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<H1><A NAME="SECTION004250000000000000000"> </A><A NAME="1460"> </A><A NAME="1461"> </A><A NAME="1462"> </A>
<BR>
The Equilibrium Continuation Window
</H1>
<BR>
<DIV ALIGN="CENTER"><A NAME="Continuation"> </A><A NAME="1506"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.26:</STRONG>
The Continuation window.</CAPTION>
<TR><TD>48#48</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
<DL>
<DD><P>
<DT><STRONG>Window title:</STRONG>
<DD>DsTool: Equilibrium Continuation
<DT><STRONG>Function:</STRONG>
<DD>The Equilibrium Continuation window allows the user to compute curves of bifurcations.
<DT><STRONG>Description:</STRONG>
<DD>The Equilibrium Continuation window is opened by selecting
the Equilibrium Continuation option from the Panels menu button
located in the Command window. The window allows the
user to compute curves of equilibrium points with one
varying parameter and curves of saddle-node and Hopf bifurcations
with two varying parameters. A brief tutorial on
continuation calculations is at the end of this section.
The different Hopf bifurcation algorithms are described in
[<A
HREF="node68.html#GM">2</A>] and [<A
HREF="node68.html#GMS">3</A>].
<P>
Color coding<A NAME="1470"> </A> for hyperbolic equlibrium points
is performed according to the
value of the Monitor switch setting. For the
bifurcation types, color coding is performed according to the choice of
colormap, as shown in the Continuation Colors window.
In the default colormap, Hopf bifurcation
points are displayed in magenta, saddle-node bifurcation points are
displayed in green, degenerate Hopf points are displayed in
orange, and resonant saddle-node points are displayed in sea
green.
<P>
<DT><STRONG>Panel items:</STRONG>
<DD>
<UL>
<LI>Iterations numeric field:
Displays the number of points along
a curve of equilibrium points that will be computed. The
type of curve that will be computed is determined by the
Mode stack setting. The default value is 10.
<LI>Monitor switch exclusive setting:
Allows the user to choose whether or not to color equilibrium points with the
DsTool convention<A NAME="1473"> </A>: blue
for sink, green for saddle and red for source.
The default value is on.
<LI>Mode stack setting:<A NAME="1474"> </A>
Allows the user to choose the system of equations to be solved with
continuation. The choices are:
<DL>
<DT><STRONG>Static Bifurcation:</STRONG>
<DD><A NAME="1476"> </A><A NAME="1477"> </A>
for continuing curves of
equilibrium points with a single active parameter.
<DT><STRONG>Saddle Node (
<!-- MATH: $\det() = 0$ -->
49#49
) :</STRONG>
<DD><A NAME="1478"> </A><A NAME="1479"> </A>
for computing curves
of saddle-node bifurcations with two active parameters.
The determinant of the Jacobian is computed and used
as an augmenting equation to the equilibrium equations.
<DT><STRONG>Saddle Node Bifurcation:</STRONG>
<DD>for computing curves
of saddle-node bifurcations with two active parameters.
<DT><STRONG>Hopf Bif ( |<I>Bp</I>=0| ) :</STRONG>
<DD><A NAME="1480"> </A><A NAME="1481"> </A>
for computing curves of
Hopf bifurcations with two active parameters, using
the bialternate product.
<DT><STRONG>Hopf Bif (
<!-- MATH: $|{\rm Bezout}|=0$ -->
50#50
) :</STRONG>
<DD>for computing curves of
Hopf bifurcations with two active parameters, using
the Bezoutian.
<DT><STRONG>Hopf Bif (JGR):</STRONG>
<DD>for computing curves of
Hopf bifurcations with two active parameters, using
the algorithm developed by Jepson, Griewank and Reddien.
<DT><STRONG>Hopf Bif (kubicek 1):</STRONG>
<DD>for computing curves of
Hopf bifurcations with two active parameters, using
the algorithm developed by Kubicek.
</DL>
<LI>Aug params listbox:
Allows the user to choose active parameters. Static Bifurcation
calculations require that the user select one active
parameter. Saddle-node and Hopf bifurcation calculations
require the user to select two active parameters. An error message will
be printed if the user tries to select too many active parameters, or
tries to initiate a calculation with too few active parameters.
<LI>Cont. param stack setting:
Allows the user to choose one of the phase space
variables or one of the active parameters to parameterize
the curve of equilibrium points.
<LI>Parameter fix switch exclusive setting:
Allows the user to choose whether the setting in Cont. param stack may be changed
during the calculation. The choice Vary allows the continuation algorithm to change
the variable used as the continuation parameter adaptively.
The choice Fix uses requires the continuation algorithm to
use only the variable selected in the Cont. param
stack setting. The default is the ``Vary'' setting.
<LI>Jacobian update switch exclusive setting:
Allows the user to choose to update the Jacobian of the augmented system
at each step or only at the initial continuation step. The default is to update the
Jacobian at each step.
<LI>Absolute error read-write text field:
Displays the absolute error tolerance used in solving the augmented equations.
The default value is 10<SUP>-5</SUP>.
<LI>Relative error read-write text field:
Displays the relative error tolerance used in
solving the augmented equations. The default value is 10<SUP>-5</SUP>.
<LI>Minimum step read-write text field:
Displays the minimum step size used by the
continuation algorithm for continuation steps of the
continuation parameter. The default value is 10<SUP>-3</SUP>.
<LI>Maximum step read-write text field:
Displays the maximum step size used by the
continuation algorithm for continuation steps of the
continuation parameter. The default value is 0.1.
<LI>Sugg. step read-write text field:
Displays the step size used by the
continuation algorithm for the initial continuation step of the
continuation parameter. The default value is 0.01.
<LI>Debug level numeric field:
Displays an integer for determining the
amount of diagnostic information printed to the terminal
window from which DsTool is run. The default value
is 0, which prints no diagnostic information to the terminal.
<LI>Forwards command button:
Initiates a continuation calculation in the direction of
increasing values of the continuation parameter for the number
of steps in the Iterations field.
<LI>Backwards command button:
Initiates a continuation calculation in the direction of
decreasing values of the continuation parameter for the number
of steps in the Iterations field.
<LI>Continue command button:
Continues the last continuation calculation for the number
of steps in the Iterations field.
<LI>Search command button:<A NAME="1487"> </A>
Uses Monte Carlo seeds to locate solutions of the augmented
equations. The number of seeds used is determiend by the
value of the Iterations text field.
<LI>State window button:<A NAME="1488"> </A><A NAME="1489"> </A>
Opens and brings to the foreground the Continuation State window.
<LI>Dismiss command button:
Closes the Equilibrium Continuation window.
</UL></DL>
<P>
<BR>
<BR>
<B>Appendix: Continuation Calculations</B>
<P>
One parameter continuation is a systematic strategy for computing
curves of solutions to <I>l</I> equations
<!-- MATH: $G: \real^{l+1} \rightarrow \real^l$ -->
51#51
in
<I>l</I>+1 variables. The mathematical foundation for continuation
algorithms is the implicit function theorem. If <I>G</I> is differentiable
and the Jacobian of <I>G</I> is surjective at a point
<!-- MATH: $x \in \real^{l+1}$ -->
52#52
where
<I>G</I> vanishes, then the solutions form a curve in a neighborhood of
<I>x</I>. Moreover, the implicit function theorem gives a formula for the
tangent vector to the solution curve. Continuation algorithms exploit
this information. They use an initial value solver for ordinary
differential equations (often just the Euler method) to step along an
approximation to the solution curve. They then appply a root finding
algorithm to significantly improve the approximation. The alternation
of root finding with numerical integration steps distinguish
continuation methods. Choices of how to parametrize the solution
curve, choose time steps and restrict the equations to a hyperplane to
obtain a square system of equations with a unique solution (locally)
need to be made. Various continuation packages take different
approaches to these matters: the continuation ``engine'' used by DsTool
is PITCON (Rheinboldt ...).
<P>
If
<!-- MATH: $f:\real^{n+k} \rightarrow \real^n$ -->
53#53
is a <I>k</I> parameter family of vector fields,
then we assemble systems of equations <I>G</I> for varied calculations.
To do so we restrict ourselves to a submanifold of dimension
<!-- MATH: $l+1 \subset \real^{n+k}$ -->
54#54.
Most frequently this submanifold is obtained by fixing <I>k</I>-<I>s</I> of the parameters of <I>f</I> (called <B>inactive parameters</B>) and varying <I>s</I> <B>active</B> parameters. In the simplest case, <I>G</I>=<I>f</I>, the number of equations is <I>n</I>,
<I>s</I>=1 and the continuation calculation computes curves of equilibria
with a single active parameter. To compute curves of bifurcations, one adds
<B>defining equations</B> to the system of equations <I>f</I>=0 to produce an
<B>augmented</B> system. The number of independent defining equations added
to <I>f</I>=0 is the <B>codimension</B> of the bifurcation.
(In some circumstances, the augmented system uses
additional <B>auxiliary</B> variables such as eigenvalues of the Jacobian
and has a corresponding number of additional equations.)
<P>
The continuation window of DsTool includes calculations for saddle-node and
Hopf bifurcations. These are the codimension one bifurcations of
equilibrium points.
For saddle-node bifurcations, the defining equation is
<!-- MATH: $\det (Df_x) = 0$ -->
55#55or an algorithm that computes another scalar quantity, such as the
smallest singular value, that vanishes precisely when <I>Df</I> is singular.
For Hopf bifurcations, the defining equation(s) compute where the
matrix <I>Df</I> has a pair of pure imaginary eigenvalues. Explicit
expressions
in the coefficients of <I>Df</I> that compute whether <I>Df</I> has a pair of
complex eigenvalues are very complicated. Alternate ways of performing
this computation are discussed in [<A
HREF="node68.html#GMS">3</A>].
There are also other algorithms that perform the calculation
by introducing auxiliary variables for the pure imaginary eigenvalues,
and in some cases the eigenvectors of <I>Df</I>.
<P>
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<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
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