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# 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:
 (61)
As before x and f take values in , while is a real parameter. Suppose that we know an equilibrium solution xc for : . If we linearize around this point (x=xc+y, ), we find the linearized system
 (62)
This system is of the form
 (63)
where the matrix A and the vector g are given by
 (64)
If A is nonsingular, equilibrium solutions of (2.7) are given by , i.e. we can solve for y as a function of .

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

Theorem 4

Let f be a smooth mapping from to . Assume that and that the matrix
 (66)
is not singular. Then there exists some neighborhood of the point
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 x1=1, . The linearized matrix A is
 (68)
which is nonsingular. According to the implicit function theorem, the system can now be solved for x in the form

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.

Next: Nonlinear stability Up: Linearization Previous: Linearization
Michael Renardy
1998-07-13