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:

(61) |

(62) |

(63) |

(64) |

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 , there should be a solution of the equation such that

(65) |

**Theorem 4**

Let *f* be a smooth mapping from to . Assume that
and that the matrix

(66) |

in which the solutions of the equation are given by a smooth function . Moreover, we have

(67) |

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

**Example 11**

Consider the system

A solution for is *x _{1}*=1, . The linearized matrix

(68) |

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