Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations,
The parameter is usually a real scalar, and the solution an n-vector. For a fixed parameter value , maps Euclidean n-space into itself.
Often the original mapping 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 .
Other forms
In some nonlinear systems, parameters are explicit. In others they are implicit, and the system of nonlinear equations is written
where is an n-vector, and its image is an n-1 vector.
This formulation, without an explicit parameter space is not usually suitable for the formulations in the following sections, because they refer to parameterized autonomous nonlinear dynamical systems of the form:
However, in an algebraic system there is no distinction between unknowns and the parameters.
Periodic motions
A periodic motion is a closed curve in phase space. That is, for some period ,
The textbook example of a periodic motion is the undamped pendulum.
If the phase space is periodic in one or more coordinates, say , with a vector , then there is a second kind of periodic motions defined by
for every integer .
The first step in writing an implicit system for a periodic motion is to move the period 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. This is necessary because any solution of the above boundary value problem can be shifted in time by an arbitrary amount (time does not appear in the defining equations—the dynamical system is called autonomous).
There are several choices for the phase constraint. If is a known periodic orbit at a parameter value near , then, Poincaré used
which states that lies in a plane which is orthogonal to the tangent vector of the closed curve. This plane is called a Poincaré section.
For a general problem a better phase constraint is an integral constraint introduced by Eusebius Doedel, which chooses the phase so that the distance between the known and unknown orbits is minimized:
Homoclinic and heteroclinic motions
Definitions
Solution component
A solution component of the nonlinear system is a set of points which satisfy and are connected to the initial solution by a path of solutions for which and .
Numerical continuation
A numerical continuation is an algorithm which takes as input a system of parametrized nonlinear equations and an initial solution , , and produces a set of points on the solution component .
Regular point
A regular point of is a point at which the Jacobian of is full rank .
Near a regular point the solution component is an isolated curve passing through the regular point (the implicit function theorem). In the figure above the point is a regular point.
Singular point
A singular point of is a point at which the Jacobian of F is not full rank.
Near a singular point the solution component may not be an isolated curve passing through the regular point. The local structure is determined by higher derivatives of . In the figure above the point where the two blue curves cross is a singular point.
In general solution components are branched curves. The branch points are singular points. Finding the solution curves leaving a singular point is called branch switching, and uses techniques from bifurcation theory (singularity theory, catastrophe theory).
For finite-dimensional systems (as defined above) the Lyapunov-Schmidt decomposition may be used to produce two systems to which the Implicit Function Theorem applies. The Lyapunov-Schmidt decomposition uses the restriction of the system to the complement of the null space of the Jacobian and the range of the Jacobian.
If the columns of the matrix are an orthonormal basis for the null space of
and the columns of the matrix are an orthonormal basis for the left null space of , then the system can be rewritten as
where is in the complement of the null space of .