An installation instruction of the program realizing the proof of the existence of uniformly hyperbolic attractor in the Kuznetsov system as described in the paper

D. Wilczak,

Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system

MINIMAL SYSTEM REQUIREMENTS:

MINIMAL RESOURCES:

After successful compilation we have two executables - solenoid and sw-solenoid. These prove the existence of uniformly hyperbolic attractor for the Smale map and for the Poincaré map in the Kuznetsov system, respectively.

COMPILATION:

It is enough to extract the archive to an empty directory, and invoke 'make' command.
  1. mkdir swhyp
  2. cp swhyp.tgz swhyp
  3. cd swhyp
  4. tar xvfz swhyp.tgz
  5. make
The last command generates two executables solenoid and sw-solenoid. The compiler and version of OS (32/64 bit and Linux/windows) are automatically recognized by the program and makefile. You do not need to take care about special compilation rules under a given platform.

RUN THE PROGRAMS:

We may run the programs
./solenoid
or
./sw-solenoid
Both programs execute in the text mode hence the graphics environment is unnecessary. We may redirect output to a given file, like
./sw-solenoid > logFile &

IMPORTANT INFORMATION ABOUT THE SW-SOLENOID PROGRAM:

The main version of the sw-solenoid program were prepared, tested and run on a cluster with 7 computers. Since the architectures of clusters are often different, it is difficult to post a program that would run on any platform. Here we present a version of the program that can be run on any single machine. It is, however, strongly recommended to run the program on a remote fast computer with as many as possible CPUs.

The program automatically recognizes the number of available CPUs and tries to use all of them. We tested the program on a single computer with 64GB of RAM and 32 CPUs (8 Quad Core). The program completed all steps of verification of uniform hyperbolicity within 2 days on this machine.

As it is mentioned in the article, the result of the Algorithm 2 for spreading of coordinate systems from periodic points is not deterministic when run in parallel mode. I can imagine that the cone condition may be not verified with computed coordinate systems. The file that contains coordinate system for which we successfully completed verification of cone condition is available from the author on request. This is a huge text file (almost 7 GB before compression). Then, the verification of the cone condition can be restarted using coordinates from this file - it is enough to comment out few lines in the file hyperbolicity/sw-solenoid.cpp.


OUTPUT:

Both programs write some log messages to the screen during the computations. Usually these are to see the progress of the computation. The most important information is at the end - the number of boxes on which the cone condition was not verified. These boxes are written to the file unverified0.

During the computations the programs save to disc enclosure, cycles, periodic points and coordinate systems. These are text files and can be used for further post-processing, if necessary.


Directory structure and source files of the programs:

Each routine in the source files is usually widely commented. Here we list the main content of each file that contains implementation of generic algorithms. The following files are specific for the Smale map and the Poincare map in the Kuznetsov system.

NOTE:

More information about libraries can be found on the web page of CAPD group: http://capd.ii.uj.edu.pl.