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1

A Neimark-Sacker bifurcation in a discrete SIS model

Choiński Marcin

Mathematica Applicanda

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2023

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Vol. 51, no. 1

109--120

EN

In this paper, we analyze a possibility of occurrence of a Neimark-Sacker bifurcation in a two-dimensional SIS discrete-time model. As a discretization method, we applied the Explicit Euler Scheme. We choose a step size of discretization method as a bifurcation parameter, what is not a typical approach. We phrase conditions giving the bifurcation appearance depending on the step size. Firstly, we determine terms on the step size enabling the eigenvalues of Jacobian matrix for the endemic stationary state of the system being complex and having absolute value equal to 1. Then we use the Center Manifold Theorem in order to exclude values of step size which disable the occurrence of bifurcation. We accomplish our results with numerical simulations.

PL

W artykule zbadano możliwość wystąpienia bifurkacji Neimerka–Sackera (BNS) w dyskretnym dwuwymiarowym modelu SIS. W celu dyskretyzacji modelu ciągłego zastosowano jawny schemat Eulera. Jako parametr bifurkacyjny wybrano długość kroku dyskretyzacji, co nie jest standardowym podejściem. Sformułowaliśmy warunki wystąpienia bifurkacji w zależności od długości kroku. Najpierw sprawdzono, dla jakich warunków wartości własne macierzy Jacobiego dla endemicznego stanu stacjonarnego są zespolone oraz ich moduł wynosi 1. Następnie zastosowano twierdzenie o rozmaitości centralnej w celu wykluczenia tych wartości kroku dyskretyzacji, dla których BNS nie występuje. Rozważania teoretyczne są uzupełnione symulacjami numerycznymi.

2

A Dynamical System Approach to Polyominoes Generation

Massazza Paolo

Fundamenta Informaticae

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2021

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Vol. 180, nr 3

251--273

EN

We describe a method which exploits discrete dynamical systems to generate suitable classes of polyominoes. We apply the method to design an algorithm that uses O(n) space to generate in constant amortized time all polyominoes corresponding to hole-free partially directed animals consisting of n sites on the square grid. By implementing the algorithm in C++ we have obtained a new sequence that does not appear in the On-Line Encyclopedia of Integer Sequences.

3

Simple discrete SIS criss-cross model of tuberculosis in heterogeneous population of homeless and non-homeless people

Bodzioch Mariusz, Choiński Marcin

Mathematica Applicanda

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2019

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Vol. 47, no. 1

103--115

EN

In this paper we propose a discrete criss-cross model of tuberculosis (TB) transmission in a heterogeneous population, which consists of two different subpopulations: homeless and non-homeless people. This criss-cross model is based on the simple continuous SIS model with bilinear transmission function and constant inflow into both subpopulations considered previously by us. We make preliminary stability analysis. We show that to control the spread of the infectious disease in a heterogeneous population it is not enough to consider the dynamics of the disease in each subpopulation separately. This result is consistent with the result for continuous model. We also fit the model to epidemic data from Warmian-Masurian Province of Poland.

PL

Zaproponowany został dyskretny krzyżowy model rozprzestrzeniania się gruźlicy w niejednorodnej populacji składającej się z bezdomnych i niebezdomnych. Model ten oparty jest na prostym modelu typu SIS z dwuliniową funkcją transmisji i stałym napływem w obu populacjach. Przeprowadzona została wstępna analiza stabilności stanów stacjonarnych. Pokazano, że aby kontrolować rozprzestrzenianie się choroby zakaźnej w niejednorodnej populacji nie jest wystarczające rozważanie dynamiki choroby w każdej podpopulacji oddzielnie. Parametry modelu zostały dopasowane do danych z województwa warmińsko-mazurskiego.

4

A New Sandpile Model with Smoothness Assumptions

Brocchi S., Massazza P.

Fundamenta Informaticae

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2016

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Vol. 143, nr 3/4

261--286

EN

We consider two new granular dynamical systems obtained from the Sand Pile Model SPM(n) by adding a smoothness condition. First, we define the Smooth Sand Pile Model SmSPM(n) and we provide a characterization of the reachable states, together with some interesting properties of the resulting lattice. Then we extend it to SmSPM*(n), a related dynamical system with a more complex lattice structure.

5

The Lattice Structure of Equally Extended Signed Partitions : A generalization of the Brylawski approach to integer partitions with a p-n junction between two semiconductors model

Cattaneo G., Chiaselotti G., Gentile T., Oliverio P. A.

Fundamenta Informaticae

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2015

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Vol. 141, nr 1

1--36

EN

Signed partitions are used in order to describe a new discrete dynamical model whose configurations have fixed sum and whose evolution rules act in balancing from left and right on the configurations of the system. The resulting model can be considered as an extension to the case of signed partitions of the discrete dynamical system introduced by Brylawski in his classical paper concerning the dominance order of integer partitions. We provide a possible interpretation of our model as a simplified description of p − n junction between two semiconductors. We also show as our model can be embedded in a specific Brylawski dynamical system by means of the introduction of a new evolution rule.

6

Computational Complexity of Avalanches in the Kadanoff Sandpile Model

Formenti E., Goles E., Martin B.

Fundamenta Informaticae

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2012

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Vol. 115, nr 1

107-124

EN

This paper investigates the avalanche problem AP for the Kadanoff sandpile model (KSPM). We prove that (a slight restriction of) AP is in NC1 in dimension one, leaving the general case open. Moreover, we prove that AP is P-complete in dimension two. The proof of this latter result is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with an initial sand distribution in KSPM. These results are also related to the known prediction problem for sandpiles which is in NC1 for one-dimensional sandpiles and P-complete for dimension 3 or higher. The computational complexity of the prediction problem remains open for the Bak’s model of two-dimensional sandpiles.

7

On the Dynamics of Social Balance on General Networks (with an application to XOR-SAT)

Istrate G.

Fundamenta Informaticae

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2009

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Vol. 91, nr 2

341-356

EN

We study nondeterministic and probabilistic versions of a discrete dynamical system (due to T. Antal, P. L. Krapivsky, and S. Redner [3]) inspired by Heider’s social balance theory. We investigate the convergence time of this dynamics on several classes of graphs. Our contributions include: 1. We point out the connection between the triad dynamics and a generalization of annihilating walks to hypergraphs. In particular, this connection allows us to completely characterize the recurrent states in graphs where each edge belongs to at most two triangles. 2. We also solve the case of hypergraphs that do not contain edges consisting of one or two vertices. 3. We show that on the so-called "triadic cycle"graph, the convergence time is linear. 4. We obtain a cubic upper bound on the convergence time on 2-regular triadic simplexes G. This bound can be further improved to a quantity that depends on the Cheeger constant of G. In particular this provides some rigorous counterparts to experimental observations in [25]. We also point out an application to the analysis of the random walk algorithm on certain instances of the 3-XOR-SAT problem.

8

Advances in Symmetric Sandpiles

Formenti E., Masson B., Pisokas T.

Fundamenta Informaticae

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2007

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Vol. 76, nr 1-2

91-112

EN

A symmetric version of the well-known SPM model for sandpiles is introduced. We prove that the new model has fixed-point dynamics. Although there might be several fixed points, a precise description of the fixed points is given. Moreover, we provide a simple closed formula for counting the number of fixed points originated by initial conditions made of a single column of grains. Bounds for the transient length are also given.