, we expect that the fully nonlinear system should have a solution which is approximated by this solution of the linear system, i.e. close to $\lambda= \lambda_c$, there should be a solution $x(\lambda)$ of the equation $f(x,\lambda)=0$ such that

$$x(\lambda)=x_c-A^{-1}g(\lambda-\lambda_c)+O((\lambda-\lambda_c)^2).$$ (65)

The implicit function theorem asserts precisely that this is the case.

Theorem 4

Let f be a smooth mapping from ${\rm I\!R}^n\times {\rm I\!R}$ to ${\rm I\!R}^n$. Assume that $f(x_c,\lambda_c)=0$ and that the matrix

$$A_{ij}={\partial f_i\over\partial x_j}(x_c,\lambda_c)$$ (66)

is not singular. Then there exists some neighborhood of the point
$(x_c,\lambda_c)$ in which the solutions of the equation $f(x,\lambda)=0$ are given by a smooth function $x=h(\lambda)$. Moreover, we have

$$h(\lambda)=x_c-A^{-1}{\partial f\over\partial\lambda}(x_c,\lambda_c)(\lambda -\lambda_c)+O((\lambda-\lambda_c)^2).$$ (67)

A proof of the implicit function theorem can be found in advanced calculus texts.

Example 11

Consider the system

   $$\begin{eqnarray} x_1^2+\sin x_2-\lambda&=&0,\nonumber\ x_1-1+\cos x_2-\lambda^2&=&0.\end{eqnarray}$$

A solution for $\lambda=0$ is x₁=1, $x_2=-\pi/2$. The linearized matrix A is

$$A=\pmatrix{2&0\cr 1&1},$$ (68)

which is nonsingular. According to the implicit function theorem, the system can now be solved for x in the form

   $$\begin{eqnarray} x&=&\pmatrix{1\cr -\pi/2}-\pmatrix{2&0\cr 1&1}^{-1} \pmatrix{-1... ...umber\ &=&\pmatrix{1+\lambda/2\cr -\pi/2-\lambda/2}+O(\lambda^2).\end{eqnarray}$$

Since x is a smooth function of $\lambda$, we can also derive higher order approximations by using a Taylor series ansatz for x and plugging into the equations.