Aledo, J. A., Goles, E., Montalva-Medel, M., Montealegre, P., & Valverde, J. C. (2023). Symmetrizable Boolean networks. Inf. Sci., 626, 787–804.
Abstract: In this work, we provide a procedure that allows us to transform certain kinds of deterministic Boolean networks on minterm or maxterm functions into symmetric ones, so inferring that such symmetrizable networks can present only periodic points of periods 1 or 2. In particular, we deal with generalized parallel (or synchronous) dynamical systems (GPDS) over undirected graphs, i. e., discrete parallel dynamical systems over undirected graphs where some of the self-loops may not appear. We also study the class of anti-symmetric GPDS (which are non-symmetrizable), proving that their periodic orbits have period 4. In addition, we introduce a class of non-symmetrizable systems which admit periodic orbits with arbitrary large periods.
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Goles, E., Montalva-Medel, M., Montealegre, P., & Rios-Wilson, M. (2022). On the complexity of generalized Q2R automaton. Adv. Appl. Math., 138, 102355.
Abstract: We study the dynamic and complexity of the generalized Q2R automaton. We show the existence of non-polynomial cycles as well as its capability to simulate with the synchronous update the classical version of the automaton updated under a block sequential update scheme. Furthermore, we show that the decision problem consisting in determine if a given node in the network changes its state is P-Hard.
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Goles, E., Montealegre, P., & Vera, J. (2016). Naming Game Automata Networks. J. Cell. Autom., 11(5-6), 497–521.
Abstract: In this paper we introduce automata networks to model some features of the emergence of a vocabulary related with the naming game model. We study the dynamical behaviour (attractors and convergence) of extremal and majority local functions.
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Vera-Damian, Y., Vidal, C., & Gonzalez-Olivares, E. (2019). Dynamics and bifurcations of a modified Leslie-Gower-type model considering a Beddington-DeAngelis functional response. Math. Meth. Appl. Sci., 42(9), 3179–3210.
Abstract: In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie-Gower model, considering a Beddington-DeAngelis functional response. It generates a complex dynamics of the predator-prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov-Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.
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