lor : Starting an Orbit from Numerical Data.

This demo illustrates how to start the computation of a family of periodic solutions from numerical data obtained, for example, from an initial value solver. As an illustrative application we consider the Lorenz equations

$$\begin{array}{cl} u_1' &= p_3 (u_2 - u_1), \\ u_2' &= p_1 u_1 - u_2 - u_1 u_3, \\ u_3' &= u_1 u_2 - p_2 u_3. \\ \end{array}$$ (14.4)

Numerical simulations with a simple initial value solver show the existence of a stable periodic orbit when $p_1=280$, $p_2=8/3$, $p_3=10$. Numerical data representing one complete periodic oscillation are contained in the file lor.dat. Each row in lor.dat contains four real numbers, namely, the time variable $t$, $u_1$, $u_2$ and $u_3$. The correponding parameter values are defined in the user-supplied subroutine stpnt. The AUTO -command us('lor') then converts the data in lor.dat to a labeled AUTO solution (with label 1) in a new file s.dat. The mesh will be suitably adapted to the solution, using the number of mesh intervals NTST and the number of collocation point per mesh interval NCOL specified in the constants-file c.lor. (Note that the file s.dat should be used for restart only. Do not append new output-files to s.dat, as the command us('lor') only creates s.dat, with no corresponding b.dat.)

Table 14.11: Commands for running demo lor.

AUTO -COMMAND	ACTION
! mkdir lor	create an empty work directory
cd lor	change directory
demo('lor')	copy the demo files to the work directory
ld('lor')	load the problem definition
us('lor')	convert lor.dat to AUTO format in s.dat
run(c='lor.1',s='dat')	compute a solution family, restart from s.dat
sv('lor')	save output-files as b.lor, s.lor, d.lor
run(c='lor.2',s='lor')	switch branches at a period-doubling detected in the first run. Constants changed : IRS, ISW, NTST
ap('lor')	append the output-files to b.lor, s.lor, d.lor