Chaos theory is an interdisciplinary area of scientific study and a branch of engineering focused on the law of dynamic systems. Even very accurate measurements of the current state of a chaotic ...system become useless indicators of where the system will be. Many approaches to nonlinear dynamics such as the Fast Fourier Transform (FFT), phase-plane diagram, Poincaré map, bifurcation diagram, and Lyapunov exponent are used to detect and suppress the chaotic motion at different degrees of freedom. Chaos can help scientists explore the sudden transition from periodic motion to non-periodic motion at any dynamical system. There is a relationship between the finite element method and chaos analysis in which the dynamic quantity is increased with the increase of mesh generation. The refinement of element size in mesh generation affects the chaotic phenomenon Chaos theory has a wide variety of applications in engineering, such as robotics, and in medical fields, such as human gait locomotion. The dynamic description of the long-term behavior of any dynamical quantity is hard or impossible to predict. The three factors that affect chaos analysis are time delay, embedding dimensions, and the distribution of that dynamical quantity against time. Periodic, non-periodic motions, and chaos are used to describe the level of nonlinear dynamics. The vibrational signal of chaos might be continuous, discontinuous, solitons, and fractals. This book contains the method of chaos to detect the level of chaotic motion such as entropy, Lyapunov exponent, Poincare map, and phase-plane diagram.
This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, ...quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition Dαft=limh⟶0ft+heα−1t−ft/h, for all t>0, and α∈0,1. If α=0, this definition coincides to the classical definition of the first order of the function f.
Abstract
This study examines the modification of the integral mean value theorem for discontinuous functions. Modification is studied by proving that a function that is not continuous at a certain ...and bounded interval can be integrated (finite integral) both in Rieman’s integral and Newton’s integral (integral as antiderivative). The discontinuity of the intended function, namely;
f
is defined on
a,b
but the value of a function and its limit are not equal at some points or infinite points and countable on (
a,b
), f is undefined on
a,b
at some points or infinite points and countable on (
a,b
) but its limit exists there. The results of this study provide a modification of the integral mean value theorem by replacing the value of
f
in the implication of the theorem with its limit value so that the integral mean value theorem is obtained for the non-continuous function.
We present a type of arithmetic called Proportional Arithmetic. The main properties and objects that emerge with this way of operating quantities are exposed. Finally, the antiderivative and the ...indefinite integral are defined in order to calculate the primitive of in the Proportional context.
The distant origins of this book are found in my lecture notes that I wrote and rewrote for several years. The traces are the constant smudges that I made from time to time when I rediscovered these ...topics, you may not like the way of presentation, but I put in the best possible effort, it is not something new either since there are many references at the end of the book; I'll be happy if it at least works for a study group, it's impossible to avoid slipping mistakes. Some frequently asked questions that we ask ourselves, What exactly is a differential equation? Where and how did they originate, what is their use? What is done with them, how are the results of such manipulations analyzed? These questions indicate guidelines to follow, study aspects: theoretical, methodological and various fields of applications. There are historical facts that inspire; for example, obtaining the equation of the tangent line to a given curve was one of the latent problems until the end of the 17th century, on the eve of the birth of differential calculus. From then on the job consisted of solving the inverse problem, finding different methods in view of the fact that very few equations can be solved by specific rules. After a little more than two centuries, it is still not possible to establish general rules for all types of differential equations The physical world is constantly changing, it is also up to differential equations to study this change as a natural phenomenon, the laws that govern it, search for a mathematical model that governs its laws and interpret it. Differential equations in general, constitute the mathematical essence for modeling and comprise a large number of events in: physics, economics, engineering, chemistry, biology, physiology and economics, medicine, mathematics, movement of celestial bodies such as planets, moons and artificial satellites, astronomy, among others. Therefore, differential equations play a crucial role for their study. The content is thinking above all to facilitate understanding by the student, providing the basic technique to face a problem. Which is why numerous examples are included and most proofs are carefully detailed. Although abstraction is important, it makes no sense to study differential equations without reference to practical problems. I want to express my appreciation: to the Faculty of the Department of Mathematics; the director of the general office of interdisciplinary research; to the Rector of the San Luis Gonzaga National University, for crystallizing my sabbatical year and allowing a happy ending for the culmination of this book; to the physicist Mg. Carlos Tenorio, for the paintings alluding to the theme and to my dear Ica; special thanks to the Editorial Fund IDICAP PACÍFICO for the publication in digital version; there are many people who read the manuscript, their comments were valuable; thank you all.
The linear Reissner–Mindlin shell theory is reformulated in the frame of the tangential differential calculus (TDC) using a global Cartesian coordinate system. The rotation of the normal vector is ...modelled with a difference vector approach. The resulting equations are applicable to both explicitly and implicitly defined shells, because the employed surface operators do not necessarily rely on a parametrization. Hence, shell analysis on surfaces implied by level-set functions is enabled, but also the classical case of parametrized surfaces is captured. As a consequence, the proposed TDC-based formulation is more general and may also be used in recent finite element approaches such as the TraceFEM and CutFEM where a parametrization of the middle surface is not required. Herein, the numerical results are obtained by isogeometric analysis using NURBS as trial and test functions for classical and new benchmark tests. In the residual errors, optimal higher-order convergence rates are confirmed when the involved physical fields are sufficiently smooth.
•Reissner–Mindlin shells are reformulated based on the tangential differential calculus (TDC).•Shell analysis on explicitly and implicitly defined surfaces is enabled.•Strong and weak forms are derived including boundary conditions and all relevant mechanical quantities.•Higher-order convergence rates are confirmed in the residual errors.
It is well known that the theory of dynamical systems is an essential mathematical tool in the analysis of various real-world processes and phenomena that evolve over time. Different aspects of this ...rich field of research are evolving all the time, including new theoretical results for qualitative investigation as well as fast numerical techniques for approximate solution. The book provides an overview of the current state of the art in this fascinating and critically important field of pure and applied mathematics, presenting recent developments in theory, modeling, algorithms, and applications.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of input. Nonlinear control systems, which are among the new ...technologies most widely used in many fields such as economic management, industrial production, technology research and development, ecological prevention and control, are at the core of worldwide automation control technology. In contrast to linear control systems, the nonlinear control system has the characteristics of a data model: stability, zero-input system response, self-excited oscillation or limit cycle, and a more complex structure, increasing the difficulty of its theoretical analysis and technical development. Nonlinear systems are common phenomena in real life and as such cannot be ignored. Analysis and research of nonlinear systems are therefore important, and researchers need to clarify their characteristics, explore scientific and effective application measures, and finally enhance their control quality. This book comprehensively investigates the main principles, core mechanisms, typical problems, and relevant solutions involved in nonlinear systems. In general, this book aims to provide advanced research on nonlinear systems and control schemes for researchers and engineers working in related fields, and thus promote future study in this research area.
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