Torus Bifurcation
The equations in this demo model an autonomous electronic circuit:
x'(t) = [-(beta+nu) x + beta y - a3 x^3 + b3 (y - x)^3]/r
y'(t) = beta x - (beta+nu) y - z - b3 (y - x)^3
z'(t) = y
To run the demo type "auto tor.auto". To view the bifurcation
diagram type "@pp tor", which shows the stationary solution
family (blue) with Hopf bifurcation (small red square) at the
bottom of the diagram, the primary family of periodic solutions
(red), and a bifurcating family of periodic solutions (purple).
The solid red diamond on the purple family represents a a torus
bifurcation point. This family ends in a homoclinic orbit. To
see the labeled solutions that have been saved click on "solution".
Next the torus bifurcation is continued with "nu" and "beta" (and
the period T) as the free problem parameters. The periodic orbit
that corresponds to the torus bifurcation has two Floquet multipliers
on the unit circle in the complex plane, i.e., e^{+i theta} and
e^{-i theta} The angle "theta" is automatically added by AUTO as
an active continuation parameter and kept track of in PAR(12), even
if this is not explicitly specified in the parameter list ICP. In
"Run 4" in the script file tor.auto, theta is actually specified
explicitly, namely,
ICP=['nu', 'beta', 'T', 'theta']
This is done to make the value of theta appear in the screen output and
in the bifurcation file (here b.tor-2p).
To plot the locus of torus bifurcations, type "@pp tor-2p". Type "d3" in
the shell window to see solid curves, and choose "beta" as the "Y"-axis.
This will show the locus of torus bifurcations in the problem parameters
nu and beta. Also choose "theta" as the "Y"-axis to see how it varies
along the locus. Note the two solutions labeled 2 and 3, where theta is
equal to 0.5 (radians). A 3D plot can be seen by choosing, for example,
"beta" as the "Y"-axis and "theta" as the "Z"-axis.
In the final run, one of the saved solutions with theta=0.5 is continued
in the three problem parameters "nu", "beta" and "gamma", keeping the angle
"theta" fixed at theta=0.5. The period "T" remains a continuation parameter,
so that the parameter list is now given as
ICP=['nu','beta','gamma','T']
The results of this run are saved in b.tor-3p and s.tor-3p, which can be
plotted by typing "@pp tor-3p". Type "d3" in the shell window to see solid
curves, and successively choose "beta", "gamma" and "T" as the "Y"-axis.
A 3D plot can be seen by choosing, for example, "beta" as the "Y"-axis
and "gamma" as the "Z"-axis.
Reference:
Freire, E., Rodriguez-Luis, A., Gamero, E. & Ponce, E. (1993), A case study
for homoclinic chaos in an autonomous electronic circuit: A trip from Takens-
Bogdanov to Hopf-Shilnikov, Physica D 62, 230253.