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This page contains a list of publications whose development was conducted within the framework of the project.
1. Paweł Pilarczyk, Pedro Real. Computation of cubical homology, cohomology, and (co)homological operations via chain contraction. Submitted.
In this paper, the authors introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique of the construction of a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre's diagonalization for cubical cells. A preprint of this paper, an implementation in C++ of the introduced algorithms, as well as some examples that illustrate this approach, are available at the ChainCon website.
2. Paweł Pilarczyk. Set-oriented numerical analysis of difference equations with parameters. Proceedings of the International Workshop Future Directions in Difference Equations (June 13-17, 2011, Vigo, Spain), E. Liz and V. Manosa Eds., Servicio de Publicacions da Universidade de Vigo, 2011, pp. 149-156. [full text]
This paper introduces an automatic method for the analysis
of dynamical systems that is an excellent ground for applications
of the theory and code developed in the framework of the project.
The purpose of this paper is to provide a possibly simple
introduction to the set-oriented algorithmic method
for determining the global dynamics of a system
of first-order ordinary difference equations with parameters,
introduced in the paper
A database schema for the analysis of global dynamics
of multiparameter systems (SIADS 2009,
Vol. 8, No. 3, pp. 757-789).
An emphasis is made on the details of input data
and on the assumptions required by the method,
as well as on the interpretation of the results of the computation.
The method is illustrated with an application
to a simple overcompensatory two-stage population model
related to modeling baleen whale populations.
3. Eduardo Liz, Paweł Pilarczyk. Global dynamics in a stage-structured discrete-time population model with harvesting. J. Theoret. Biol., Vol. 297 (2012), 148-165. DOI: 10.1016/j.jtbi.2011.12.012.
This paper constitutes an application of an automatic method
for the analysis of dynamical systems that uses parts of theory and code
developed in the framework of the project.
The purpose of this paper is to analyze the effect
of constant effort harvesting
upon global dynamics of a discrete-time population model
with juvenile and adult stages.
We consider different scenarios, including adult-only mortality,
juvenile-only mortality, and equal mortality of juveniles and adults.
In addition to analytical study of equilibria of the system,
we analyze global dynamics by means of an automated
set-oriented rigorous numerical method.
We obtain a comprehensive overview of the dynamics
as the harvest rate and survival probability change.
In particular, we determine the range of parameters
for which the population abundance gets larger
in spite of an increase in the harvest rate (so-called hydra effect),
and for which subsequent increases in harvesting effort
can magnify fluctuations in population abundance (destabilize it)
and then stabilize it again (so-called bubble effect).
4. Paweł Pilarczyk, Luis García, Benjamin A. Carreras, Irene Llerena. A dynamical model for plasma confinement transitions. J. Phys. A: Math. Theor., Vol. 45, No. 12 (2012), 125502. DOI: 10.1088/1751-8113/45/12/125502.
This paper constitutes an application of an automatic method
for the analysis of dynamical systems that uses parts of theory and code
developed in the framework of the project.
A three equation model describing the evolution of the turbulence
level, averaged shear flow and sheared zonal flow is analyzed
using topological properties of the asymptotic solutions.
An exploration in parameter space is done identifying the
attractor sets, which are fixed points and limit cycles. Then
a more detailed analysis of all Morse sets is conducted using
topological-combinatorial computations. This model allows the
description of different types of transition to improved plasma
confinement regimes.
5. Justin Bush, Marcio Gameiro, Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Ippei Obayashi, Paweł Pilarczyk. Combinatorial-Topological Framework for the Analysis of Global Dynamics, Chaos, Vol. 22, No. 4 (2012), 047508. DOI: 10.1063/1.4767672.
This paper describes a computational framework
for an automatic method for the analysis of dynamical systems
that is an excellent ground for applications of the theory and code
developed in the framework of the project.
We discuss an algorithmic framework based on efficient graph algorithms
and algebraic-topological computational tools. The framework is aimed
at automatic computation of a database of global dynamics
of a given m-parameter semidynamical system with discrete time
on a bounded subset of the n-dimensional phase space.
We introduce the mathematical background,
which is based upon Conley's topological approach to dynamics,
describe the algorithms for the analysis of the dynamics
using rectangular grids both in phase space and parameter space,
and show two sample applications.
[More publications to come in the nearest future.]
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