Cisternas, J., & Concha, A. (2024). Searching nontrivial magnetic equilibria using the deflated Newton method. Chaos Solitons Fractals, 179, 114468.
Abstract: Nonlinear systems that model physical experiments often have many equilibrium configurations, and the number of these static solutions grows with the number of degrees of freedom and the presence of symmetries. It is impossible to know a priori how many equilibria exist and which ones are stable or relevant, therefore from the modeler's perspective, an exhaustive search and symmetry classification in the space of solutions are necessary. With this purpose in mind, the method of deflation (introduced by Farrell as a modification of the classic Newton iterative method) offers a systematic way of finding every possible solution of a set of equations. In this contribution we apply deflated Newton and deflated continuation methods to a model of macroscopic magnetic rotors, and find hundreds of new equilibria that can be classified according to their symmetry. We assess the benefits and limitations of the method for finding branches of solutions in the presence of a symmetry group, and explore the high dimensional basins of attraction of the method in selected 2 dimensional sections, illustrating the effect of deflation on the convergence.

Goles, E., Medina, P., Montealegre, P., & Santivanez, J. (2022). Majority networks and consensus dynamics. Chaos Solitons Fractals, 164, 112697.
Abstract: Consensus is an emergent property of many complex systems, considering this as an absolute majority phenomenon. In this work we study consensus dynamics in grids (in silicon), where individuals (the vertices) with two possible opinions (binary states) interact with the eight nearest neighbors (Moore’s neighborhood). Dynamics emerge once the majority rule drives the evolution of the system. In this work, we fully characterize the subneighborhoods on which the consensus may be reached or not. Given this, we study the quality of the consensus proposing two new measures, namely effectiveness and efficiency. We characterize attraction basins through the energylike and magnetizationlike functions similar to the Ising spin model.

MontalvaMedel, M., Rica, S., & Urbina, F. (2020). Phase space classification of an Ising cellular automaton: The Q2R model. Chaos Solitons Fractals, 133, 14 pp.
Abstract: An exact classification of the different dynamical behaviors that exhibits the phase space of a reversible and conservative cellular automaton, the socalled Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a cycle. In this work, we classify them in four types accordingly to well differentiated topological characteristics. Three of them which we call of type SI, SII, and SIII share a symmetry property, while the fourth, which we call of type AS does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous types of cycles. Moreover, at a combinatorial level, we can determine the number of cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space and, in addition, we realize an exhaustive study of a small Ising system (4 x 4) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation. (C) 2020 Elsevier Ltd. All rights reserved.

Ruivo, E. L. P., MontalvaMedel, M., de Oliveira, P. P. B., & Perrot, K. (2018). Characterisation of the elementary cellular automata in terms of their maximum sensitivity to all possible asynchronous updates. Chaos Solitons Fractals, 113, 209–220.
Abstract: Cellular automata are fullydiscrete dynamical systems with global behaviour depending upon their locally specified state transitions. They have been extensively studied as models of complex systems as well as objects of mathematical and computational interest. Classically, the local rule of a cellular automaton is iterated synchronously over the entire configuration. However, the question of how asynchronous updates change the behaviour of a cellular automaton has become a major issue in recent years. Here, we analyse the elementary cellular automata rule space in terms of how many different onestep trajectories a rule would entail when taking into account all possible deterministic ways of updating the rule, for one time step, over all possible initial configurations. More precisely, we provide a characterisation of the elementary cellular automata, by means of their onestep maximum sensitivity to all possible update schedules, that is, the property that any change in the update schedule causes the rule's onestep trajectories also to change after one iteration. Although the onestep maximum sensitivity does not imply that the remainder of the timeevolutions will be distinct, it is a necessary condition for that. (C) 2018 Elsevier Ltd. All rights reserved.
