Practicum 2, March 4, 2010, 14:00-16:00

0. Boot Linux. Open this file in a web-browser under Linux.

1. Install and test your personall copy of AUTO97:
    (a) copy the file auto.tar in your home directory.
    (b) open Terminal and execute the following commands:

tar -xvf auto.tar
cd auto/97/cmds
source ./auto.env
cd ../src
make

        This would compile AUTO97.

cd ../..
mkdir exp
cd exp
@dm exp

        Examine the files in the directory exp.

make first

        You should get some AUTO97 output on the screen, as well as new files in directory exp. Find and examine them.

2. Install your personal copy of the MATLAB toolbox mplaut [De Feo, 1999] to vew AUTO97 files:
    (a) copy the file mplaut.tar in your home directory.   
    (b) in the Terminal window, execute

tar -xvf mplaut.tar

       The directory mplaut will be created that contains all necessary files.

    (c) start MATLAB by executing

matlab &

     and choose mplaut to be your current directory.

    (d) add the current  directory to MATLAB Path.
    (e) make the directory auto/exp the current MATLAB directory and input the following MATLAB commands:

trj=readq('exp');
for i=1:5
    plot(trj(:,2,i),trj(:,3,i));
    hold on
end;
hold off

      This will produce a solution family.

bif=readp('exp');
plot(bif(1:50,5),bif(1:50,7));

      This should generate a bifurcation diagram with one LP point.

3. Install and your personall copy of SlideCont2.0:
    (a) copy the file slidecont.tar in your home directory.
    (b) execute the following commands in the Terminal window:

tar -xvf slidecont.tar
cd slidecont/cmds
source ./slc.env
cd ../src
make

        This would compile SlideCont2.0.


4. Study the Dry Friction Example:
    (a) examine files in the directory slidecont/examples/dryf; consult - if necessary - Section 6.4 of the User Guide to SlideCont 2.0 by Dercole & Kuznetsov (2005).
    (b) reproduce the results presented in Section 6.1 of the paper by Dercole & Kuznetsov (2005) by exectuting the command

make

     in the directory slidecont/examples/dryf; consult -if necessary - Section 7.1 of the User Guide to SlideCont 2.0.

    (c) reproduce Figs.5,6,7 of that paper using the MATLAB toolbox mplaut.

HOME ASSIGNMENT:

Consider the following planar Filippov system:
x'= x - f(x) y
y'= -m y + e f(x) y
where f(x)=0 if x<R and f(x)=x/(1+h x) if x>R. This is a prey-predator model with a prey refuge [Gause, 1936].

1. Fix m=0.2, e=0.5 and simulate the system for
R=0.2 and h=0.1, 0.5, 0.7
as well as for
R=0.6 and h=0.1, 0.7.
Consider only nonnegative values of x,y.

2. Produce as complete as possible bifurcation diagram of the system in the (h, R)-plane for m=0.2, e=0.5.  Focus on the domain: 0<R<1,0<h<1.
Describe all generic phase portraits and their bifurcations.