phs : Effect of the Phase Condition.

This demo illustrates the effect of the phase condition on the computation of periodic solutions. We consider the differential equation

$$\begin{array}{cl} u_1'&= \lambda u_1 - u_2, \\ u_2'&= u_1 (1-u_1) . \\ \end{array}$$ (14.14)

This equation has a Hopf bifurcation from the trivial solution at $\lambda=0$. The bifurcating family of periodic solutions is vertical and along it the period increases monotonically. The family terminates in a homoclinic orbit containing the saddle point $(u_1,u_2)=(1,0)$. Graphical inspection of the computed periodic orbits, for example $u_1$ versus the scaled time variable $t$, shows how the phase condition has the effect of keeping the ``peak'' in the solution in the same location.

Table 14.21: Commands for running demo phs.

AUTO -COMMAND	ACTION
! mkdir phs	create an empty work directory
cd phs	change directory
demo('phs')	copy the demo files to the work directory
ld('phs')	load the problem definition
run(c='phs.1')	detect Hopf bifurcation
sv('phs')	save output-files as b.phs, s.phs, d.phs
run(c='phs.2',s='phs')	compute periodic solutions. Constants changed : IRS, IPS, NPR
ap('phs')	append output-files to b.phs, s.phs, d.phs