A systematic search method for obtaining multiple solutions of simultaneous nonlinear equations

A systematic search method has been developed for obtaining the multiple solutions of a nonlinear equation of the form
f(x) = 0
, where f is a continuously differentiable function from
R^n \rightarrrow R^n
. The method is based on numerical integration of the associated system of differential equations
\dot{f}_i = -f_i
for
i = 1, 2,\cdots , n - 1
and
\dot{f}_n = \pm f_n
along the space curve
l
of intersection
f_i(x) = 0, i = 1, 2,\cdots, n - 1
. The plus or minus sign is chosen so as to make
\{x_k\}
move in the desired direction on
l
.