symbolic dynamics and abundance of (symmetric) homoclinic and heteroclinic connections in the Michelson system

as described in two papers by D. Wilczak

Journal of Differential Equations, Vol.194, 433-459, (2003)

SIAM Journal on Applied Dynamical Systems, Vol.4, No.3, 489-514, (2005)

The original programs written in 2003 and 2005 were very difficult to maintain nowadays. Since then we developed far more advanced computational techniques that make it possible shortening and simplifying algorithms. Therefore in 2015 I decided to completely rewrite the program.

Data for the proof (i.e. coordinates of triple sets) remain the same as given in the papers. We changed, however, the algorithms that check covering relations and hyperbolicity. Here is list of major changes

- Taylor method of order 20 with automatic step control is used. Therefore all tables in the articles with data related to time steps and orders are out of date.
- Automatic subdivision of edges is implemented. Therefore all tables in the articles with data on uniform subdivisions of edges are out of date.
- The cone condition that guaranties convergence to Lyapunov orbits is implemented as proposed in the article

H. Kokubu, D. Wilczak and P. Zgliczynski,*Rigorous verification of cocoon bifurcations in the Michelson system,*Nonlinearity 20, No.9, 2147-2174, (2007).

The present version of the program is written in C++11. It has been tested under gcc-5.2 compiler.

The program uses the CAPD library as a main numerical tool for rigorous computation of Poincare map and its derivative. It has been tested with CAPD v.4.2.89 version which can be downloaded from here: CAPD. There is also a short tutorial on how the library should be compiled.

The program that checks all required covering relations and hyperbolicity of the Lyapunov orbits can be compiled by the following call from the command line.

- extract the archive:
**tar xfvz ks.tgz** - enter program directory:
**cd ks** - compile the program:
**make CAPD=relativeOrAbsolutePathToCAPD_BIN/**

make CAPD=/home/daniel/capd/bin/ make CAPD=/usr/local/bin/ make

If the CAPD/bin directory is on system search path then just **make** can be called as shown in the last example.

**Warning:** Do not forget last **slash** character at the end of the path.

./main

The program runs less than 1 second on my laptop and writes to the screen logs from verification of the covering relations. Moreover, two text files **H1.dat** and **H2.dat** are created.

They contain bounds on the derivative of Poincare map computed on the sets H1 and H2, respectively. The derivatives are also given in the coordinates of triple sets H1, H2. Then all computations needed for verification of the cone condition are given.

Daniel Wilczak

December 16, 2015