Francisco Balibrea

This article focuses on the study of local truncation errors (LTEs) in the Network Simulation Method (NSM), specifically when using the trapezoidal method and Gear’s methods. The NSM, which represents differential equations through electrical circuit elements, offers advantages in solving nonlinear dynamic systems such as the van der Pol equation. The analysis compares the performance of these numerical methods in terms of their stability and error minimization, with particular emphasis on LTE. By leveraging circuit-based techniques prior to numerical application, the NSM improves convergence. This study evaluates the impact of step size on LTE and highlights the trade-offs between accuracy and computational cost when using the trapezoidal and Gear methods.

In this paper we solve two open problems concerning distributional chaos in non-autonomous discrete dynamical systems stated in [4] and [17]. In the first problem it is wondered if the limit function of pointwise convergent non-autonomous system with positive topological entropy is DC2. We show that the answer to this problem depends on the given metric and can be both, positive or negative. In the second open problem it is wondered if to be DC1 is a generic property of pointwise convergent non-autonomous systems. We prove that the answer is negative for convergent systems on the Cantor set. Concerning interval systems, we show that DC1 chaotic systems form dense, but not open (nor closed) set in the space of non-autonomous convergent systems on the interval, independently of the metric we use.

Dry friction cannot be understood on a macroscopic scale without knowing what happens at the contact of sliding surfaces on an atomic scale. Tests on this scale are very expensive and very sensitive to the effects of contamination or inaccurate fittings. On the other hand, the sample dimensions are small because of the requirements of the test equipment, which makes it difficult to generalise the conclusions drawn. This work reviews the models used to analyse friction processes, and proposes the application of one of the models, the Frenkel–Kontorova–Tomlinson (FKT) model, to study the dry frictional behaviour of doped surfaces. The study shows that for concreted types of doped pattern, the behaviour can change from chaotic to periodic depending on the stiffness, which in turn are associated with temperature.

In this paper, we deal with some applications of the network simulation method (NMS) to the non-linear differential equations derived of a parametric family associated to stated problems by Newton in and others like the parabolic mirror and van der Pol non-linear equation. We underly the efficientcy of the (NMS) method, compare it with Matlab procedures and present figures of solutions of the equations obtained by it on the mentioned problems. Additionally, we introduce also the electric-electronic circuits we have designed to be able of obtaining the solutions of the referred equations.

The Frenkel-Kontorova-Tomlinson (FKT) model represents mechanical systems in which the atomic smooth surfaces of two bodies slide against each other. The model is very sensitive to changes of the system parameters, and ranges from simple stable harmonic to chaotic solutions. The design of the model between two bodies for the dynamic problem, following the network method rules, is explained with precision and run on standard electrical circuit simulation software. It provides the phase diagrams of atom displacement for each atom and the total friction force by the summation of all the atom displacements. This article is focused on studying the effect of the selected time step on the result and in the lack of sensitivity of Lyapunov exponents to assess chaotic behaviour.