Enzyme-Model
This demo computes bifurcation diagrams for a 2D system
of ODEs that models an activator-inhibitor enzyme model.
The primary parameter is called rho, and the secondary
parameter is s0.
To run this demo type "auto enz.auto". This will result
in a basic bifurcation diagram for s0=110 displaying a
stationary solution family having two Hopf bifurcations,
interconnected by a family of periodic solutions with
two folds. These results are saved in AUTO files b.s110,
s.s110 and d.s110. The free parameter is rho. To plot the
bifurcation diagram type "@pp s110".
The Hopf bifurcations and folds are subsequently continued
in 2 parameters, rho and s0. The results, saved in b.loci,
s.loci, and d.loci, show that for increasing s0 the Hopf
bifurcations disappear first, while the folds persist to a
larger value of s0. This fact suggests the existence of an
isola of periodic solution, which is subsequently verified
for s0=111, with results saved in b.s111, s.s111, d.s111.
To plot the loci of Hopf bifurcation points and folds, type
"@pp loci" and select "s0" as the "Y" axis. Type "d0" in the
shell window to change to solid curves, and type "d3" to see
the labels of the periodic solutions that have been saved.
(The shell commands "d0", "d3", etc. are defined in the file
autorc.)
To plot the isola at s0=111, type "@pp s111".
Reference: E. J. Doedel, H. B. Keller, J. P. Kernevez,
Numerical Analysis and Control Of Bifurcation Problems,
Part I: Bifurcation in Finite Dimensions, IJBC 1 #3, 1991,
493-520; Part II: Bifurcation in Infinite Dimensions,
IJBC 1 #4. 1991, 745-772.