Numerical continuation
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, The parameter
from the boundary conditions to the ODE: The second step is to add an additional equation, a phase constraint, that can be thought of as determining the period.
A numerical continuation is an algorithm which takes as input a system of parametrized nonlinear equations and an initial solution
One advantage of natural parameter continuation is that it uses the solution method for the problem as a black box.
There has been a lot of work in the area of large scale continuation on applying more sophisticated algorithms to black box solvers (see e.g. LOCA).
With these two operations this continuation algorithm is easy to state (although of course an efficient implementation requires a more sophisticated approach.
Pseudo-arclength continuation was independently developed by Edward Riks and Gerald Wempner for finite element applications in the late 1960s, and published in journals in the early 1970s by H.B.
A detailed account of these early developments is provided in the textbook by M. A. Crisfield: Nonlinear Finite Element Analysis of Solids and Structures, Vol 1: Basic Concepts, Wiley, 1991.
Crisfield was one of the most active developers of this class of methods, which are by now standard procedures of commercial nonlinear finite element programs.
[2] It is extremely insightful as to the presence of stable solutions (attracting or repelling) in the study of nonlinear differential equations where time stepping in the form of the Crank Nicolson algorithm is extremely time consuming as well as unstable in cases of nonlinear growth of the dependent variables in the system.
Also, research using these techniques has provided the possibility of finding stable manifolds and bifurcations to invariant-tori in the case of the restricted three-body problem in Newtonian gravity and have also given interesting and deep insights into the behaviour of systems such as the Lorenz equations.
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