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
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...
, or
fixed point, of a
parameterized family of
flows or
maps
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
are of this form, and by
discretizing trajectories of a flow or iterating a map,
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
s and
heteroclinic orbit
In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end o ...
s 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
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
.
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
An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
:
:
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
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. ...
(
singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
,
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
: