Math Physiology: COMPUTER LAB II

Brusselator.ode is the code for the chemical oscillator examined in class.

SubHopf.ode is a planar system which has a subcritical Hopf bifurcation at the parameter value a=0.

Brusselator.ode Lab

1) Download Brusselator.ode.set . This file contains graphical and numerical parameter values used to initiate xppaut.

2) xppaut Brusselator.ode &

3) Under the "File" menu, select "Read set" and then "OK". This will load the contents of "Brusselator.ode.set" into xppaut. Axes will have changed, numerical integration parameters under the "nUmerics" menu will have changed, etc. When using xppaut on a new problem, you can save the current settings to a file with the "Write Set" command under the "File" menu.

4) Change parameter a=0.5, add nullclines and select some initial condition near the equilibria. You should have a picture that looks something like:

5) Erase (E) the figure, change a=0.1, add nullclines and select new initial conditions. The phase portrait should now indicate the presence of a larger amplitude limit cycle oscillation. Your figure should look something like:

6) Now we will use the "aut" part of "xppaut" to generate a bifurcation diagram showing equilibria and periodic orbits eminating from a Hopf point. "aut" is short for the acronymn AUTO which refers to a Fortran library of routines (written by E. Doedel) for creating bifurcation diagrams in a variety of different dynamical system settings. If an equilibria x(a)=(u(a),v(a)) of a system is known for some a=a0 value then AUTO will numerically compute x(a) for other "a" values. This process is called "continuation". Before you can even get the "aut" part of "xppaut" to work, you must first find an equilibria value and enter it in as an initial conditions. For our model, we calculated the equilibria as:

u = (a+b)

v = b/(a+b)^2

Set a=0.5 and b=0.5 and then select "Initialconds", then "New" and then initial conditions for (u,v) to values of (a,b) corresponding to the above formulae. We are now ready to use AUTO!!!!

7) In the "xppaut" window select "File" then "Auto". The following window should appear:

The diagram is set up for plotting the u coordinate of equilibria and periodic solutions. To change the appearance of the window, select "Axes", then "hI-lo" and set the range of a and u as follows:

XMIN=0, XMAX=0.5

YLOW=0, YMIN=4

Now select the "Numerics" menu. Change the parameter values so they match the values below:

The meaning of each of these are documented in the PDF file you can download from Lab 1 on xppaut. You should have changed 4 parameter values:

Ds = a suggested "arclength" increment to continue with.

Dsmax = the maximum allowable "arclength" increment

Par min = lower bound of your parameter range

Par max = upper bound of your parameter range

Here we are wanting to "continue" the equilibria over the range of a in (0.,0.5). This range is DIFFERENT than those set under the "Axes" menu (which are just the window range values). Aslo, Ds is negative here because we are starting at a=0.5 and decreasing to a=0.

Now...hold on.....select "Run" then "Steady State". You should get:

The thick line indicates stable equilibria. The thin line indicates unstable equilibria. Note the change of stability at a point labelled "2". AUTO keeps track of special points, including bifurcation points where there is a change in stability.

8) Now comes the cool part...."Switching branches". In AUTO, you can continue limit cycles and other equilibria branches from bifurcation points. Here, because of the analysis in class we suspect that point "2" is a Hopf point from which a "branch" of periodic solutions should eminate. Select "Grab" so we can grab a point to switch branches from. By using the "Tab" key you can tab through all the labelled special points. Tab thru until you reach point "2". Then you should see:

Note that point "2" has "Ty=HB" which is short for Hopf Bifurcation. The (a,u(a)) values at that Hopf point are indicated along with the minimal period for periodic solutions eminating from it! To actually "Grab" the point you must now hit the "Return" key. Having done this AUTO now is ready to switch branches.

Select "Run", then "Periodic". You should now have:

The dark dots indicate a stable limit cycle (periodic orbit). If they turned out to be unstable, AUTO would have drawn small open circles. Since the stable periodic orbits surround unstable equilibria the bifurcation is "supercritical", i.e. a supercritical Hopf bifurcation.

SubHopf.ode Lab

1) exit xppaut. 2) Download SubHopf.ode.set , then "xppaut SubHopf.ode &".

3) Repeat the above procedures to generate a bifurcation diagram showing a Subcritical Hopf bifurcation of the new system at a=0, u=v=0.