in the Planar Restricted Circular Three Body Problem (PCR3BP) as described in two papers by D. Wilczak and P. Zgliczyński

Communications in Mathematical Physics, Vol.234, No.1, 37-75, (2003).

Communications in Mathematical Physics, Vol. 259, No.3, 561-576, (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 Therefore all tables in the articles with data related to time steps and orders are out of date.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 Therefore all tables in the articles with data on uniform subdivisions of edges are out of date.
- All covering relations are checked by forward computations. Therefore we do not need to check monotonicity conditions.
- 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 pcr3bp.tgz** - enter program directory:
**cd pcr3bp** - 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 circa 37 seconds 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