Rigorous methods for Delay Differential Equations
This page contains supplementary materials for all publications related to the topic of rigorous numerical methods for Delay Differential Equations, which is an ongoing project that was started with my PhD Thesis. The materials are ordered from newer to older.
(Apparently) unstable periodic orbits in Mackey-Glass equation and symbolic dynamics in delay-perturbed Rossler ODE
In this work we prove existence of three periodic orbits in the Mackey-Glass equation, for which the base periods of the associated poincare map to some section is 1, 2 and 4. The parameters for the Mackey-Glass are the classic ones for which the chaos was numerically observed. Also, we prove the existence of symbolic dynamics in the Rossler system with small perturbation that depends on the past. The perturbation is small in amplitude, but the delay is macroscopic.
Description of the source codes and the source codes can be found here
When refering to this work please consider citing:
Szczelina, R.; Zgliczyński, P.;
High-order Lohner-type algorithm for rigorous computation of Poincare maps in systems of Delay Differential Equations with several delays.,
Submitted,
[arxiv preprint]
(Apparently stable) periodic orbits in Mackey-Glass equation
Description of the source codes and the source codes can be found here
When refering to this work please consider citing:
Szczelina, R.; Zgliczyński, P.;
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation,
Foundations of Computational Mathematics (2018), Vol. 18, Iss 6, Pages 1299--1332,
[Open Access, doi:10.1007/s10208-017-9369-5]
[arxiv preprint]
Scalar DDE from J. Losson, M. C. Mackey, A. Longtin, Chaos 3(1993), No. 2, 167–176
Description of the source codes and the source codes can be found here
When refering to this work please consider citing the following article:
Szczelina, R.
A computer assisted proof of multiple periodic orbits in some first order non-linear delay differential equation,
Electronic Journal of Qualitative Theory of Differential Equations (2016), No. 83, 1-19,
[open-access]
PhD Dissertation
Dissertation can be downloaded from ssdnm project web page, or directly from this link [.pdf].
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.
Description of the source codes and the source codes can be found here This page also contains all the data of the preformance analysis done in the PhD Thesis.