Essentials of Nonlinear Circuit Dynamics with MATLAB® and Laboratory Experiments

Arturo Buscarino, Luigi Fortuna, Mattia Frasca

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351849722
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

This book deals with nonlinear dynamics of electronic circuits, which could be used in robot control, secure communications, sensors and synchronized networks. The genesis of the content is related to a course on complex adaptive systems that has been held at the University of Catania since 2005. The efforts are devoted in order to emulate with nonlinear electronic circuits nonlinear dynamics. Step-by-step methods show the essential concepts of complex systems by using the Varela diagrams and accompanying MATLAB® exercises to reinforce new information. Special attention has been devoted to chaotic systems and networks of chaotic circuits by exploring the fundamentals, such as synchronization and control. The aim of the book is to give to readers a comprehensive view of the main concepts of nonlinear dynamics to help them better understand complex systems and their control through the use of electronics devices.

1

Introduction to nonlinear systems

CONTENTS

1.1 Classification of complex systems

1.2 First-order systems

1.2.1 Graphical analysis of equilibrium points

1.3 Numerical solutions of differential equations

1.3.1 Runge–Kutta methods

1.4 Exercises

Further reading

The first chapter of this book aims to introduce the readers to the basic notions of complex systems. Reviewing the classifications commonly adopted in literature, a new perspective over complex systems is proposed that allows us to discuss the concepts of nonlinearity, uncertainty, and system dimension unveiling the role that the interplay between them has in the emergence of complex behaviors. Furthermore, some numerical and analytical tools useful for the study of complex systems are introduced.

1.1 Classification of complex systems

The study of complex systems has fascinated people from a wide range of scientific fields, since these systems became, in the last decades, a well-established paradigm to describe both natural and artificial phenomena. In order to provide the necessary guidelines to study and classify complex systems, let us start considering the qualitative scheme reported in Figure 1.1. This outstanding diagram, originally proposed by Weinberg [93], classifies complex systems with respect to the level of uncertainty and complexity. The clusters that can be retrieved in such a diagram show an increasing level of disorder when uncertainty grows, ranging from ordered to random structures. The increase of the complexity level allows us to identify a region in the diagram characterized by the emergence of patterns and organization as a result of the interplay between complexity and uncertainty. This range is called homeodynamic and is characterized by the existence of processes where self-adaptation and self-organization play a crucial role.

FIGURE 1.1
Systems at different levels of complexity and uncertainty.

Along with the conceptual diagram by Weinberg, an abstract representation and synthesis of complex systems can be schematized as in Figure 1.2 by means of a hierarchical aggregation graph, including also the dynamic relationship between each level, leading to emergence [24]. Complex systems, in fact, attain different levels of organization as size scales. It often occurs that the higher the number of elements the higher the level of organization, which may occur as a consequence of the self-organization process, or as the effect of a control strategy.

This book will present the different topics related to the dynamics of complex adaptive systems following a route through the scheme reported in Figure 1.3. This represents a modified version of the diagram first introduced by Varela et al. [91] and reported also by Schreiber [76]. The original diagram was meant to merge the two paradigms of nonlinear determinism and linear stochasticity in a unified framework, since they were considered in the ′90s the only ones with a solid mathematical background. Hence, the two paradigms represent the extreme positions of the original diagram. The modified version reported in Figure 1.3 considers a further axis which takes into account the role of the number of state variables. This modification of the Varela diagram allows us to combine the classification presented by the diagrams in Figures 1.1 and 1.2. In fact, there is an intrinsic relationship among the three figures as they give a global view of complex systems classification.

The third dimension added in the Varela diagram allows us to introduce the concept of “networks of dynamical elements” which mirrors the organized complexity depicted in Figure 1.1. Therefore, in the following we will refer to the diagram in Figure 1.3 to introduce the various topics discussed in the book chapters. We will introduce the essential elements that allow us to establish a dichotomy between each element of Figure 1.3 and the various items reported in the book.

A further fundamental point is that, in this book, we aim to establish and exploit a strict link between dynamical complex systems, mathematical models, and electronic circuits and devices. Under this perspective, complex systems will be investigated on the basis of specifically designed electronic ...