A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization and Its Digital Implementation

YesIn this paper, a new fractional order chaotic system without equilibrium is proposed, analyti-cally and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigation were used to describe the system dynamical behaviors including, the system equilibria, the chaotic attractors, the bifurcation diagrams and the Lyapunov expo-nents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attrac-tors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive con-trol theory has been developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state varia-bles for the master and slave. Consequently, the update laws of the slave parameters are ob-tained, where the slave parameters are assumed to be uncertain and estimate corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results are obtained by MATLAB and the Arduino Due boards respectively, where a good consistent between the simulation results and the ex-perimental results. indicating that the new fractional order chaotic system is capable of being employed in real-world applications

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Fractional order chaotic systems and their electronic design

"Con el desarrollo del cálculo fraccionario y la teoría del caos, los sistemas caóticos de orden fraccionario se han convertido en una forma útil de evaluar las características de los sistemas dinámicos. En esta dirección, esta tesis es principalmente relacionada, es decir, en el estudio de sistemas caóticos de orden fraccionario, basado en sistemas disipativos de inestables, un sistema disipativo de inestable de orden fraccionario es propuesto. Algunas propiedades dinámicas como puntos de equilibrio, exponentes de Lyapunov, diagramas de bifurcación y comportamientos dinámicos caóticos del sistema caótico de orden fraccionario son estudiados. Los resultados obtenidos muestran claramente que el sistema discutido presenta un comportamiento caótico. Por medio de considerar la teoría del cálculo fraccionario y simulaciones numéricas, se muestra que el comportamiento caótico existe en el sistema de tres ecuaciones diferenciales de orden fraccionario acopladas, con un orden menor a tres. Estos resultados son validados por la existencia de un exponente positivo de Lyapunov, además de algunos diagramas de fase. Por otra parte, la presencia de caos es también verificada obteniendo la herradura topológica. Dicha prueba topológica garantiza la generaci´n de caos en el sistema de orden fraccionario propuesto. En orden de verificar la efectividad del sistema propuesto, un circuito electrónico es diseñado con el fin de sintetizar el sistema caótico de orden fraccionario.""With the development of fractional order calculus and chaos theory, the fractional order chaotic systems have become a useful way to evaluate characteristics of dynamical systems and forecast the trend of complex systems. In this direction, this thesis is primarily concerned with the study of fractional order chaotic systems, based on an unstable dissipative system (UDS), a fractional order unstable dissipative system (FOUDS) is proposed. Dynamical properties, such as equilibrium points, Lyapunov exponents, bifurcation diagrams and phase diagrams of the fractional order chaotic system are studied. The obtained results shown that the fractional order unstable dissipative system has a chaotic behavior. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the fractional order three dimensional system with order less than three. The lowest order to yield chaos in this system is 2.4. The results are validated by the existence of one positive Lyapunov exponent, phase diagrams; Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order unstable dissipative system. In order to verify the effectiveness of the proposed system, an electronic circuit is designed with the purpose of synthesize the fractional order chaotic system, the fractional order integral is realized with electronic circuit utilizing the synthesis of a fractance circuit. The realization has been done via synthesis as passive RC circuits connected to an operational amplifier. The continuos fractional expansion have been utilized on fractional integration transfer function which has been approximated to integer order rational transfer function considering the Charef Method. The analogue electronics circuits have been simulated using HSPICE.

Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Symmetry in Chaotic Systems and Circuits

Symmetry can play an important role in the field of nonlinear systems and especially in the design of nonlinear circuits that produce chaos. Therefore, this Special Issue, titled “Symmetry in Chaotic Systems and Circuits”, presents the latest scientific advances in nonlinear chaotic systems and circuits that introduce various kinds of symmetries. Applications of chaotic systems and circuits with symmetries, or with a deliberate lack of symmetry, are also presented in this Special Issue. The volume contains 14 published papers from authors around the world. This reflects the high impact of this Special Issue

Mathematical Economics

This book is devoted to the application of fractional calculus in economics to describe processes with memory and non-locality. Fractional calculus is a branch of mathematics that studies the properties of differential and integral operators that are characterized by real or complex orders. Fractional calculus methods are powerful tools for describing the processes and systems with memory and nonlocality. Recently, fractional integro-differential equations have been used to describe a wide class of economical processes with power law memory and spatial nonlocality. Generalizations of basic economic concepts and notions the economic processes with memory were proposed. New mathematical models with continuous time are proposed to describe economic dynamics with long memory. This book is a collection of articles reflecting the latest mathematical and conceptual developments in mathematical economics with memory and non-locality based on applications of fractional calculus

Low-Cost Inventions and Patents

Inventions have led to the technological advances of mankind. There are inventions of all kinds, some of which have lasted hundreds of years or even longer. Low-cost technologies are expected to be easy to build, have little or no energy consumption, and be easy to maintain and operate. The use of sustainable technologies is essential in order to move towards a greater global coverage of technology, and therefore to improve human quality of life. Low-cost products always respond to a specific need, even if no in-depth analysis of the situation or possible solutions has been carried out. It is a consensus in all industrialized countries that patents have a decisive influence on the organization of the economy, as they are a key element in promoting technological innovation. Patents must aim to promote the technological development of countries, starting from their industrial situations

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio