pen : Rotations of Coupled Pendula.

This demo illustrates the computation of rotations, i.e., solutions that are periodic, modulo a phase gain of an even multiple of $\pi$. AUTO checks the starting data for components with such a phase gain and, if present, it will automatically adjust the computations accordingly. The model equations, a system of two coupled pendula, (DoArOt:91 DoArOt:91), are given by

$$\begin{array}{cl} & \phi_1'' + \epsilon \phi_1' + \sin \phi_1... ...hi_2' + \sin \phi_2 = I + \gamma(\phi_1-\phi_2) ,\\ \end{array}$$ (14.11)

or, in equivalent first order form,

$$\begin{array}{cl} & \phi_1' = \psi_1, \\ & \phi_2' = \psi_2, ... ...\psi_2 - \sin \phi_2 + I + \gamma(\phi_1-\phi_2).\\ \end{array}$$ (14.12)

Throughout $\gamma=0.175$. Initially, $\epsilon =0.1$ and $I=0.4$.

Numerical data representing one complete rotation are contained in the file pen.dat. Each row in pen.dat contains five real numbers, namely, the time variable $t$, $\phi_1$, $\phi_2$, $\psi_1$ and $\psi_2$. The correponding parameter values are defined in the user-supplied subroutine STPNT.

Actually, in this example, a scaled time variable $t$ is given in pen.dat. For this reason the period (PAR(11)) is also set in STPNT. Normally AUTO would automatically set the period according to the data in pen.dat.

The AUTO -command us('pen') converts the data in pen.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.pen. (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('pen') only creates s.dat, with no corresponding b.dat.)

The first run, with $I$ as free problem parameter, starts from the converted solution with label 1 in pen.dat. A period-doubling bifurcation is located, and the period-doubled family is computed in the second run. Two branch points are located, and the bifurcating families are traced out in the third and fourth run, respectively. The fifth run generates starting data for the subsequent computation of a locus of period-doubling bifurcations. The actual computation is done in the sixth run, with $\epsilon$ and $I$ as free problem parameters.

Table 14.19: Commands for running demo pen.

AUTO -COMMAND	ACTION
! mkdir pen	create an empty work directory
cd pen	change directory
demo('pen')	copy the demo files to the work directory
ld('pen')	load the problem definition
us('pen')	convert pen.dat to AUTO format in s.dat
run(c='pen.1',s='dat')	locate a period doubling bifurcation; restart from s.dat
sv('pen')	save output-files as b.pen, s.pen, d.pen
run(c='pen.2',s='pen')	a family of period-doubled (and out-of-phase) rotations. Constants changed : IPS, NTST, ISW, NMX
ap('pen')	append output-files tp b.pen, s.pen, d.pen
run(c='pen.3',s='pen')	a secondary bifurcating family (without bifurcation detection). Constants changed : IRS, ISP
ap('pen')	append output-files to b.pen, s.pen, d.pen
run(c='pen.4',s='pen')	another secondary bifurcating family (without bifurcation detection). Constants changed : IRS
ap('pen')	append output-files to b.pen, s.pen, d.pen
run(c='pen.5',s='pen')	generate starting data for period doubling continuation. Constants changed : IRS, ICP, ICP, ISW, NMX
sv('t')	save output-files as b.t, s.t, d.t
run(c='pen.6',s='t')	compute a locus of period doubling bifurcations; restart from s.t. Constants changed : IRS
sv('pd')	save output-files as b.pd, s.pd, d.pd