For determining the line characteristic impedance Z ₁ of uniform transmission lines, it is typically necessary to use a calibrated vector network analyzer (VNA), whose calibration reference impedance Z ₂ is known. Using the eight-error term model for modeling a VNA and based on the fact that the values of the error terms are unique and independent of the calibration structures used to determine them, a new method for determining the line characteristic impedance of uniform transmission lines is presented. The main feature of the new method is that it only uses S-parameter measurements of a thru-line and a thru-load combination measured with an uncalibrated VNA.

In the study of dynamic systems, bifurcation diagrams are a very popular graphical tool for studying stability and nonlinear changes in behavior. They are instrumental in analyzing the system’s response to parameter changes. These diagrams show the system’s various dynamic patterns and phase transitions by plotting the relationship between the system response and the parameters. This paper presents a computational algorithm for studying bifurcations in dynamic systems, especially for systems with chaotic behavior that depends on parameter changes. Depending on the type of system to be analyzed, the following two strategies for computing bifurcation diagrams are described: (i) detecting crossing points through the Poincaré plane and (ii) the identification of local maxima (or minima) in one of the system solutions. In addition, this paper presents a method for implementing parallel computation in MATLAB using the Parallel Computing Toolbox from MathWorks, which significantly reduces the computational time required to generate bifurcation diagrams. This work contributes to the study of dynamic systems by providing the reader with accessible tools for studying any dynamic system under established standards and reducing the computational time required for these types of studies by implementing these algorithms utilizing the multi-core processors found in modern computers.

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

We propose a strategy to solve the tracking and regulation problem for a 2DOF underactuated mass-spring-damper system with backlash on the underactuated joint, parametric uncertainties, and partial measurement of the state vector. The design of the controller is divided into two stages; in the first stage, it is assumed that the full state vector and all perturbations in the system are available. The model is divided into one actuated subsystem and one underactuated subsystem. The position of the actuated mass is defined as the control input of the underactuated subsystem, which is designed as an ideal controller that solves the tracking and regulation control problems for the underactuated mass. Finally, the control input of the actuated subsystem is designed to solve the tracking problem considering as a reference signal the control signal of the underactuated subsystem. The second stage solves the problem of the implementation of the previously designed ideal controller using the active disturbances rejection control structure (ADRC). Here state observers estimate the nonmeasured state variables and, at the same time, estimate perturbations and auxiliary signals for their compensation. The performance of the closed-loop system is illustrated by numerical simulations and experimental results.

Complex systems pervade nature and form the core of many technological applications. An exciting feature of these systems is that they exhibit a wide range of temporal behaviors, ranging from collective motion, synchronization, pattern formation, and chaos, among others. This has not only caught the attention of scientists, but also the interest of a wider audience. Consequently, our goal in this work is to provide a simple but descriptive explanation of some concepts related to complex systems. Specifically, the reader embarks on a journey that begins in the 17th century with the discovery of synchronization by Dutch scientist Christiaan Huygens and ends in the chaotic world explored by meteorologist Edward Lorenz around 1963. The journey is filled with examples, including synchronized clocks and metronomes, electronic fireflies that flash harmoniously, and even a chaotic dress.

Fractional calculus (or arbitrary order calculus) refers to the integration and derivative operators of an order different than one and was developed in 1695. They have been widely used to study dynamical systems, especially chaotic systems, as the use of arbitrary-order operators broke the milestone of restricting autonomous continuous systems of order three to obtain chaotic behavior and triggered the study of fractional chaotic systems. In this paper, we study the chaotic behavior in fractional systems in more detail and characterize the geometric variations that the dynamics of the system undergo when using arbitrary-order operators by asking the following question: is the Lyapunov exponent sufficient to describe the dynamical variations in a chaotic system of fractional order? By quantifying the convex envelope generated by the 2D projection of the system into all its phase portraits, the changes in the area of the system, as well as the volume of the attractor, are characterized. The results are compared with standard metrics for the study of chaotic systems, such as the Kaplan–Yorke dimension and the fractal dimension, and we also evaluate the frequency fluctuations in the dynamical response. It is found that our methodology can better describe the changes occurring in the systems, while the traditional dimensions are limited to confirming chaotic behaviors; meanwhile, the frequency spectrum hardly changes. The results deepen the study of fractional-order chaotic systems, contribute to understanding the implications and effects observed in the dynamics of the systems, and provide a reference framework for decision-making when using arbitrary-order operators to model dynamical systems.