# PhD Thesis

This page contains supplementary materials to the PhD Thesis titled Rigorous integration of Delay Differential Equations by Robert Szczelina (me) and supervised by prof. dr hab. Piotr Zgliczyński.

### Pictures

Figure 1. This is the numerical approximation of the supposed Large Amplitude Slowly Oscilatory Periodic (LSOP) solution to the smooth version of the system investigated by Krisztin and Vas. The system is of the form $$x'(t) = -x(t) + f(x(t-1))$$, where $$f(x)$$ is a 5th degree polynomial. More details can be found in the PHD Thesis and in the documentation of the source code.

Figure 2. This is the numerical approximation of the supposed stable periodic solution to the simplified system, where $$f(x)$$ is a 3rd degree polynomial. This solution (and also a stationary solution $$x = 0$$ to the same equation) is used in the PERFORMANCE tests section. More details can be found in the PHD Thesis and in the documentation of the source code.

Figure 3. Approximate eigenbase for the the LSOP solution from Fig. 1.

### Dissertation

An animated presentation (LaTex, Beamer) of the construction of a (p,n)-representation (a basic concept in the thesis) for some exemplary function can be found under this link [.pdf]. The presentation should be viewed in Presentation Mode (usually View -> Presentation or F5 hotkey).

### Source codes

Codes for rigorous integration are implemented as template-based C++ classes and routines. It heavily uses CAPD library, for which source codes can be downloaded here. The programs were tested with CAPD v3.0. Below you may find two versions of the source code - with and without CAPD. The verison with CAPD is relatively simpler to compile and run, but the compilation time is very long. Version without CAPD is for those who have CAPD and do not want to compile it again.

### Performance analysis

PHD Thesis contains the analysis of the performance (in terms of the quality of the interval arithmetics) and several figures related to this issue. Here we present all the data used to test the performance of the DDE rigorous integrator, its dependence on the choosen (p,n)-prepresentation and Lohner set representation.

There are three files related to this: first is the .pdf file containing all the figures together with short descriptions, as in the PhD Thesis. The second one is the .zip file containing all the input files that were used to produce results. The third one is the .zip file that contains all results from the computations. For each test the results are stored inside a separate directory that contain several files. Under this link you can read the description of the input files and output directory.

The files were generated by the program test_1_periodic which is contained in the source codes under programs/test_1_periodic. Despite the name of the test, which is somehow confusing in this case, it run both tests - for periodic and stable solutions. The name of the test will be changed in the future releases of the library.

• All figures from numerical experiments: [.pdf]
• Input files: [.zip]
• Output files: [.zip]
• Documentation on the input and output file formats is here

### Computer assisted proofs

Mackey-Glass equation:

• Description of the source codes can be found here
• Source codes with CAPD library bundled (~27MB) [.zip]
• Source codes without CAPD library (~7MB) [.zip]
Please consider citing arxiv preprint ArXiv:1607.01080 when reffering to this work.