CAP4KAM_nDOF: Computer-Assisted Proofs of existence of KAM tori in Hamiltonian systems with n (>=2) Degrees Of Freedom
Description
In the folder you can produce by uncompressing the attached zipped file (namely, that folder is called "CAP4KAM_nDOF"), you should find everything you need in order to perform a complete computer-assisted proof of existence of invariant tori for a Hamiltonian that satifies three assumptions: (i) it describes a (Hamiltonian) system with n>=2 degrees of freedom and its canonical coordinates are n pairs of action-angle variables; (ii) it is close enough to a Kolmogorov normal form (so fulfilling also both the non-resonance and the non-degeneracy conditions usually adopted in the framework of KAM theory); (iii) its expansion in Taylor series (with respect to the actions) is finite, while its Fourier expansions (in the angles) can be infinite. The software included in the present folder "CAP4KAM_nDOF" is an extension of a first public release that was a sort of supplementary material of the paper [VL], i.e., Locatelli, Ugo (2021), “CAP4KAM2D: Computer-Assisted Proofs For demonstrating the existence of 2-Dimensional KAM tori”, Mendeley Data, V1, doi: 10.17632/jdx22ysh2s.1 The software included in the folder "CAP4KAM_nDOF" is designed to be in a "easy-to-use" layout. Moreover, it is probably not too difficult to be modified for people expert in programming (in C). Everything about the files included in the folder "CAP4KAM_nDOF" is widely described in the README.txt, that contains also careful explanations that should be useful for running the codes, monitoring the results, modifying the input files, etc. REMARK: this is the second release of the software included in the present folder "CAP4KAM_nDOF". With respect to the first release some modifications have been introduced here. None of them has a remarkable impact on the final results apart the correction of a mistake in the expl_transf.c file. Indeed, in the previous version of such a code, a functional norm of the generating function was computed in such a way to be similar to the "max-norm" (for what concerns the dependency on the actions), while the "l1-norm" has to be used instead. This mistake has been removed in the present new release; such a correction induces a worsening effect on the final (rigorous) results that are provided by the Computer Assisted Proofs. The results that are described in the folder "docs" have been updated accordingly. Eventual corrections or remarks about the software package included in the folder "CAP4KAM_nDOF" are more than welcome and can be sent to the author (Ugo Locatelli) at the following e-mail address: locatell@mat.uniroma2.it
Files
Steps to reproduce
1- Uncompress the CAP4KAM_nDOF.zip file 2- Read carefully the README.txt file The software included in the folder "CAP4KAM_nDOF" is designed to be in a "easy-to-use" layout. Moreover, it is probably not too difficult to be modified for people expert in programming (in C). Everything about the files included in the folder "CAP4KAM_nDOF" is widely described in the README.txt, that contains also careful explanations that should be useful for running the codes, monitoring the results, modifying the input files, etc. As a first test, you can execute everything you need in order to complete the whole procedure implementing the computer-assisted proof in the case of a simple application to a simple model which includes two coupled pendula that are also subject to an external forcing. On a Linux/Unix system this should be done by typing two commands in a shell that is positioned on the same directory where the the CAP4KAM_nDOF.zip file has been uncompressed. First, you should digit cd CAP4KAM_nDOF in such a way to move in the folder where all the codes are included; then, just write the command ./Batch Such a Batch file will start the compilation of the codes (by referring also to the Makefile), so as to create the executable files that will be launched. WARNING! Check that Batch is an executable file before trying to launch it. If not, you have to preliminarly digit the command chmod +x Batch