An installation instruction of the program realizing the proof of the existence periodic orbits
for Kuramoto-Shivashinky PDE on the line with odd and periodic boundary conditions as described in paper
P. Zgliczyński
"Rigorous Numerics for Dissipative PDEs III. An
effective algorithm for rigorous integration of dissipative PDEs"
IMPORTANT:
The program
should complie and run under windows or linux with 32b or 64b Intel or AMD processors.
The program requires gcc-3.4 or newer (we tested it on gcc-3.4.2(windows), gcc-4.2.1(linux) and gcc-4.4.3(linux)). Earlier versions of gcc are not supported.
NO GRAPHICS:
The program must be run in the text mode.
Installation instruction under linux
- please unpack the archive
unzip --a ksper.zip
-
change directory to 'ks' and call 'make'.
cd ks
make
The last command generates an executable file ksper in the current directory.
-
We may run the program
./ksper
Installation instruction under MS Windows with MinGW compiler:
The program has been tested under MS Windows XP Professional with gcc 3.4.4 compiler.
We use the MinGW compiler (Minimalist GNU for Windows - http://www.mingw.org).
- please unpack the archive ksper.zip (using for example Total Commander)
-
change directory to 'ks' and call 'make'.
cd ks
make
The last command generates an executable file ksper.exe in the current directory.
-
We may run the program
ksper
STRUCTURE OF THE DIRECTIONS:
- ks - source code, the location of the executable file
- data/temp - here the reports files will be placed:
- ksihp.txt - for attracting periodic orbits
- ksunstb.txt - for unstable periodic orbits
- data/ks - contains initial data for the proofs. It must contain file orbit.txt
- obj - the directory to store the object files from the compilation
- capd - the files from the CAPD library (version from April 2010)
Initial data for the proofs:
In the directory data/ks there must be the file "orbit.txt"
which contains the an approximate orbit used to guess the the initial tail.
The structure of the file is:
- first line: the dimension
- next lines: each line contains points from the orbit, consequitive coordinates separated by
one space or more
We generated several files containing orbits:
- orbit02991-10.txt - for mu=0.02991 +- e-10, symmetric unstable orbit
- orbit02991.txt - for mu=0.02991 , symmetric unstable orbit
- orbit032.txt - mu=0.032, symmetric stable orbit
- orbit1212.txt - mu=0.1212, symmetric unstable orbit
- orbit1215.txt - mu=0.1215, symmetric unstable orbit
- orbit1215atr.txt - mu=0.1215, non-symmetric attracting orbit
- orbit125.txt - mu=0.125, symmetric attracting orbit, complex eigenvalues
- orbit127.txt - mu=0.127, symmetric attracting orbit,
Hence for example to reproduce the proof for mu=0.127 it is recomended to enter the directory
"data/ks" and to copy the file "orbit127.txt" to "orbit.txt"
What can go wrong with initial data
In case when the orbit file "orbit.txt" is not the one correspoding to the mu selected, then this does not need to result in the failure to complete the proof. Basically it should work when the value of mu does not differ to much from
the one for the orbit. Usually more iterates in the proof will be needed.
For the attracting periodic points case the program
looks for an attracting orbit by allowing to furter evolve for the point from the file "orbit.txt" until it settles onto the attracting periodic orbit. In this situation, when wrong "orbit.txt" is used, it can happen that
our initial condition is not in the basin of attraction of a periodic orbit, but rather of some fixed point. This will result usually in the impossibility
For unstable periodic point - the program ask for initial condition from the list of several points,
independently from the current content of the file "orbit.txt" - which is used as the guess for the initial
tail only. It can happen that the approximate periodic point cannot be found by the program.
Intermediate messages during the proof
The most important ones about the ratios (quotients):
diameter(P(N)[i])/diameter(N)[i], i=1,..,m.
For stable (apparently attracting) orbits we want all these ratio to be less than 1 .
For unstable orbits we want just one ratio to be bigger than 1 (we are after orbits with just
one unstable direction)
Output:
It is located in the directory
data/temp
- ksihp.txt - for attracting periodic orbits
- ksunstb.txt - for unstable periodic orbits
NOTE:
More informations about CAPD libraries can be found on the web page of CAPD group:
http://capd.ii.uj.edu.pl.