Vertical-Hopf
This demo computes a "vertical" family of periodic
solutions, i.e., a family along which the "bifurcation
parameter" remains constant. This poses no difficulty
for the pseudo-arclength continuation in AUTO. In fact,
the current demo is mostly intended as an illustration
of the very simple numerical technique that AUTO uses
for following solutions of conservative systems, namely
by introducing an "unfolding term" with "unfolding
parameter" lambda, where lambda is one of the unknowns
solved for in each continuation step.
The equations of this demos are
u'(t) = lambda*u - v
v'(t) = u*(1-u)
which have a family of periodic solutions when lambda=0
(and none when lambda is nonzero). Here the unfolding
term is "lambda*u", with unfolding parameter lambda.
To run this demo type "auto vhb.auto". In the screen
output note that lambda remains indeed very small.
To plot the output files b.vhb, s.vhb, type "@pp vhb".
This will show the bifurcation diagram, with the zero
stationary solution family in blue, the vertical family
of periodic solutions represented by a red curve, and
the Hopf bifurcation by a red square. To see the periodic
solutions in the phase plane, click on "solution". Note
that the periodic solutions approach a homoclinic orbit;
in fact the period of the last orbit computed is 10^6.
To appreciate the power of mesh adaption, together with
continuation, plot the last computed orbit as a function
of t, and blow up the region of the "spike" until a smooth
transition can be seen. It is especially noteworthy that
this computation only used 50 mesh intervals (NTST=50).