\section{The Periodic Orbits Window}\index{Periodic Orbits window}\index{window, Periodic Orbits}
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\begin{figure}[H]
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\centerline{\psfig{figure=ps/porbit.ps,height=137pt}}
\caption{\label{Periodic Orbits}
The Periodic Orbits window.
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\end{figure}

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\begin{description}

\item[Window title:] DsTool: Periodic Orbits
\item[Function:] The Periodic Orbits window allows the 
	user to find periodic orbits and to do continuation\index{continuation}
	calculations for periodic orbits of vector fields.
\item[Description:] The Periodic Orbits window is opened by selecting the 
	Periodic Orbits option from the Panels menu button located in the Command
	window.	  The window contains parameters for controlling the finding of 
	periodic orbits and doing continuation calculations. 
	This window should not be used for mappings.  Instead, the Fixed Points window should be used,
	with its Period field set to a positive integer.

	Colorcoding is performed according to the stability type of objects:
	saddle orbits are colored green, attracting orbits are blue, and repelling orbits are red.

%
%	Colorcoding\index{color, fixed points} is performed according to the stability type of 
%	objects: saddle orbits are marked as green crosses, attracting orbits are
%	blue triangles, and sources are red squares.  Furthermore,
%	unstable manifolds\index{color, manifolds} are plotted in red and stable manifolds
%	are plotted in blue.
%\item[Window type:] pop-up
%\item[Window attributes:] pinnable(in), non-resizable, fixed/variable template
\item[Panel items:] \mbox{}
	\begin{itemize}
	\item Algorithm stack setting:
		Allows the user to choose which algorithm will be used to 
		find periodic orbits.   The current options are:
		\begin{description}
		\item[Newton:]\index{Newton's method} Use Newton's method applied to the Poincar\'e map
			defined by the subsequent variables.   Derivative calculations
			are done by finite differencing.  This is the default option.
		\item[Attracting PO only:]  Use the fact that attracting periodic orbits
			attract nearby trajectories to find the periodic orbit.  That is,
			we integrate forward in time until either we've come close enough 
			to the periodic orbit (\ie, the difference between subsequent iterates 
			of the Poincar\'e map is less than the convergence criterion defined in 
			the Fixed Points window), or we give up (reach the value of Stop in 
			the Orbits window).   
		\end{description}
	\item Period numeric field:
%		If the current dynamical system is a vector field, the value for this text field is a 
% 		floating point number which indicates the period of a periodic trajectory. 
		Displays the period of the fixed point of the Poincar\'e map to search for.  In most
		cases, having this field set to the default value (which is $1$) will be adequate.
	\item Variable for section stack setting:
		Allows the user to choose the variable which is used to create the Poincar\'e 
		section\index{Poincar\'e section}.
		Allowable options are the dependent variables of the current dynamical system.
	\item Variable value of section read-write text field:
		Displays the value of the section variable which is used to create the Poincar\'e section.
		That is, if the Variable for section is ``x'', and this field is set 
		to ``a'', then the Poincar\'e section is defined as the $n-1$ dimensional hyperplane
		$x = a$.
	\item Parameter to vary stack setting:
		Allows the user to choose a parameter to perform continuation calculations with.
		Allowable options are the parameters of the current dynamical system.
	\item Number of steps numeric field:
		Displays the (maximum) number of steps of size parameter increment to take in the 
		parameter specified by the Parameter to vary stack.   For each step, we attempt to 
		find a periodic orbit.  If we fail to find a periodic orbit for a parameter
		value, the calculation ceases.   Otherwise, we increment the parameter by the value
		of parameter increment and continue.  If this field is set to $0$ (the default value),
		the routines search for periodic orbits with the given parameter values.
	\item Parameter increment read-write text field:
		Displays the size of the increment of the specified parameter at each step of the 
		continuation calculation.  The default value is $0.01$.
	\item Go command button:
		Initiates the periodic orbit (continuation) calculation, using
		the current values of the displayed fields.
	\item Dismiss command button:
		Closes the Periodic Orbits window.  
% The fields will not revert to their default values.
	\end{itemize}
\end{description}
