\% ********** Class maps **********
\topic map_Class {iterated maps}
The class {\bf generic iterated maps} comprises
discrete-time dynamical systems 
defined by a map
\par
\qc{$x\; \rightarrow \;f(x,\alpha),$}
\par
where $x\in {\f b* R}^n$ is a vector of phase variables,
$\alpha\in {\f b* R}^m$ is a vector of numerical parameters, and
$f$ is a vector-function (right-hand sides). 
Map $f$ is assumed to be sufficiently smooth but has no special properties
(like symmetry, etc.).
\par
The system is often written as
\par
\qc{$x\prime =\;f(x,\alpha)\;=\;f_{\alpha}(x),$}
\par
where $x\prime \in {\f b* R}^n$ is the image of $x$ under $f_{\alpha}$.
\par
A map is specified or edited
in the \jump kh_RHSWin {System specification dialog box}
which is opened by the {\bf Select|System} command from the
\jump kh_MainWin {Main window.}\par
There are following class-specific fields in the dialog box.\par
{\desc 3
{\bf Coordinates}\>is where you list names of phase (state) variables.
See \jump kh_RHSWin {System specification dialog box} on how to
fill in the fields. \par
{\bf Parameters}\>is where you list names of parameters your system
depends on.\par
{\bf Time}\>is where you may type in the name of the discrete time variable
which must be a simple name. You may want to specify it in
a case you intend to monitor time series.\par
}
The following constants are defined and may be used
in RHS, its derivatives and local functions:\par
{\desc 2
{\bf COORDINATESDIM}\>the number of coordinates (counting array
components).\par
{\bf PARAMETERSDIM}\>the numer of parameters.\par
}
\par
{\bf Example 1:} For the scalar map
\par
\qc{$u\prime=\lambda e^{-u},$}
\par
the following specification is valid.
\par
{\li 5
\desc 4
{\bf Coordinates:} \>$U$\par
{\bf Parameters:} \>$lambda$\par
{\bf Time:}\> $t$\par
{\bf RHS:} \>$U\prime=lambda*exp(-U);$\par
}
\par
{\bf Example 2:} For the two-dimensional map
\par
\qc{$
\system 2 rcl 1.2 1 
{\xi_1\prime}{=}{\xi_1+\xi_2}
{\xi_2\prime}{=}{\xi_2+a\xi^{2}_{1}+ b \xi_{1} \xi_{2}}
$}
\par
the following specification is valid.
\par
{\li 5
\desc 5
{\bf Coordinates:} \>$x[2]$\par
{\bf Parameters:} \>$a,b$\par
{\bf Time:}\> $t$\par
{\bf RHS:} \>$x\prime [0]=x[0]+x[1];$\par
{} \>$x\prime [1]=x[1]+a*x[0]*x[0]+b*x[0]*x[1];$\par
}

\end
\% map_Class

\topic point_map {point}
You can compute \jump map_or {an orbit} starting from a {\bf point} in the phase-parameter space.
\end 

\topic map_or {orbit}
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime =\;F(x),\: \: x \in {\f b* R}^n,
$
}
\par
an {\bf orbit} starting at $x_0 \in {\f b* R}^n$ is a sequence of points
\par
\qc{$x_0,\;x_1,\;x_2,\;x_3,\ldots$}
\par
such that 
\par
\qc{
$
x_{k+1}=\;F(x_k),\;\;k=0,1,2,\ldots.
$
}
\par
Orbits are generated by \jump iter {iterator}.
\end 

\topic iter {iterator}
An {\bf iterator} generates \jump map_or {orbits} of an \jump map_Class {iterated map} $F$
as following:
\par
\qc{$x_{k+1}=\;F^{(M)}(x_k),\;\;k=0,1,2,\ldots,N_{max}$}
\par
where $M$ defines the number of superpositions of $F$, while $N_{max}$ specifies the length 
of the computed orbit. 
\par
The parameters $M$ and $N_{max}$ can be fixed via 
the \jump kh_StarterWin {starter window} and \jump kh_GeneratorWin {generator window}, respectively.
\end


\topic iterpar {iterator parameters}
The \jump iter {iterator} has only one parameter:
\par
\par
\desc 2
{\bf Iteration data}\>\par
{\li 1 Number of iterations} \> Maximal number of points in the orbit $N_{max}$\par
\par
The parameters can be modified via the \jump kh_GeneratorWin {generator window}.
\end

\topic map_start_or {starter parameters for orbit}
The \jump kh_Starter {starter} which sets necessary data to \jump kh_Generator {generate} 
the \jump map_or {orbit} has the following {\bf parameters} that can be modified via 
the \jump kh_StarterWin {starter window.}
\par
{
\desc 9
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $time$} \> The time name and its integer value at the initial point.\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point.\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}).\par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point. Otherwise, CONTENT
 uses the values listed in the window.  \par
}
\par
\include map_init
\end \% map_start_or

\topic map_init {-}
Here is an example of a function SetInitPoint:
\par
\verbatim 2
     X=A;
     Y=A*A;
\par
In this example, X and Y are phase variables and A is a parameter, as
specified in the \jump kh_RHSWin {system specification dialog box.}
\par
In general, you write a body of a C function that assigns values to the coordinates
of the initial point and/or the corresponding parameter values.
\end


\topic map_Types {types of curves}
CONTENT currently supports the continuation of the following curves:
\par
\par
\jump map_or {Orbit}
\par
\jump map_fp {Fixed point curve}
\par
\jump map_lp {Limit point curve}
\par
\jump map_pd {Period-doubling (flip) curve}
\par
\jump map_ns {Neimark-Sacker curve}
\end \% map_Types


\topic map_fp {fixed point curve}
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime\; =\; f(x,\alpha)\;=\;f_{\alpha}(x),\: \: x \in {\bf R}^n,\: \alpha \in {\bf R}^1,
$
}
\qr {(1)}
\par
an {\bf fixed point curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{n+1}$ endowed with coordinates $(x,\alpha)$
defined by\par
\qc{
$
f(x,\alpha)-x=0.
$
}\qr {(2)}
\par
If one introduces $y=(x,\alpha) \in {\bf R}^{n+1},$ the manifold
(2) is given by\par
\qc{
$
F(y)=0,
$
} \qr {(3)}\par
where $F:\: {\bf R}^{n+1} \rightarrow {\bf R}^n,$ $F(y)=f(x,\alpha)-x,$ are the \jump kh_DefFunc 
{defining functions}. The Jacobian matrix of (3) involved in the continuation
has the form 
\par
\qc{$
F_y(y)=(f_x(x,\alpha)-I_{n}|f_{\alpha}(x,\alpha)),
$}\par
where $I_{n}$ is the unit matrix.
No special properties of Jacobian matrix of (3) $F_y(y)$ are assumed and the {\bf generic linear
algebra library} is used to solve appearing linear system of equations.
\par
Several \jump test_fp {test functions} can be computed along the fixed point curve to
detect and process \jump test_fp {fixed point singularities}.
\par
\par
A fixed point curve can be continued form a user-supplied \jump point_fp {point}, 
a \jump fp_fp {fixed point} (including a \jump LP_fp {limit point} and a 
\jump NS_fp {Neimark-Sacker point}), a \jump PD_fp {period-doubling point}  
or a \jump branch_fp {branching point} on the computed 
fixed point curve. To \jump conti {continue} of the fixed point curve, one needs to set relevant
{\bf parameters to start from}
\par
\par
$\;\;\;$ \jump start_fpp {a point}\par
$\;\;\;$ \jump start_fpfp {a fixed point}\par
$\;\;\;$ \jump start_fpbp {a branching point}\par
$\;\;\;$ \jump start_fppd {a period-doubling point}\par
\par
\par
via the \jump kh_StarterWin {starter window}.
\par
The continuation of \jump fp_fp {period-M cycles} of (1) is reduced to the continuation of
fixed points of its $M$-th iterate (superposition):
\par
\qc{
$
f^{M}_{\alpha}(x)-x=0.
$
}
\end \% map_fp

\topic start_fpp {starter parameters (fixed point curve from a point)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_fp {fixed point curve} from a user-supplied {\bf point} has the following parameters
that can be modified via the \jump kh_StarterWin {starter window.}
\par
\par
{
\desc 20
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Corrector data}\>\par
{\li 1 MaxIter} \>  Maximal number of corrections to locate the first point.\par
{\li 1 MaxNewtonIter} \>  Maximal number of corrections with recomputation of the Jacobian
			matrix.\par
{\li 1 VarTolerance} \>  Tolerance with respect to phase variables and parameters.\par
{\li 1 FuncTolerance} \>  Tolerance with respect to defining functions.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_fp {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the starter before computing the first point. Otherwise, CONTENT
 uses the values listed in the window. \par
}
\par
\include map_init
\end \% start_fpp

\topic start_fpfp {starter parameters (fixed point curve from a fixed point)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_fp {fixed point curve} from an \jump fp_fp {fixed point} has the following parameters
that can be modified via the \jump kh_StarterWin {starter window.}
\par
\par
{
\desc 15
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\li 1 $x$} \> A list of coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_fp {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the starter before computing the first point. Otherwise, CONTENT
 uses the values listed in the window.  \par
}
\par
\include map_init
\end \% start_fpfp

\topic start_fpbp {starter parameters (fixed point curve form a branching point)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
an \jump map_fp {fixed point curve} from a {\bf branching point} has the following parameters
that can be modified via the \jump kh_StarterWin {starter window.}
\par
{
\desc 15
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.
\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Branch switch data}\> \par
{\li 1 Branch} \> Switch toggle ({\bf primary}|{\bf secondary}).\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_fp {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
}
\par
\end \% start_fpbp

\topic start_fppd {starter parameters (fixed point curve form a flip point)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
an \jump map_fp {fixed point curve} (original or double-period) from a {\bf period-doubling(flip) point} has the following parameters
that can be modified via the \jump kh_StarterWin {starter window.}
\par
{
\desc 15
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.
\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Branch switch data}\> \par
{\li 1 Branch} \> Switch toggle ({\bf original}|{\bf double-period}).\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_fp {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
}
\par
\end \% start_fpbp

\topic test_fp {test functions and singularities along the fixed point curve}

The following \jump kh_TestFunc {test-functions} can be computed along the \jump map_fp {fixed point curve}
$F(y)=f(x,\alpha)-x=0$:
\par
\qc{
$
\table 4 rcl 0 0 
{\psi_1}{\:=\:}{det \matrix 2 c 0 0 {F_y}{v^T},}
{\psi_2}{\:=\:}{v_{n+1},}
{\psi_3}{\:=\:}{det (f_x + I_n).}
{\psi_4}{\:=\:}{det (f_x \otimes f_x - I_m).}
$}
\par
Here $v=(v_1,v_2,\ldots,v_{n+1})\in {\bf R}^{n+1}$ is the normalized tangent vector to the curve
at a point $y=(x,\alpha)$ on it; $I_n$ is the unit $n\times n$ matrix; and $\otimes$ means the
\jump bialt {bialternate product} of two matrices, $m=(1/2)n(n-1)$.
\par
\par
The following {\bf singularities} can be detected and located as regular zeroes of
the fixed point test-functions:
\par
{\li 2
\desc 4 
\jump branch_fp {Branching point}:\> $\psi_1=0.$
\par
\jump LP_fp {Limit point}:\> $\psi_2=0,\; \psi_1 \neq 0.$
\par
\jump PD_fp {Period-doubling (flip) bifurcation}:\> $\psi_3=0.$
\par
\jump NS_fp {Neimark-Sacker bifurcation}:\> $\psi_4=0.$
\par
}
\end \% test_fp

\topic point_fp {point}
You can compute 
\par
{\li 2
\desc 2
\jump map_fp {a fixed point curve}\>
 \par
\jump map_or {an orbit}\>
 \par
}
\par
starting from a {\bf point} in the phase-parameter space. In the former case, 
the point should be rather close to an \jump fp_fp {fixed point}.

\end 

\topic fp_fp {fixed point}
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime = F(x),\: \: x \in {\bf R}^n,
$
}
\qr{(1)}
\par
a {\bf fixed point} is a point $x_0 \in {\bf R}^n$ such that $F(x_{0})=x_{0}.$
\par
{\bf Multipliers} $\mu_{1},\mu_{2},\ldots,\mu_{n}$ of a fixed point $x_0$
are the roots of the {\bf characteristic equation}
\par
\qc{
$
h(\lambda) = det (A-\mu I_n) = 0,
$
}
\par
where 
\par
\qc{$A=F_x(x_0)$}
\par
is the {\bf Jacobian matrix} and $I_n$ is the $n\; \times \;n$ unit matrix. 
\par
The fixed point is called {\bf hyperbolic}
if there is no multiplier at the unit circle, i.e. with $|\mu|=1$. 
A hyperbolic fixed point has invariant
{\bf stable and unstable manifolds} of dimension equal to the number of 
multipliers with $|\mu|<1$ and $|\mu|>1$, respectively. 
If there is a multiplier at the unit circle, the fixed point is called {\bf nonhyperbolic}.
A nonhyperbolic fixed point has a {\bf center manifold} of dimension equal to
the number of multipliers with $|\mu|=1$. The invariant manifolds are tangent to
the corresponding generalized eigenspaces at $x_0$.
\par
A {\bf period-M cycle} is a periodic orbit:
\par
\qc{
$
x_0,\; F(x_0),\; F^{2}(x_0),\; \ldots,\; F^{M-1}(x_0),\; F^{M}(x_0)=x_0.
$
}
par
Any period-M cycle corresponds to a nontrivial fixed point of $M-th$ 
iterate (superposition) of (1), $F^{M}(x_{0})=x_{0}$ 
\end %\fp_fp


\topic branch_fp {branching point}
Generically, two branches $M_{1,2}$ of the \jump map_fp {fixed point curve} 
intersect transversally at a {\bf branching point}:
\par
\qc{\picture 0.5 fig10_10.fig}
\par
Let $v \in {\bf R}^{n+1}$ be the normalized tangent
vector to the primary (traced) branch of the fixed point curve defined by
\par
\qc{$
F(y)=f(x,\alpha)-x=0.
$}
\par
Consider the $(n+1)\;\times\; (n+1)$ matrix
\par
\qc{$
D=\matrix 2 c 0 0 {F_y(y_0)}{v^T} .
$} 
\par
A tangent vector $V$ to the secondary branch of the fixed point curve
at a branching point $y_0$ can be computed as following. A unit vector $q\in {\bf R}^{n+1}$
that is orthogonal to $v$ is a normalized null-eigenvector of $D$:
\par
\qc{$
Dq=0,\,\,\, \langle q,q\rangle = 1.
$}
\par
A normalized null-eigenvector $p\in {\bf R}^{n+1}$ of the transposed matrix $D^T$,
\par
\qc{$
D^{T}p=0,\,\,\, \langle p,p\rangle = 1,
$}
\par
has the form
\par
\qc{$
p=\matrix 2 c 0 0 {\phi}{0},
$}
\par
where $\phi \in  {\bf R}^{n}$. A tangent vector $V$ to the secondary branch is then
given by $V=q+kv$, where
\par
\qc{$
k=-\frac{b_{22}}{2b_{12}}=-\frac{\langle \phi,B(q,q)\rangle}{2\langle \phi,B(v,q)\rangle},
$}
\par
provided $b_{12}\neq 0$. Here the bilinear function $B(v,q)$ is given by
\par
\qc{$
B_i(v,q)= \Sum {j,k=1} {n+1} {{\brackets.| {\frac{\partial^2
	 F_i(y_0)}{\partial y_j\partial y_k}}}_{y=y_0}} \;
       v_j q_k, 
      \, i=1,2,\ldots,n.
$}
\par
\par
The vector $V$ is then normalized and stored together with the tangent vector $v$. 
This allows to switch branches. If $b_{12}=0$, the branching point is {\bf non-simple} 
and the switching is not supported. The number
$k=ctg\, \varphi$, where $\varphi$ is the angle between the branches, is displayed
in the \jump kh_MainWin {message field}.\par

\end \% branch_fp

\topic LP_fp {limit point}
At a {\bf limit point}, the tangent vector $v$ to 
the \jump map_fp {fixed point curve} has the form
\par
\qc{$v=\matrix 2 c 0 0 {q}{0},
$}
\par
where $q\in {\bf R}^{n}$ is the eigenvector, $Aq=q$, corresponding to the simple unit
\jump fp_fp {multiplier} $\mu=1$ of the Jacobian matrix $A=f_x(x_0,\alpha_0)$. 
\par
\par
\qc{\picture 0.5 fig10m_4.fig}
\par
\par
The \jump map_Class {map} restricted to the one-dimensional
\jump fp_fp {center manifold} at $\alpha=\alpha_0$
has the form
\par
\qc{$
\xi\prime = \xi + \frac{1}{2}a \xi^2 + O(|\xi|^3),\; \xi \in {\bf R}^1.
$}\par
where the quadratic normal form coefficient $a$ is computed by\par
\qc{$
a=\langle p,B(q,q) \rangle,
$}\par
where the eigenvectors $q$ and $p$ satisfy the equations\par
\qc{$
Aq=q,\; A^T p=p,\; \langle q,q\rangle = 1,\;  \langle p,q\rangle  = 1,
$}\par
and\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
      \, i=1,2,\ldots,n.
$}\par
If $a \neq 0$ and the system depends generically on the parameter, 
two fixed points collide and disappear at the critical parameter value $\alpha_0$.
\par
\par
\qc{\picture 0.9 fig4_2.fig}
\par

\par
The normalized eigenvector $q$ corresponding to the unit multiplier is stored for
two-parameter \jump map_lp {continuation of limit points}.
The value of $a$ can be read in the \jump kh_MainWin {message field}.\par
\end \% LP_fp

\topic map_lp {limit point curve}
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}
\par
a {\bf limit point curve} is a one-dimensional manifold $\Gamma$ in ${\bf R}^{N}$ endowed 
with coordinates $(x,y,\ldots,\alpha)$,
whose projection onto the $(x,\alpha)$-space gives \jump LP_fp {limit points} 
of (1) having a simple multiplier $\mu_1=1.$
\par
\par
The \jump kh_DefFunc {defining functions} used in CONTENT for the limit point 
curve continuation can be specified 
by
\par
\par
$\;\;\;$\jump map_lp0 {the standard augmented system,}
\par
$\;\;\;$\jump map_lp1 {the minimal bordered system.}
\par
\end %map_lp

\topic map_lp0 {limit point curve (standard)}
This method to define the \jump map_lp {limit point curve} is implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf standard limit point curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{2n+2}$ endowed with coordinates $y=(x,v,\alpha)$
defined by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 3 rcl 1.2 1 
{f(x,\alpha) - x}{=}{0}
{f_{x}(x,\alpha)v - v}{=}{0}
{\langle v,v\rangle - 1}{=}{0}
$
}\qr {(2)}\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear
algebra library} is used to solve appearing linear system of equations.
\par
Several \jump test_lp0 {test functions} can be computed along the limit point curve to
detect and process \jump test_lp0 {limit point singularities}.
\par
\par
The limit point curve can be continued from a regular \jump LP_fp {limit point} as well as 
from all \jump test_lp0 {limit point singularities}. 
To start \jump conti {continuation} of the limit point curve, 
one needs to set relevant \jump map_start_lp0 {starter parameters} via the \jump kh_StarterWin {starter window}.
\end \% map_lp0

\topic map_lp1 {limit point curve (bordered)}
This method to define the \jump map_lp {limit point curve} is implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime =f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf bordered limit point curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{n+2}$ endowed with coordinates $y=(x,\alpha)$
defined by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 2 rcl 1.2 1 
{f(x,\alpha) - x}{=}{0}
{g(x,\alpha)}{=}{0,}
$
}\qr {(2)}
\par
where the scalar function $g=g(x,\alpha)$ is obtained from solving 
the single bordered $(n+1)-$dimensional system 
\par
\qc{$
\matrix 2 cc 2 1 {A(x,\alpha)-I_n}{W}
		 {V^{T}}{0} 
		  \matrix 2 c 2 0 {v(x,\alpha)}{g(x,\alpha)}$ $=$ 
		  $\matrix 2 c 2 0 {0}{1},
$}
\par
where $A(x,\alpha)=f_x(x,\alpha)$ and 
the vectors $V,W \in {\bf R}^n$ are adapted along the curve to make the above system
nonsingular.
\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear
algebra library} is used to solve appearing linear system of equations.
\par
Several \jump test_lp1 {test functions} can be computed along the limit point curve to
detect and process \jump test_lp1 {limit point singularities}.
\par
\par
The limit point curve can be continued from a regular \jump LP_fp {limit point} as well as
from all \jump test_lp1 {limit point singularities}. 
To start \jump conti {continuation} of the limit point curve, 
one needs to set relevant \jump map_start_lp1 {starter parameters} via the \jump kh_StarterWin {starter window}.
\end \% map_lp1

\topic test_lp0 {test functions and singularities along the standard limit point curve}
The \jump kh_TestFunc {test functions} that can be computed along the 
\jump map_lp0 {standard limit point curve} are the following:
\par
\par
\qc{
$
\table 4 rcl 0 0 
{\psi_1}{\:=\:}{\langle \tilde{w},v\rangle,}
{\psi_2}{\:=\:}{det (A + I_n).}
{\psi_3}{\:=\:}{det (A \otimes A - I_m).}
{\psi_4}{\:=\:}{\langle w,B(v,v)\rangle,}
$}
\par
Here $A=f_x(x,\alpha)$ and eigenvectors $v$ and $w(\tilde{w})$ satisfy the equations
\par
\qc{$
Av=v,\; A^T w=w,\; A^T \tilde{w}=\tilde{w},\; \langle v,v\rangle = \langle \tilde{w},\tilde{w}\rangle = 1,\; 
\langle w,v\rangle = 1,
$}\par
and\par
\qc{$
B_i(v,w)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       v_j w_k, 
      \, i=1,2,\ldots,n;
$}\par
and $\otimes$ means the \jump bialt {bialternate product} of two matrices, $2m=n(n-1).$
\par 
\par
The following {\bf singularities} can be detected and located as regular zeroes of
the limit point test-functions:
\par
{\li 2
\desc 4 
\jump R1 {1:1 Resonance}:\> $\psi_1=0$ 
\par
\jump FF {Fold-flip}:\> $\psi_2=0$
\par
\jump FN {Fold-NS}:\> $\psi_3=0$
\par
\jump CP {Cusp}:\> $\psi_4=0$
\par
}
\end \% test_lp0

\topic test_lp1 {test functions and singularities along the bordered limit point curve}
The \jump kh_TestFunc {test functions} that can be computed along the 
\jump map_lp1 {bordered limit point curve} are the following:
\par
\par
\qc{
$
\table 4 rcl 0 0 
{\psi_1}{\:=\:}{g_{\lambda},}
{\psi_2}{\:=\:}{det (A + I_n),}
{\psi_3}{\:=\:}{det (A \otimes A- I_m),}
{\psi_4}{\:=\:}{\langle w,B(v,v)\rangle,}
$}
\par
where $g_{\lambda}$ is obtained from solving the single bordered $(n+1)-$dimensional system 
\par
\qc{$
\matrix 2 cc 2 1 {A-I_n}{W}
		 {V^{T}}{0} 
		  \matrix 2 c 2 0 {v_{\lambda}(x,\alpha)}{g_{\lambda}(x,\alpha)}$ $=$ 
		  $\matrix 2 c 2 0 {v(x,\alpha)}{0}.
$}
\par
Here $A=f_x(x,\alpha)$ and eigenvectors $v$ and $w$ satisfy the equations
\par
\qc{$
Av=v,\; A^T w=w,\; \langle v,v\rangle = \langle w,v\rangle = 1,
$}\par
and\par
\qc{$
B_i(v,w)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       v_j w_k, 
      \, i=1,2,\ldots,n;
$}\par
and $\otimes$ means the \jump bialt {bialternate product} of two matrices; $2m=n(n-1).$.
\par 
\par
The following {\bf singularities} can be detected and located as regular zeroes of
the limit point test-functions:
\par
{\li 2
\desc 4 
\jump R1 {1:1 Resonance}:\> $\psi_1=0$ 
\par
\jump FF {Fold-flip}:\> $\psi_2=0$
\par
\jump FN {Fold-NS}:\> $\psi_3=0$
\par
\jump CP {Cusp}:\> $\psi_4=0$
\par
}
\end \% test_lp1

\topic map_start_lp0 {starter parameters (standard limit point curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_lp0 {standard limit point curve} has the following {\bf parameters}:
\par
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_lp0 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Multipliers}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.

\end \% map_start_lp0

\topic map_start_lp1 {starter parameters (bordered limit point curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_lp1 {bordered limit point curve} has the following {\bf parameters}:
\par
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_lp1 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Multipliers}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.

\end \% map_start_lp1


\topic PD_fp {period-doubling (flip) bifurcation}
At a {\bf period-doubling (flip)} bifurcation, the  
the \jump fp_fp {fixed point} $x_0$ has a simple \jump fp_fp {multiplier}
$\mu=-1$ of the Jacobian matrix $A=f_x(x_0,\alpha_0)$, and no other
multipliers with $|\mu|=1$.
\par
The \jump map_Class {map} restricted to the one-dimensional
\jump fp_fp {center manifold} at $\alpha=\alpha_0$
can be transformed to the normal form
\par
\qc{$
\xi\prime = -\xi + c \xi^3 + O(|\xi|^4),\; \xi \in {\bf R}^1.
$}\par
where the cubic normal form coefficient $c$ is computed by\par
\qc{$
c= \frac{1}{6} \langle p,C(q,q,q)\rangle
 + \frac{1}{2} \langle p, B(q,(I-A)^{-1}B(q,q))\rangle ,
$}\par
where the eigenvectors $q$ and $p$ satisfy the equations
\par
\qc{$
Aq=-q,\; A^T p=-p,\; \langle q,q\rangle = 1,\;  \langle p,q\rangle  = 1,
$}\par
and\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
$}
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n$.
\par
If $c \neq 0$ and the map depends generically on the parameter, 
a double-period cycle $\{x_{1},x_{2}\}$ bifurcates from the fixed point 
$x_0$ at the critical parameter value $\alpha_0$, while the fixed point changes its stability.
\par
\par
\qc{\picture 0.9 fig4_5.fig}
\par
\par
The normalized eigenvector $q$ corresponding to the multiplier $\mu=-1$ is stored for
the switching to the continuation of the double-period cycle, as well as
for the two-parameter continuation of flip points.
The value of $c$ can be read in the \jump kh_MainWin {message field}.\par
\end \% PD_fp

\topic map_pd {period-doubling (flip) curve}
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}
\par
a {\bf period-doubling (flip) curve} is a one-dimensional manifold $\Gamma$ in ${\bf R}^{N}$ endowed 
with coordinates $(x,y,\ldots,\alpha)$,
whose projection onto the $(x,\alpha)$-space gives \jump PD_fp {period-doubling points} 
of (1) having a simple multiplier $\mu_1=-1.$
\par
\par
The \jump kh_DefFunc {defining functions} used in CONTENT for the limit point 
curve continuation can be specified 
by
\par
\par
$\;\;\;$\jump map_pd0 {the standard augmented system,}
\par
$\;\;\;$\jump map_pd1 {the minimal bordered system.}
\par
\end %map_pd

\topic map_pd0 {period-doubling curve (standard)}
This method to define the \jump map_pd {period-doubling curve} is implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf standard period-doubling curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{2n+2}$ endowed with coordinates $y=(x,v,\alpha)$
defined by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 3 rcl 1.2 1 
{f(x,\alpha) - x}{=}{0}
{f_{x}(x,\alpha)v + v}{=}{0}
{\langle v,v\rangle - 1}{=}{0}
$
}\qr {(2)}\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear
algebra library} is used to solve appearing linear system of equations.
\par
Several \jump test_pd0 {test functions} can be computed along the period-doubling curve to
detect and process \jump test_pd0 {period-doubling singularities}.
\par
\par
The period-doubling curve can be continued from a regular \jump PD_fp {period-doubling point} as well as 
from all \jump test_pd0 {period-doubling singularities}. 
To start \jump conti {continuation} of the period-doubling curve, 
one needs to set relevant \jump map_start_pd0 {starter parameters} via the \jump kh_StarterWin {starter window}.
\end \% map_pd0

\topic map_pd1 {period-doubling curve (bordered)}
This method to define the \jump map_pd {period-doubling curve} is implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime =f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf bordered period-doubling curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{n+2}$ endowed with coordinates $y=(x,\alpha)$
defined by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 2 rcl 1.2 1 
{f(x,\alpha) - x}{=}{0}
{g(x,\alpha)}{=}{0,}
$
}\qr {(2)}
\par
where the scalar function $g=g(x,\alpha)$ is obtained from solving 
the single bordered $(n+1)-$dimensional system 
\par
\qc{$
\matrix 2 cc 2 1 {A(x,\alpha)+I_n}{W}
		 {V^{T}}{0} 
		  \matrix 2 c 2 0 {v(x,\alpha)}{g(x,\alpha)}$ $=$ 
		  $\matrix 2 c 2 0 {0}{1},
$}
\par
where $A(x,\alpha)=f_x(x,\alpha)$ and 
the vectors $V,W \in {\bf R}^n$ are adapted along the curve to make the above system
nonsingular.
\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear
algebra library} is used to solve appearing linear system of equations.
\par
Several \jump test_pd1 {test functions} can be computed along the limit point curve to
detect and process \jump test_pd1 {period-doubling singularities}.
\par
\par
The period-doubling curve can be continued from a regular \jump PD_fp {period-doubling point} as well as
from all \jump test_pd1 {period-doubling singularities}. 
To start \jump conti {continuation} of the period-doubling curve, 
one needs to set relevant \jump map_start_pd1 {starter parameters} via the \jump kh_StarterWin {starter window}.
\end 
\% map_pd1

\topic test_pd0 {test functions and singularities along the standard period-doubling curve}
The \jump kh_TestFunc {test functions} that can be computed along the 
\jump map_pd0 {standard period-doubling curve} are the following:
\par
\par
\qc{
$
\table 4 rcl 0 0 
{\psi_1}{\:=\:}{\langle \tilde{w},v\rangle,}
{\psi_2}{\:=\:}{det (A - I_n).}
{\psi_3}{\:=\:}{det (A \otimes A - I_m).}
{\psi_4}{\:=\:}{\langle w,C(v,v,v)\rangle + 3 \langle w, B(v,(I-A)^{-1}B(v,v))\rangle,}
$}
\par
Here $A=f_x(x,\alpha)$ and eigenvectors $v$ and $w(\tilde{w})$ satisfy the equations
\par
\qc{$
Av=-v,\; A^T w=-w,\; A^T \tilde{w}=-\tilde{w},\; \langle v,v\rangle = \langle \tilde{w},\tilde{w}\rangle = 1,\; 
\langle w,v\rangle = 1,
$}\par
and\par
\qc{$
B_i(v,w)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       v_j w_k, 
$}
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n,$ and $\otimes$ means the 
\jump bialt {bialternate product} of two matrices, $2m=n(n-1).$
\par 
\par
The following {\bf singularities} can be detected and located as regular zeroes of
the limit point test-functions:
\par
{\li 2
\desc 4 
\jump R2 {1:2 Resonance}:\> $\psi_1=0$ 
\par
\jump FF {Fold-flip}:\> $\psi_2=0$
\par
\jump PN {Flip-NS}:\> $\psi_3=0$
\par
\jump DF {Degenerate flip}:\> $\psi_4=0$
\par
}
\end \% test_pd0

\topic test_pd1 {test functions and singularities along the bordered period-doubling curve}
The \jump kh_TestFunc {test functions} that can be computed along the 
\jump map_pd1 {bordered period-doubling curve} are the following:
\par
\par
\qc{
$
\table 4 rcl 0 0 
{\psi_1}{\:=\:}{g_{\lambda},}
{\psi_2}{\:=\:}{det (A - I_n),}
{\psi_3}{\:=\:}{det (A \otimes A- I_m),}
{\psi_4}{\:=\:}{\langle w,C(v,v,v)\rangle + 3 \langle w, B(v,(I-A)^{-1}B(v,v))\rangle,}
$}
\par
where $g_{\lambda}$ is obtained from solving the single bordered $(n+1)-$dimensional system 
\par
\qc{$
\matrix 2 cc 2 1 {A+I_n}{W}
		 {V^{T}}{0} 
		  \matrix 2 c 2 0 {v_{\lambda}(x,\alpha)}{g_{\lambda}(x,\alpha)}$ $=$ 
		  $\matrix 2 c 2 0 {v(x,\alpha)}{0}.
$}
\par
Here $A=f_x(x,\alpha)$ and eigenvectors $v$ and $w$ satisfy the equations
\par
\qc{$
Av=-v,\; A^T w=-w,\; \langle v,v\rangle = \langle w,v\rangle = 1,
$}\par
and\par
\qc{$
B_i(v,w)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       v_j w_k, 
$}
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n,$ and $\otimes$ means the \jump bialt {bialternate product} 
of two matrices; $2m=n(n-1).$
\par 
\par
The following {\bf singularities} can be detected and located as regular zeroes of
the period-doubling test-functions:
\par
{\li 2
\desc 4 
\jump R2 {1:2 Resonance}:\> $\psi_1=0$ 
\par
\jump FF {Fold-flip}:\> $\psi_2=0$
\par
\jump PN {Flip-NS}:\> $\psi_3=0$
\par
\jump DF {Degenerate flip}:\> $\psi_4=0$
\par
}
\end \% test_pd1

\topic map_start_pd0 {starter parameters (standard period-doubling curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_pd0 {standard period-doubling curve} has the following {\bf parameters}:
\par
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_pd0 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Multipliers}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.

\end \% map_start_pd0

\topic map_start_pd1 {starter parameters (bordered period-doubling curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_pd1 {bordered period-doubling curve} has the following {\bf parameters}:
\par
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_pd1 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Multipliers}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.

\end \% map_start_pd1



\topic NS_fp {Neimark-Sacker bifurcation}
If the critical fixed point $x_0$ has a simple pair of complex
multipliers
\par
\qc{ 
$\mu_{1,2}=e^{\pm i\theta_0},\: 0\;<\;\theta_0\;<\;\pi,$
}
\par
then, generically, a unique {\bf closed invariant curve} bifurcates from the fixed points as the parameter
crosses the critical value $\alpha_0$ in one or another direction, while the fixed point point
changes stability. 
\par
The \jump map_Class {map} restricted to the two-dimensional \jump fp_fp {center manifold} 
has the complex form
\par
\qc{$
z\prime = e^{i\theta_0} z+\frac{1}{2}g_{20}z^2 +
	   g_{11}z\bar z + \frac{1}{2}g_{02}{\bar z}^2  +
	   \frac{1}{2}g_{21}z^2\bar z + \ldots,\; z \in {\bf C}^1.
$}\par
This map can be transformed by an invertible smooth change of
coordinates to the Poincare normal form
\par
\qc{$
w\prime = e^{i\theta_0}w(1 + d(0) |w|^2 ) + O(|w|^4), \; w \in {\bf C}^1.
$}\par
where $a(0)=Re\, d(0)$ can be computed at the critical parameter value by the formula\par
\qc{$
a(0)=\frac{1}{2} Re\; e^{-i\theta_0}\ \brackets[. {
\langle p,C(q,q,\bar q)\rangle + 2\langle p,B(q,(I_n - A)^{-1}B(q,\bar q))\rangle
							   }
\, + \, \brackets.] {
\langle p,B(\bar q,{(e^{2i\theta_0} I_n - A)}^{-1}B(q,q))\rangle
			      },
$}\par
where the complex eigenvectors $q,p \in {\bf C}^n$ satisfy\par
\qc{$
Aq=e^{i\theta_0} q,\; A^T p=e^{-i\theta_0} p,\;  \langle Re\,q,Im\, q\rangle = 0,
\; \langle p,q\rangle  = 1,
$}\par
the multilinear functions $B(q,p)$ and $C(p,q,r)$ are defined by
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
$}
\par
and
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n$.
\par
Provided the critical multipliers cross the imaginary axis at a nonzero velocity with respect to
the parameter away from {\bf strong resonances}, i.e.
\par
\qc{$
e^{ik\theta_0} \neq 1,\; \; for\;  k=1,2,3,\; and\;  4,
$}
\par
a unique stable closed invariant curve bifurcates if $a(0) <0$, 
\par
\qc{\picture 0.8 supNS.fig}
\par
while a unique unstable closed invariant curve exists for close parameter value if  $a(0) >0$: 
\par
\qc{\picture 0.8 subNS.fig}
\par
If the critical fixed point has simple real
eigenvalues $\mu_{1,2},\: \mu_1\mu_2 = 1,$ then it is
a {\bf neutral saddle} which does not bifurcate. The fixed point has 
two real linearly independent eigenvectors $q,p \in {\bf R}^n$ satisfy
\par
\qc{$
Aq=\mu_1 q,\; Ap=\mu_2 p,\;  \langle q,q,\rangle = \langle p,p\rangle  = 1,
$}\par
\par
The critical eigenvalues and the normalized eigenvectors $p,q$
are stored for two-parameter continuation of the Neimark-Sacker (neutral saddle) curve, 
\par
The value
$\theta_0$ (and $l_1$) can be read in  the \jump kh_MainWin {message field}.\par

\end

\topic map_NS {Neimark-Sacker curve}
Given a \jump ode_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}
\par
a {\bf Neimark-Saker curve} is a one-dimensional manifold $\Gamma$ in ${\bf R}^{N}$ endowed 
with coordinates $(x,y,\ldots,\alpha)$,
whose projection onto the $(x,\alpha)$-space gives \jump NS_fp {Neimark-Sacker points} 
of (1) having a pair of \jump fp_fp {multipliers} with unit product: $\mu_1 \mu_2=1.$
If the critical multipliers are complex $(\lambda_{1,2}=exp(\pm i\theta_0),\; \theta_0>0)$,
the fixed point is a \jump fp_fp {nonhyperbolic} {\bf neutral focus}. If the multipliers are real
$(\lambda_{1}\lambda_{2}=1)$, the fixed point is a {\bf neutral saddle}.
\par
\par
The \jump kh_DefFunc {defining functions} used in CONTENT for Neimark-Sacker curve continuation 
can be specified by
\par
\par
$\;\;\;$\jump map_NS0 {the standard augmented system,}
\par
$\;\;\;$\jump map_NS1 {the bordered biproduct system,}
\par
$\;\;\;$\jump map_NS2 {the bordered squared Jacobian system.}
\par
\par
Supported {\bf methods} of Neimark-Sacker curve computation differ by \jump kh_DefFunc {defining}, 
\jump kh_TestFunc {test}, and \jump kh_ProcFunc {processing functions}. 
\end %map_NS

\topic map_NS0 {Neimark-Sacker curve (standard)}
Given a \jump ode_Class {map}, 
\par
\qc{
$
x\prime =f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf standard} \jump map_NS {Neimark-Sacker curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{2n+3}$ endowed with coordinates $y=(x,v,\kappa,\alpha)$
specified by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 4 rcl 1.2 1 
{f(x,\alpha)-x}{=}{0}
{([f_{x}(x,\alpha)]^{2}-2\kappa f_{x}(x,\alpha)+I_n)v}{=}{0}
{\langle v,v\rangle - 1}{=}{0}
{\langle w,v\rangle}{=}{0,}
$
}\qr {(2)}
\par
where $\kappa= cos (\theta)$ ($exp(\pm i\theta_0)$ is the pair of Neimark-Sacker
multipliers), and $w\in {\bf R}^n$ is a vector that is adapted along the curve to be {\bf not
orthogonal} to the null-eigenspace of the matrix $[f_{x}(x,\alpha)]^{2}-2\kappa f_{x}(x,\alpha)+I_n$.
Namely, to compute the next point on the curve, vector $w$ is taken to be orthogonal
to the vector $v$ at the found point and such that $\{v,w\}$ span the eigenspace $T^{c}$ 
corresponding to the eigenpair with eigenvalues having unit product.
\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear algebra library} is used to solve appearing linear systems.
\par
Several \jump test_NS0 {test functions} can be computed along the standard Neimark-Sacker curve to
detect and process \jump test_NS0 {Neimark-Sacker singularities}.
\par
\par
The standard Neimark-Sacker curve can be continued from a 
regular \jump NS_fp {Neimark-Sacker point} (as well as from all 
\jump test_NS0 {Neimark-Sacker singularities}. To \jump conti {continue} of the 
standard Neimark-Sacker curve, one 
has to set relevant \jump start_NS0 {starter parameters}
via the \jump kh_StarterWin {starter window}.
\end \% map_NS0

\topic map_NS1 {Neimark-Sacker curve (bordered biproduct)}
This method to define the \jump map_NS1 {Neimark-Sacker curve} is developed and implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime =f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf biproduct} \jump map_NS1 {Neimark-Sacker curve} is a one-dimensional manifold $\Gamma$
in ${\bf R}^{n+2}$ endowed with coordinates $y=(x,\alpha)$
specified by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 2 rcl 1.2 1 
{f(x,\alpha)}{=}{0}
{det\; G(x,\alpha)}{=}{0,}
$
}\qr {(2)}\par
where the components $g_{ij}$ of the $(2\times 2)-$matrix $G$ are obtained from solving 
the double bordered $(m+2)-$dimensional system with $2m =n(n-1)$
\par
\qc{$
\matrix 3 lrr 2 1 {B}{W_{1}}{W_{2}}
		  {V^{T}_{1}}{d_{11}}{d_{12}} 
		  {V^{T}_{2}}{d_{21}}{d_{22}}\matrix 3 cc 2 0 {Q_{1}}{\;Q_{2}}
						    {g_{11}}{\;g_{12}}
						    {g_{21}}{\;g_{22}}$ $=$ $\matrix 3 cc 3 0 {0}{\;0}
											    {1}{\;0}
											    {0}{\;1}.
$}
\par
Here the matrix $B=f_{x}(x,\alpha) \otimes f_{x}(x,\alpha) - I_{m}$ is a $(m\;\times\;m)-$matrix , where 
$\otimes$ stands for the \jump bialt {bialternate product} of two
$(n\;\times\;n)$ matrices. The vectors $V_1,V_2,W_1,W_2\in {\bf R}^m$
and scalars $d_{11}, d_{12}, d_{21}, d_{22}$ 
are adapted along the curve to make the above system {\bf nonsingular}. 
The defining system (2) actually determines
Jacobian matrices with a unit-product pair of eigenvalues.
So the solution branch may contain Neimark-Sacker points, strong resonances, and neutral saddle points.
\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear algebra library} is used to solve appearing linear systems.
\par
Several \jump test_NS1 {test functions} can be computed along the biproduct Neimark-Sacker curve to
detect and process \jump test_NS1 {Neimark-Sacker singularities}.
\par
\par
The biproduct Neimark-Sacker curve can be continued from a regular \jump NS-fp {Neimark-Sacker point} 
(as welll as from \jump test_NS1 {Neimark-Sacker singularities}. To \jump conti {continue} 
the biproduct Neimark-Sacker curve, one 
has to set relevant \jump start_NS1 {starter parameters}
via the \jump kh_StarterWin {starter window}.
\end \% map_NS1

\topic map_NS2 {Neimark-Sacker curve (bordered squared Jacobian)}
This method to define the \jump map_NS {Neimark-Sacker curve} is developed and implemented
in cooperation with {\bf W. Govaerts} and {\bf B. Sijnave} (Vakgroep Toegepaste Wiskunde en
Informatica, Universiteit Gent, Belgium).
\par
Given a \jump map_Class {map}, 
\par
\qc{
$
x\prime=f(x,\alpha),\: \: x\prime,x \in {\bf R}^n,\: \alpha \in {\bf R}^2,
$
}
\qr {(1)}\par
a {\bf bordered squared Jacobian} \jump map_NS {Neimark-Sacker curve} is a 
one-dimensional manifold $\Gamma$
in ${\bf R}^{n+3}$ endowed with coordinates $y=(x,\alpha,\kappa)$
specified by the following \jump kh_DefFunc {defining functions} \par
\qc{
$
\system 3 rcl 1.2 1 
{f(x,\alpha)}{=}{0}
{g_{i_{1},j_{1}}(x,\alpha)}{=}{0,}
{g_{i_{2},j_{2}}(x,\alpha)}{=}{0,}
$
}\qr {(2)}\par
where the components $g_{ij}$ of the $(2\times 2)-$matrix $G$ are obtained from solving 
the double bordered $(n+2)-$dimensional system
\par
\qc{$
\matrix 3 lrr 2 1 {B}{W_{1}}{W_{2}}
		  {V^{T}_{1}}{0}{0} 
		  {V^{T}_{2}}{0}{0}\matrix 3 cc 2 0 {Q_{1}}{\;Q_{2}}
						    {g_{11}}{\;g_{12}}
						    {g_{21}}{\;g_{22}}$ $=$ $\matrix 3 cc 3 0 {0}{\;0}
											    {1}{\;0}
											    {0}{\;1}.
$}
\par
Here the matrix $B=[f_{x}(x,\alpha)]^2 -2 \kappa f_{x}(x,\alpha) + I_{n}$, and 
$\kappa= cos \theta$ ($exp(\pm i\theta_0)$ is the pair of Neimark-Sacker
multipliers). 
The vectors $V_1,V_2,W_1,W_2\in {\bf R}^n$
are adapted along the curve to make the above system {\bf nonsingular}. 
The defining system (2) actually determines
Jacobian matrices with a zero-sum pair of eigenvalues.
So the solution branch may contain Neimark-Sacker points, strong resonances, and neutral saddle points.
\par
No special properties of the Jacobian matrix of (2) are assumed in the \jump conti {continuation}
and the {\bf generic linear algebra library} is used to solve appearing linear systems.
\par
Several \jump test_NS2 {test functions} can be computed along the biproduct Neimark-Sacker curve to
detect and process \jump test_NS2 {Neimark-Sacker singularities}.
\par
\par
The bordered squared Neimark-Sacker curve can be continued from a regular \jump NS_fp {Neimark-Sacker point} 
as well as from  \jump test_NS2 {Neimark-Sacker singularities}. 
To \jump conti {continue} of the bordered square Neimark-Sacker curve, one 
has to set relevant \jump start_NS2 {starter parameters}
via the \jump kh_StarterWin {starter window}.

\end \% map_NS2

\topic test_NS0 {test functions and singularities along the standard Neimark-Sacker curve}
The \jump kh_TestFunc {test-functions} that can be computed along the 
\jump map_NS0 {standard Neimark-Sacker curve} are the following:
\par
\qc{
$
\table 8 rcl 0 0 
{\psi_1}{\:=\:}{\frac{1}{2} Re\; e^{-i\theta_0}\ \brackets[. {
\langle p,C(q,q,\bar q)\rangle + 2\langle p,B(q,(I_n - A)^{-1}B(q,\bar q))\rangle
							   }
\, + \, \brackets.] {
\langle p,B(\bar q,{(e^{2i\theta_0} I_n - A)}^{-1}B(q,q))\rangle
			      },}
{\psi_2}{\:=\:}{\kappa-1,}
{\psi_3}{\:=\:}{det(A-I_n),}
{\psi_4}{\:=\:}{det(A+I_n),}
{\psi_5}{\:=\:}{det(A|_{N^{c}}\otimes A|_{N^{c}} - I_{m}),}
{\psi_6}{\:=\:}{\kappa + 1,}
{\psi_7}{\:=\:}{\kappa + \frac{1}{2},}
{\psi_8}{\:=\:}{\kappa,}
$}
\par
where $2m=(n-2)(n-3).$
Here $A = f_x(x,\alpha)$, the complex eigenvectors $q,p \in {\bf C}^n$ satisfy\par
\qc{$
Aq=e^{i\theta_0} q,\; A^T p=e^{-i\theta_0} p,\;  \langle Re\,q,Im\, q\rangle = 0,
\; \langle p,q\rangle  = 1,
$}\par
$N^{c} \subset {\bf R}^{n}$ is the orthogonal complement to the critical two-dimensional 
eigenspace corresponding to
the eigenpair of $A$ with unit product, 
while the multilinear functions $B(q,p)$ and $C(p,q,r)$ are defined at $(x_0,\alpha_0)$ by
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
$}
\par
and
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n$. The test function $\psi_1$ is only defined when the critical
multipliers are {\bf complex}, i.e. $-1<\kappa< 1.$
\par
The following {\bf singularities} can be detected and located as regular zeros of
the above defined test-functions:
\par
{\li 2
\desc 8 
\jump DN {Generalized Neimark-Sacker}:\> $\psi_1=0$
\par
\jump FN {Fold-NS}:\> $\psi_2 \neq 0,\;\psi_3=0$
\par
\jump PN {Flip-NS}:\> $\psi_4=0,\; \psi_6 \neq 0$
\par
\jump NN {Double Neimark-Sacker}:\> $\psi_5=0$
\par
\jump R1 {1:1 Resonance}:\> $\psi_2=\psi_3=0$
\par
\jump R2 {1:2 Resonance}:\> $\psi_4=\psi_6=0$
\par
\jump R3 {1:3 Resonance}:\> $\psi_7=0$
\par
\jump R4 {1:4 Resonance}:\> $\psi_8=0$
\par
}
\end \% test_NS0

\topic test_NS1 {test functions and singularities along the biproduct Neimark-Sacker curve}
The \jump kh_TestFunc {test-functions} that can be computed along 
the \jump map_NS1 {biproduct Neimark-Sacker curve}
are the following:
\par
\qc{
$
\table 8 rcl 0 0 
{\psi_1}{\:=\:}{\frac{1}{2} Re\; e^{-i\theta_0}\ \brackets[. {
\langle p,C(q,q,\bar q)\rangle + 2\langle p,B(q,(I_n - A)^{-1}B(q,\bar q))\rangle
							   }
\, + \, \brackets.] {
\langle p,B(\bar q,{(e^{2i\theta_0} I_n - A)}^{-1}B(q,q))\rangle
			      },}
{\psi_2}{\:=\:}{\kappa-1,}
{\psi_3}{\:=\:}{det(A-I_n),}
{\psi_4}{\:=\:}{det(A+I_n),}
{\psi_5}{\:=\:}{det \matrix 2 lr 2 1 {A \otimes A -I_m}{W_{1}}
		  {V^{T}_{1}}{0},}
{\psi_6}{\:=\:}{\kappa + 1,}
{\psi_7}{\:=\:}{\kappa + \frac{1}{2},}
{\psi_8}{\:=\:}{\kappa,}
$}
\par
where $2m=n(n-1).$
Here $A = f_x(x,\alpha)$, $\otimes$ 
stands for the \jump bialt {bialternate product} of two matrices), and
the complex vectors $q,p \in {\bf C}^n$ satisfy\par
\qc{$
Aq=e^{i\theta_0} q,\; A^T p=e^{-i\theta_0} p,\;  \langle Re\,q,Im\, q\rangle = 0,
\; \langle p,q\rangle  = 1,
$}\par
$N^{c} \subset {\bf R}^{n}$ is the orthogonal complement to the critical two-dimensional 
eigenspace corresponding to the eigenpair of $A$ with unit product,
while the multilinear functions $B(q,p)$ and $C(p,q,r)$ are defined at $(x_0,\alpha_0)$ by
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
$}
\par
and
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n$. The test function $\psi_1$ is only defined when the critical
multipliers are {\bf complex}, i.e $$.
\par
The following {\bf singularities} can be detected and located as regular zeros of
the above defined test-functions:
\par
{\li 2
\desc 8 
\jump DN {Generalized Neimark-Sacker}:\> $\psi_1=0$
\par
\jump FN {Fold-NS}:\> $\psi_2 \neq 0,\;\psi_3=0$
\par
\jump PN {Flip-NS}:\> $\psi_4=0,\; \psi_6 \neq 0$
\par
\jump NN {Double Neimark-Sacker}:\> $\psi_5=0$
\par
\jump R1 {1:1 Resonance}:\> $\psi_2=\psi_3=0$
\par
\jump R2 {1:2 Resonance}:\> $\psi_4=\psi_6=0$
\par
\jump R3 {1:3 Resonance}:\> $\psi_7=0$
\par
\jump R4 {1:4 Resonance}:\> $\psi_8=0$
\par
}
\end \% test_NS1

\topic test_NS2 {test functions and singularities along the squared Neimark-Sacker curve}
The \jump kh_TestFunc {test-functions} that can be computed along 
the \jump map_NS2 {squared Neimark-Sacker curve}
are the following:
\par
\qc{
$
\table 8 rcl 0 0 
{\psi_1}{\:=\:}{\frac{1}{2} Re\; e^{-i\theta_0}\ \brackets[. {
\langle p,C(q,q,\bar q)\rangle + 2\langle p,B(q,(I_n - A)^{-1}B(q,\bar q))\rangle
							   }
\, + \, \brackets.] {
\langle p,B(\bar q,{(e^{2i\theta_0} I_n - A)}^{-1}B(q,q))\rangle
			      },}
{\psi_2}{\:=\:}{\kappa-1,}
{\psi_3}{\:=\:}{det(A-I_n),}
{\psi_4}{\:=\:}{det(A+I_n),}
{\psi_5}{\:=\:}{det(A|_{N^{c}}\otimes A|_{N^{c}} - I_{m}),}
{\psi_6}{\:=\:}{\kappa + 1,}
{\psi_7}{\:=\:}{\kappa + \frac{1}{2},}
{\psi_8}{\:=\:}{\kappa,}
$}
\par
where $2m=(n-2)(n-3).$ Here $A = f_x(x,\alpha)$, $\otimes$ 
stands for the \jump bialt {bialternate product} of two matrices), and
the complex vectors $q,p \in {\bf C}^n$ satisfy\par
\qc{$
Aq=e^{i\theta_0} q,\; A^T p=e^{-i\theta_0} p,\;  \langle Re\,q,Im\, q\rangle = 0,
\; \langle p,q\rangle  = 1,
$}\par
$N^{c} \subset {\bf R}^{n}$ is the orthogonal complement to the critical two-dimensional 
eigenspace corresponding to the eigenpair of $A$ with unit product,
while the multilinear functions $B(q,p)$ and $C(p,q,r)$ are defined at $(x_0,\alpha_0)$ by
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k, 
$}
\par
and
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}\par
where $i=1,2,\ldots,n$. The test function $\psi_1$ is only defined when the critical
multipliers are {\bf complex}, i.e $$.
\par
The following {\bf singularities} can be detected and located as regular zeros of
the above defined test-functions:
\par
{\li 2
\desc 8 
\jump DN {Generalized Neimark-Sacker}:\> $\psi_1=0$
\par
\jump FN {Fold-NS}:\> $\psi_2 \neq 0,\;\psi_3=0$
\par
\jump PN {Flip-NS}:\> $\psi_4=0,\; \psi_6 \neq 0$
\par
\jump NN {Double Neimark-Sacker}:\> $\psi_5=0$
\par
\jump R1 {1:1 Resonance}:\> $\psi_2=\psi_3=0$
\par
\jump R2 {1:2 Resonance}:\> $\psi_4=\psi_6=0$
\par
\jump R3 {1:3 Resonance}:\> $\psi_7=0$
\par
\jump R4 {1:4 Resonance}:\> $\psi_8=0$
\par
}
\end \% test_NS2


\topic start_NS0 {starter parameters (standard Neimark-Sacker curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_NS0 {standard Neimark-Sacker curve} has the following {\bf parameters}:
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_NS0 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.
\end \% start_NS0

\topic start_NS1 {starter parameters (biproduct Neimark-Sacker curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_NS1 {biproduct Neimark-Sacker curve} has the following {\bf parameters}:
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_NS1 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.
\end \% start_NS1

\topic start_NS0 {starter parameters (standard Neimark-Sacker curve)}
The \jump kh_Starter {starter} which produces necessary data to \jump kh_Generator {generate} 
the \jump map_NS0 {standard Neimark-Sacker curve} has the following {\bf parameters}:
\par
{
\desc 17
{\bf Iteration data}\>\par
{\li 1 Superposition} \>  Number $M$ of superpositions of the iterated map\par
{\bf Initial point}\>\par
{\li 1 $x$} \> A list of phase coordinate names and their values at the initial point.\par
{\li 1 $\alpha$} \> A list of parameter names and their values at the initial point. 
 The names of active parameters are highlighted.\par
{\bf Singular Solver data}\>\par
{\li 1 SingTolerance} \>  Tolerance for the singularity of a matrix.\par
{\bf Jacobian data}\> \par
{\li 1 Increment} \> Increment to approximate partial derivatives by finite differences.\par
{\bf Monitor singularities}\>\par
{\li 1 \jump test_NS2 {Singularity types}}\> Monitor toggle ({\bf yes}|{\bf no}).\par
{\bf User defined functions}\>\par
{\li 1 \jump kh_UserFunc {Function names}}\> Process toggle ({\bf ignore}|{\bf monitor}|{\bf detect}|{\bf append}).\par
{\bf Eigenvalues}\> \par
{\li 1 Compute} \> Compute toggle ({\bf yes}|{\bf no}). \par
{\bf Set initial point}\> \par
{\li 1 SetInitPoint} \> \jump kh_FuncPar {Functional parameter} assigning initial values to
 phase variables and parameters. When it is specified and
 activated, it is called by the started before computing the first point.  \par
}
\par
The parameters can be modified via the \jump kh_StarterWin {starter window}.
\end \% start_NS2


\topic bialt {bialternate product}
The {\bf bialternate product} $(A \otimes B)$ of two $n\, \times\, n$ matrices $A$ and $B$ 
is an $m\, \times\, m$ matrix, $m=\frac{n(n-1)}{2},$ 
whose elements are enumerated by the multiindex $\{(p,q),(r,s)\}$, 
where $p=2,3,\ldots,\,n;$ $q=1,2,\ldots,\,p-1$, and
$r=2,3,\ldots,\,n;$ $s=1,2,\ldots,\,r-1$, and are given by the formula
\par
\qc{
$
(A\otimes B)_{(p,q),(r,s)}=\frac{1}{2}
{\brackets()
{\brackets||\table 2 lr 0 0 {a_{pr}}{\;a_{ps}}
		{b_{qr}}{\:b_{qs}} \; + \; 
{\brackets||\table 2 cc 0 0 {b_{pr}}{\:b_{ps}}
		{a_{qr}}{\:a_{qs}}}
},
$
}
\par
where $a_{jk}$ are elements of $A$, and $b_{jk}$ are elements of $B,\:j,k=1,2,\ldots,n$. 
There are two important special cases of the bialternate product:\par
\qc{$
(2A\otimes I)_{(p,q),(r,s)}=
  \system 6 cl 1.2 1
      {-a_{ps}}       {{\f * if}\: r=q,}
      {a_{pr}}        {{\f * if}\: r\neq p \: {\f * and}\: s=q,}
      {a_{pp}+\:a_{qq}} {{\f * if}\: r=p \: {\f * and}\: s=q,}
      {a_{qs}}        {{\f * if}\: r=p \: {\f * and}\: s \neq q,}
      {-a_{qr}}       {{\f * if}\: s=p,}
      {0}             {\f * otherwise,}
$}
\par
and
\par
\qc{
$
(A\otimes A)_{(p,q),(r,s)}={\brackets||\table 2 lr 0 0 {a_{pr}}{\:a_{ps}}{a_{qr}}{\:a_{qs}}}=
a_{pr}a_{qs}-\:a_{ps}a_{qr}.
$
}
\par
One has 
\par
\qc{$
 det (2A \otimes I_n) = \prod {j>k}{}{(\lambda_j+\:\lambda_k)},
$}\par
and
\par
\qc{$
 det (A \otimes A \:-\: I_m) = \prod {j>k}{}{(\lambda_{j}\lambda_{k} \: - \: 1)},
$}\par
where $\lambda_k$ are the eigenvalues of matrix $A$.

\end \% bialt

\topic R1 {strong 1:1 resonance}
The {\bf strong 1:1 resonance} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to a double multiplier $\mu =1.$ The map restricted to the two-dimensional
center manifold at the critical parameter values can be transformed to the normal form
\par
\qc{$
\system 2 rcl 1.2 1 
{{\xi\prime}_1}{=}{\xi_1+\xi_2}
{{\xi\prime}_2}{=}{\xi_2+a\xi^{2}_{1} + b \xi_{1} \xi_{2} + O(||\xi||^3).}
$}
\par
Here $\xi = (\xi_{1},\xi_{2})^{T} \in {\bf R}^2$, and
\par
\qc{$
a=\frac{1}{2}\langle r,B(q,q)\rangle,\;\;b=\langle s,B(q,q)\rangle + \langle r,B(q,p)\rangle,
$}\par
where the (generalized) eigenvectors $q,p,r,s \in {\bf R}^n$ are defined by\par
\qc{$
Aq=q,\; Ap=p+q,\; A^T s=s,\; A^T r=r+s,\;  
$}
\par
and satisfy
\par
\qc{$
\langle q,q \rangle = \langle s,p \rangle = \langle r,q \rangle = 1,\;
\langle q,p \rangle = \langle s,q \rangle = \langle r,p \rangle = 0,\;
$}
\par
The multilinear function $B(q,p)$ is defined at $(x_0,\alpha_0)$ in terms  
of the map components $f_i(x,\alpha),\; i=1,\ldots,n$ by the formulas
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k. 
$}
\par
Under the nongegeneracy conditions
\par
\qc{$
A=a \neq 0,\;\;B=b-2a \neq 0,
$}\par
a generic map near the strong 1:1 resonance exhibits both fold and Neimark-Sacker
bifurcations, as well as an intersection of invariant manifolds of the saddle fixed
point ({\bf homoclinic structure}). See detils, for example, in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\par
\par
The  eigenvectors $p,q$ are stored for
switching to the continuation of the \jump map_NS {Neimark-Sacker} and
\jump map_lp {limit point curves} from the 1:1 point.
The values $A$ and $B$ can be read in the \jump kh_MainWin {message field}.\par
\end \%R1

\topic R2 {strong 1:2 resonance}
The {\bf strong 1:2 resonance} is a codim 2 singularity of the \jump fp_fp {fixed point}
corresponding to a double multiplier $\mu =-1.$ See detils, for example, in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\par
\end \%R2

\topic R3 {strong 1:3 resonance}
The {\bf strong 1:3 resonance} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers 
\par
\qc{$
\mu_{1,2} = e^{\pm i\theta_0},\; \theta_0=\frac{2\pi}{3}.
$}
\par 
See detils, for example, 
in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\par
\end \%R3

\topic R4 {strong 1:4 resonance}
The {\bf strong 1:4 resonance} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers 
\par
\qc{$\mu_{1,2} = \pm i.
$}
\par 
See detils, for example, 
in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\par
\end \%R4

\topic FF {fold-flip bifurcation}
The {\bf fold-flip} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers 
\par
\qc{$\mu_{1} = 1,\; \mu_{2} = -1.
$}
\par
\end \%FF

\topic FN {fold-NS  bifurcation}
The {\bf fold$\;-\;$Neimark-Sacker} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers 
\par
\qc{$\mu_{1} = 1,\; \mu_{2}\mu_{3} = 1,
$}
\par
in particular, 
\par
\qc{$\mu_{2,3} = e^{\pm i\theta_0},
$}
\par
where $\theta_{0} \neq 0,\pi,2\pi/3,\pi/4$.
\end \%FN

\topic PN {flip-NS bifurcation}
The {\bf flip$\;-\;$Neimark-Sacker} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers 
\par
\qc{$\mu_{1} = -1,\; \mu_{2}\mu_{3} = 1,
$}
\par
in particular, 
\par
\qc{$\mu_{2,3} = e^{\pm i\theta_0},
$}
\par
where $\theta_{0} \neq 0,\pi,2\pi/3,\pi/4$.
\end \%PN

\topic NN {double Neimark-Sacker bifurcation}
The {\bf double Neimark-Sacker} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the multipliers
\par
\qc{$\mu_{1}\mu_{2} = \mu_{3}\mu_{4} = 1,
$}
\par
in particular, 
\par
\qc{$\mu_{1,2} = e^{\pm i\theta_0},\; \mu_{3,4} = e^{\pm i\theta_1},
$}
\par
where $\theta_{0,1} \neq 0,\pi,2\pi/3,\pi/4$.
\end \%NN

\topic CP {cusp bifurcation}
The {\bf cusp} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the simple multiplier $\mu_{1} = 1$ and $a=0$ in the 
\jump LP_fp {normal form  for the limit point bifurcation}. 
At $\alpha=\alpha_0 \in {\bf R}^2$ the restricted to the one-dimensional
\jump fp_fp {center manifold} generic map has the form
\par
\qc{$
\xi\prime =\xi + c \xi^3 + O(|\xi|^4),\; \xi \in {\bf R}^1.
$}
\par
If $c\neq 0$ and the map depends generically on active parameters, its restriction
to the parameter-dependent center manifold is locally topologically equivalent
to the normal form
\par
\qc{$
\xi\prime\; =\; \xi\; +\; \beta_1 +\; \beta_2 \xi\; \pm \; \xi^3.
$}
\par
where $\sigma = sign(c)=\pm 1$.
Therefore, generically, three fixed points collide at the cusp point and the projection of
the \jump map_lp {limit point curve} onto the parameter plane forms a {\bf semicubic parabola}.
\par
The {\bf cusp normal form coefficient} can be computed at the critical parameter values 
by the formula
\par
\qc{$
c=\frac{1}{6}\langle p,C(q,q,q)\rangle - \frac{1}{2} \langle p,B(q,s)\rangle,
$}\par
where the vectors $q,p \in {\bf R}^n$ are defined by
\par
\qc{$
Aq=q,\;\; A^T p=p, \;\; \langle q,q \rangle = \langle p,q \rangle = 1,\;
$}
\par
while the vector $s \in {\bf R}^n$ is extracted from the solution of the bordered 
$(n+1)$-dimensional system
\par
\qc{$
\matrix 2 lr 2 1 {A-I_n}{q}{p^{T}}{0} \matrix 2 c 2 0 {s}{u}$ $=$ $\matrix 2 c 2 0 {-\langle p, B(q,q) \rangle}{0}.
$}
\par
Here the multilinear vector-functions $B(q,p)$ and $C(p,q,r)$ are defined at 
the critical point $(x_0,\alpha_0)$ in terms of 
the map components $f_i(x,\alpha),\; i=1,\ldots,n$ by
\par
\qc{$
B_i(q,p)= \Sum {j,k=1} {n} {{\brackets.| {\frac{\partial^2
	 f_i(x,\alpha_0)}{\partial x_j\partial x_k}}}_{x=x_0}} \;
       q_j p_k 
$}
\par
and
\par
\qc{$
C_i(p,q,r)= \Sum {j,k,l=1} {n} {{\brackets.| {
    \frac{\partial^3
   f_i(x,\alpha_0)}{\partial x_j\partial x_k\partial x_l}}}_{x=x_0}}
   p_j q_k r_l, 
$}
\par
where $i=1,\ldots,n$. See detils, for example, in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\par
\par
The normalized eigenvector $q$ is stored for starting two-parameter continuation of 
the passing \jump map_lp {limit point curve}. The value of the normal form coefficient $c$ can be read 
in  the \jump kh_MainWin {message field}.
\end \%CP

\topic DF {degenerate flip bifurcation}
The {\bf degenerate flip bifurcation} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the simple multiplier $\mu_{1} = -1$ and $c=0$ in the   
\jump PD_fp {normal form for the flip bifurcation}. The local topological normal form of the 
map restricted to the one-dimensional center manifold near the bifurcation is
\par
\qc{$
x\prime \;=\; -(1+\beta_1)x\; +\; \beta_2 x^3\; \pm \;x^5.
$}
\par
See detils, for example, in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\end \%DF

\topic DN {degenerate Neimark-Sacker bifurcation}
The {\bf degenerate NS (Chenciner) bifurcation} is a codim 2 singularity of the \jump fp_fp {fixed point} 
corresponding to the simple multipliers 
\par
\qc{$\mu_{1,2} = e^{\pm i\theta_0},
$}
\par
where $\theta_0 \neq 0,\pi,2\pi/3,\pi/4,$ and $a(0)=0$ in the 
\jump NS_fp {normal form for Neimark-Sacker bifurcation}.
See detils, for example, in \jump Ku95 {[Kuznetsov, 1995,1998]}.
\end \%DN

\topic Ku95 {book on bifurcations}
Kuznetsov, Yu.A. "Elements of Applied Bifurcation Theory", Springer-Verlag, New York, 1995,1998.
\end