CAP4KAM2D: Computer-Assisted Proofs For demonstrating the existence of 2-Dimensional KAM tori
Description
In the folder you can produce by uncompressing the attached zipped file (namely, that folder is called "CAP4KAM"), 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 two degrees of freedom; (ii) it is close enough to a Kolmogorov normal form (so fulfilling also both the non-resonance and non-degeneracy conditions usually adopted in the framework of KAM theory); (iii) its expansion in Taylor-Fourier series (in its action-angle canonical coordinates) is finite. The software included in the folder "CAP4KAM" refers to the following paper: [VL] Valvo L., Locatelli U.: "Hamiltonian Control of Magnetic Field Lines: Computer Assisted Results Proving the Existence of KAM Barriers", preprint (2021); available both at the web address https://arxiv.org/abs/2101.07785 and in a pdf copy you can find in the attachments. That work should not be too difficult to read for people working in the field of Hamiltonian perturbation theory and, in particular, it should clarify the three assumptions listed above. The software included in the folder "CAP4KAM" 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" is widely described in the README.txt, that contains also a careful explanations that is useful for running the codes, monitoring the results, modifying the input files, etc. Eventual corrections or remarks about the software package included in the folder "CAP4KAM" 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.zip file 2- Read carefully the README.txt file Everything about the files included in the folder "CAP4KAM" is widely described in the README.txt, that contains also a careful explanations that is 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 forced pendulum model. 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.zip file has been uncompressed. First, you should digit cd CAP4KAM 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