Numerical continuation

Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations,

${\displaystyle F(\mathbf {u} ,\lambda )=0.}$^[1]

The parameter ${\displaystyle \lambda }$ is usually a real scalar and the solution ${\displaystyle \mathbf {u} }$ is an n-vector. For a fixed parameter value ${\displaystyle \lambda }$, ${\textstyle F(\cdot ,\lambda )}$ maps Euclidean n-space into itself.

Often the original mapping ${\displaystyle F}$ is from a Banach space into itself, and the Euclidean n-space is a finite-dimensional Banach space.

A steady state, or fixed point, of a parameterized family of flows or maps are of this form, and by discretizing trajectories of a flow or iterating a map, periodic orbits and heteroclinic orbits can also be posed as a solution of ${\displaystyle F=0}$.