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In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

For Hilfer derivatives of the product of two functions, we present equations and inequalities, generalizing well‐known results for Caputo and Riemann‐Liouville derivatives. Using the Laplace transformation, we introduce a generalized distributed Mittag‐Leffler‐Hilfer stability and show two results for like‐Lyapunov stability. We also extend equations and inequalities for the product of two functions of Hilfer derivatives of distributed order. Finally, we give some consequences and examples that illustrate the theory.

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M-derivative for the Mittag–Leffler function. In the exact solutions derived in this manuscript, two new parameters play an important role in the convergence towards, or the divergence from, the steady state of capital and per capita product. The economical dynamics of these nations are influenced by the intensity of the capital and labor factors, as well as the level of depreciation, the labor force rate and the level of saving.

The pandemic caused by the SARS-CoV-2 virus spreads more rapidly in densely populated areas. The number of confirmed cases is counted by the millions in some countries, such as USA, Brazil, and Mexico. These three countries also report the world’s highest cumulative death tolls caused by the disease as of February 2021. In this study, a comparative analysis of urban development, economic level, and the number of COVID-19 cases in Mexico City, is conducted. Mexico City, the capital city of Mexico, is among the most densely populated metropolitan areas and one of the largest financial centers in the continent. Among the sixteen municipalities, in which Mexico City is divided, there exist enormous economic and urban development gaps. Based in a comparability index (CI), this study found a correlation between the number of confirmed cases of the COVID-19 disease with the population density, the per capita income, and the dwelling occupancy index in each municipality.

Car-following is an approach to understand traffic behavior restricted to pairs of cars, identifying a “leader” moving in front of a “follower”, which at the same time, it is assumed that it does not surpass to the first one. From the first attempts to formulate the way in which individual cars are affected in a road through these models, linear differential equations were suggested by author like Pipes or Helly. These expressions represent such phenomena quite well, even though they have been overcome by other more recent and accurate models. However, in this paper, we show that those early formulations have some properties that are not fully reported, presenting the different ways in which they can be expressed, and analyzing them in their stability behaviors. Pipes’ model can be extended to what it is known as Helly’s model, which is viewed as a more precise model to emulate this microscopic approach to traffic. Once established some convenient forms of expression, two control designs are suggested herein. These regulation schemes are also complemented with their respective stability analyses, which reflect some important properties with implications in real driving. It is significant that these linear designs can be very easy to understand and to implement, including those important features related to safety and comfort.

ABSTRACT

Helly's Model is a very simple and well-known differential equation that describes the car-following phenomenon very accurately with a very intuitive approach. It involves relative distances and velocities between two cars (one in front of the other) which are the variables of the model and which are related through constants which give proportionality and consistency to it. Those constants are parameters that must be calculated in order to achieve the necessary similarity to the real behaviour that they are modelling. In this work, an identification scheme for such estimation is presented. A set of measured data taken from real driving experiments are used to calculate the values of such parameters. Then, a simulation of the velocity developed by one of the cars is performed in order to compare this simulated behaviour against the data directly measured from that same car. The results show that there the differences of both sets of data are minimal, and that the model is very well adjusted.