arXiv subjects

Jacek Kubica

Publications and source records attributed to Jacek Kubica.

4 recordsLinked to original sources

arXiv2024

Pitchfork bifurcation and heteroclinic connections in the Kuramoto--Sivashinsky PDE

We present a method for the complete analysis of the dynamics of dissipative Partial Differential Equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto--Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation.

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arXiv2020

Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces

We generalize and extend the Conley-Morse-Forman theory for combinatorial multivector fields introduced in \cite{Mr2017}. The generalization consists in dropping the restrictive assumption in \cite{Mr2017} that every multivector has a unique maximal element. The extension is from the setting of Lefschetz complexes to the more general situation of finite topological spaces. We define isolated invariant sets, isolating neighbourhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities.

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arXiv2019

Lefschetz Complexes as Finite Topological Spaces

We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.

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arXiv2018

Persistent Homology of Morse Decompositions in Combinatorial Dynamics

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics directly from the sample. We study the homological persistence of {\em Morse decompositions} of such systems, an important descriptor of the dynamics, as a tool for validating the reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to "vector cloud" data. We present experimental results on two numerical examples.

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