\% ********* List of classes of dynamical systems *********
\topic ClassList {list of classes of dynamical systems}
CONTENT currently supports the following classes:\par
{
 \li 1
 {\bf generic autonomous ODEs}\par
 {\bf generic iterated maps}\par
 {\bf generic PDEs on the interval}\par
 {\bf generic autonomous DAEs}\par
}
\par
\end	\% ClassList

\% ********** Classes of dynamical systems **********
\topic Classes {class of dynamical system}
The term {\bf class of dynamical systems} refers to
particular common properties of dynamical systems.
In {\bf Content} the notion of class is used to establish
a link between the properties and a set of numerical algorithms
for analysis of dynamical systems. Class also defines
the way in which functions specified by the user
are interpreted by {\bf Content}.
For example, by saying that you will study a dynamical
system belonging to the class of {\bf generic autonomous ODEs}
you implicitly specify that functions you give
are right-hand sides of ODEs and that algorithms
for bifurcation analysis of autonomous ODEs are
applicable to you dynamical system.\par
You change the class by the \jump kh_menuSelect {Select|Class}
command from the \jump kh_MainWin {Main window.}
\include ClassList \par
See description of the \jump *curclass {Current class.}
\end	\% Classes

\% ********** Dynamical system **********
\topic DynSys {dynamical system}
A dynamical system in {\bf Content} can be
specified as belonging to one of the supported
\jump Classes {class} of dynamical systems.
The name of the current class is shown in the
\jump kh_MainWin {Class field} in the Main window.
You may change current class through the
\jump kh_menuSelect {Select|Class} menu in the Main window.\par
\include *curclass \par
\end	\% DynSys

\% *********** Continuer generator **********
\topic conti {continuer}
A {\bf continuer} generates a sequence of points
\par
\qc{$y^{(1)},y^{(2)},\ldots,y^{(k)},\ldots$}
\par
approximating a curve that is implicitly defined by the system
of $n$ scalar equations in $n+1$ dimensional space:
\par
\qc{$F(y)=0,\;\; F:{\f b* R}^{n+1} \rightarrow  {\f b* R}^{n}.$}
\par
Given a point $y^{(k)}$ located on the curve
and the normalized tangent vector $v^{(k)}$ to the curve at this point,
the next point $y^{(k+1)}$ and its associated tangent vector $v^{(k+1)}$
are computed in CONTENT as following.
\par
{\bf Prediction:} 
\par
\qc{$Y^{0}=y^{(k)}+h_{k}v^{(k)},\:\: V^{0}=v^{(k)}.$}
\par
where $h_{k}$ is the current {\bf step size}.
\par
\qc {\picture 0.5 conti.fig}
\par
{\bf Corrections:}
\par
Iterate for $i=0,1,2,\ldots$, I_{max}:
\par
\qc{$A=F_{y}(Y^{i}),\:\: B=\matrix 2 c 0 0 {A}{[V^{i}]^{T}},$}
\par
\qc{$R=\matrix 2 c 0 0 {AV^{i}}{0},$}
\par
\qc{$W=V^{i}-B^{-1}R,\:\: V^{i+1}=\frac{W}{||W||},$}
\par
\qc{$R=\matrix 2 c 0 0 {F(Y^{i})}{0},$}
\par
\qc{$Y^{i+1}=Y^{i}-B^{-1}R.$}
\par
If
\par
\qc{$||F(Y^{i})|| < \varepsilon_{F}$ and $||Y^{i+1}-Y^{i}|| < \varepsilon_Y$}
\par
then 
\par
\qc{$y^{(k+1)}=Y^{i+1},\:\: v^{(k+1)}=V^{i+1}$.}
\par
Recomputation of the Jacobian matrix $A$ and update of the vector $V$ can be suppressed
after $I_{maxNewt}$ iterations.
\par
\par
{\bf Step size control:} Set, $h_{k+1}=sh_{k},$
where
\par
\qc{$
s=  \system 3 cl 1.2 1
      {1.3}           	{i<3,}
      {1.0}      	{3\leq i < I_{max},}
      {0.5} 		{i=I_{max}.}
$}
\par
If $h_{k+1} > H_{max}$, $h_{k+1}=H_{max}$. If $h_{k+1}<H_{min}$, the continuation
is terminated with the corresponding \jump kh_MainWin {message.} The continuation is also 
terminated if the curve appears to be {\bf closed} or a given number of
points along the curve ($k=N_{maxPoint}$) is computed.
\par
\par
{\bf Zero of test and user functions:}
\par
Scalar functions $g_m(y)$ can be {\bf monitored} along the curve. If such a function
changes sign between $y^{(k)}$ and $y^{(k+1)}$, its zero can be {\bf detected} and 
computed by bi-sections.
\par
\par
Values of \jump conpar {parameters of the continuer} can be modified via the corresponding
\jump kh_GeneratorWin {generator window}.
\end \%conti

\topic conpar {continuation parameters}
The \jump conti {continuer} has the following numerical parameters:
\par
\par
\desc 18
{\bf Continuation data}\>\par
{\li 1 InitStepsize} \>  Initial stepsize\par
{\li 1 CurrentStepsize} \> Size of the current step along the curve\par
{\li 1 MinStepsize} \> Minimal stepsize $H_{min}$\par
{\li 1 MaxStepsize} \> Maximal stepsize $H_{max}$\par
{\bf Corrector data}\>\par
{\li 1 MaxIter} \> Maximal number $I_{max}$ of corrections\par
{\li 1 MaxNewtonIter} \> Maximal number $I_{maxNewt}$ of corrections with the
       recomputation of the Jacobian matrix\par
{\li 1 VarTolerance} \> Accuracy $\varepsilon_Y$ with respect to variables\par
{\li 1 FuncTolerance} \> Accuracy $\varepsilon_F$ with respect to defining functions\par
{\li 1 TestTolerance} \> Accuracy for zero location of test and user functions\par
{\li 1 MaxIterTest} \> Maximal number of bi-sections for zero location\par
{\bf Stop data}\>\par
{\li 1 MaxNumPoints} \> Maximal number $N_{maxPoint}$ of points along the curve 
in one direction\par
{\li 1 BeforeClosureCheck} \> Number of points to skip before closure check\par
{\bf Output data}\>\par
{\li 1 BetweenAdaptation} \>Curve type dependent parameter meaning
the number of points between adaptation.\par
{\li 1 OutLevel}\> Output index: 0 - no output during corrections; 1 - output at each
     correction.\par
\par
The parameters can be modified via the \jump kh_GeneratorWin {generator window}.\end \%conpar