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<H1><A NAME="SECTION004240000000000000000"> </A><A NAME="1419"> </A><A NAME="1420"> </A>
<BR>
The Periodic Orbits Window
</H1>
<BR>
<DIV ALIGN="CENTER"><A NAME="Periodic_Orbits"> </A><A NAME="1436"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.25:</STRONG>
The Periodic Orbits window.</CAPTION>
<TR><TD>47#47</TD></TR>
</TABLE>
</DIV>
<BR>
<P>
<DL>
<DD><P>
<DT><STRONG>Window title:</STRONG>
<DD>DsTool: Periodic Orbits
<DT><STRONG>Function:</STRONG>
<DD>The Periodic Orbits window allows the
user to find periodic orbits and to do continuation<A NAME="1426"> </A>
calculations for periodic orbits of vector fields.
<DT><STRONG>Description:</STRONG>
<DD>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.
<P>
Colorcoding is performed according to the stability type of objects:
saddle orbits are colored green, attracting orbits are blue, and repelling orbits are red.
<P>
<DT><STRONG>Panel items:</STRONG>
<DD>
<UL>
<LI>Algorithm stack setting:
Allows the user to choose which algorithm will be used to
find periodic orbits. The current options are:
<DL>
<DT><STRONG>Newton:</STRONG>
<DD><A NAME="1430"> </A> Use Newton's method applied to the Poincaré map
defined by the subsequent variables. Derivative calculations
are done by finite differencing. This is the default option.
<DT><STRONG>Attracting PO only:</STRONG>
<DD>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 (, the difference between subsequent iterates
of the Poincaré 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).
</DL>
<LI>Period numeric field:
Displays the period of the fixed point of the Poincaré map to search for. In most
cases, having this field set to the default value (which is 1) will be adequate.
<LI>Variable for section stack setting:
Allows the user to choose the variable which is used to create the Poincaré
section<A NAME="1432"> </A>.
Allowable options are the dependent variables of the current dynamical system.
<LI>Variable value of section read-write text field:
Displays the value of the section variable which is used to create the Poincaré section.
That is, if the Variable for section is ``x'', and this field is set
to ``a'', then the Poincaré section is defined as the <I>n</I>-1 dimensional hyperplane
<I>x</I> = <I>a</I>.
<LI>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.
<LI>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.
<LI>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.
<LI>Go command button:
Initiates the periodic orbit (continuation) calculation, using
the current values of the displayed fields.
<LI>Dismiss command button:
Closes the Periodic Orbits window.
</UL></DL><HR>
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<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
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