arXiv subjects

Michał Palczewski

Publications and source records attributed to Michał Palczewski.

3 recordsLinked to original sources

arXiv2026

Topological-numerical analysis of global dynamics in the discrete-time two-gene Andrecut-Kauffman model

We conduct a topological--numerical analysis of global dynamics in a discrete-time two-gene Andrecut--Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use a numerical method to construct Morse decomposition of the system across a wide range of parameters at a fixed finite resolution in the state space and in the parameter space, both spaces split into uniform rectangular grids (a technique also called "pixelation"). We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We represent the Morse decomposition and connecting orbits by a directed acyclic graph. We introduce pictograms to convey the information provided by the Conley index in an easy to understand schematic way. We show and analyze bifurcations captured using this technique. We call this method CMAD for short (Conley-Morse graphs for the Analysis of Dynamics). The main advantage of our method is that it finds isolating neighborhoods of both stable and unstable invariant sets and that it provides validated (rigorous) numerical results: we actually obtain computer assisted proof that the constructed sets are indeed isolating neighborhoods of Morse sets in a certain Morse decomposition of the system. In particular, this means that we have captured all the interesting dynamics within the analyzed range of the phase space perceived at the given finite resolution. We also conduct numerical simulations aimed at showing the location of attractors in the isolating neighborhoods found. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics at finite (coarse) resolution in an applied dynamical system depending on a few parameters, like the gene regulatory model that we analyze.

math.DS↗

arXiv2026

Rigorous computation of expansion in one-dimensional dynamics

We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted proof). The method uses efficient graph algorithms and an iterative approach for optimal performance. A software implementation of the method is made publicly available. This is an example of a quantitative result in the theory of dynamical systems, as opposed to many qualitative results whose assumptions may be difficult to verify and the conclusions may have limited use in practical models that describe natural phenomena. We discuss and illustrate the effectiveness of our method and apply it to the quadratic map family.

math.DS↗

arXiv2025

Bistability and chaos in the discrete two-gene Andrecut-Kauffman model

We conduct numerical analysis of the 2-dimensional discrete-time gene expression model originally introduced by Andrecut and Kauffman (Phys. Lett. A 367: 281-287, 2007). In contrast to the previous studies, we analyze the dynamics with different reaction rates $α_1$ and $α_2$ for each of the two genes under consideration. We explore bifurcation diagrams for the model with $α_1$ varying in a wide range and $α_2$ fixed. We detect chaotic dynamics by means of the positive maximum Lyapunov exponent and we scan through selected parameters to detect those combinations for which chaotic dynamics can be found in the model. Moreover, we find bistability in the model, that is, the existence of two disjoint attractors. Both situations are interesting from the point of view of applications, as they imply unpredictability of the dynamics encountered. Finally, we show some specific values of parameters of the model in which the two attractors are of different kind (a periodic orbit and a chaotic attractor) or of the same kind (two periodic orbits or two chaotic attractors).

nlin.CD↗