Introduction to Nonlinear Oscillations

Vladimir I. Nekorkin

• Publisher: Wiley-VCH
• Year: 2015
• Print ISBN: 9783527413300
• eBook ISBN: 9783527685424
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

A systematic outline of the basic theory of oscillations, combining several tools in a single textbook. The author explains fundamental ideas and methods, while equally aiming to teach students the techniques of solving specific (practical) or more complex problems.

Following an introduction to fundamental notions and concepts of modern nonlinear dynamics, the text goes on to set out the basics of stability theory, as well as bifurcation theory in one and two-dimensional cases. Foundations of asymptotic methods and the theory of relaxation oscillations are presented, with much attention paid to a method of mappings and its applications.

With each chapter including exercises and solutions, including computer problems, this book can be used in courses on oscillation theory for physics and engineering students. It also serves as a good reference for students and scientists in computational neuroscience.

1
Introduction to the Theory of Oscillations

1.1 General Features of the Theory of Oscillations

Oscillatory processes and systems are so widely distributed in nature, technology, and society that we frequently encounter them in our everyday life and can, apparently, formulate their basic properties without difficulty. Indeed, when we hear about fluctuations in temperature, exchange rates, voltage, a pendulum, the water level, and so on, we understand that it is in relation to processes in time or space, which have varying degrees of repetition and return to their original or similar states. Moreover, these base properties of the processes do not depend on the nature of systems and Can, therefore, be described and studied from just the point of view of a general interdisciplinary approach. This is exactly the approach that the theory of oscillations explores, the subject of which are the oscillatory phenomena and the processes in systems of different nature. The theory of oscillations gets its oscillatory properties from the analysis of the corresponding models. As a result of such an analysis, a connection between the parameters of the model and its oscillatory properties is established.

The theory of oscillations is both an applied and fundamental science. The applied character of the theory of oscillations is determined by its multiple applications in physics, mechanics, automated control, radio engineering and electronics, instrumentation, and so on. In these spheres of science, a large amount of research of different systems and phenomena was carried out, using the methods of the theory of oscillations. Furthermore, new technical directions have arisen on the basis of the theory of oscillations, namely, vibrational engineering and vibrational diagnostics, biomechanics, and so on. The fundamental characteristic of the theory of oscillations is based on the studied models themselves. They are the so-called dynamical systems, with the help of which one can describe any determinate evolution in time or in time and space. It is exactly the study of dynamical systems that allowed the theory of oscillations to introduce the concepts and conditions, develop the methods, and achieve the results that exert a large influence on other natural sciences. Here, we only mention the linearized stability theory, the concept of self-sustained oscillations and resonance, bifurcation theory, chaotic oscillations, and so on.

1.2 Dynamical Systems

Consider the system, the state of which is determined by the vector x(t) ∈ Rⁿ. Assume that the evolution of the system is determined by a single parameter family of operators G^t, t ∈ R (or t ∈ R₊) or t ∈ Z (or t ∈ Z₊), such that the state of the system at the instant t

(1.1)

where x₀ is its initial state (initial point). We also assume that the evolutionary operators satisfy the following two properties, which reflect the de...

Table of contents

• Cover
• Related Titles
• Title Page
• Copyright
• Contents
• Preface
• 1 Introduction to the Theory of Oscillations
• 2 One-Dimensional Dynamics
• 3 Stability of Equilibria. A Classification of Equilibria of Two-Dimensional Linear Systems
• 4 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems
• 5 Linear and Nonlinear Oscillators
• 6 Basic Properties of Maps
• 7 Limit Cycles
• 8 Basic Bifurcations of Equilibria in the Plane
• 9 Bifurcations of Limit Cycles. Saddle Homoclinic Bifurcation
• 10 The Saddle-Node Homoclinic Bifurcation. Dynamics of Slow–Fast Systems in the Plane
• 11 Dynamics of a Superconducting Josephson Junction
• 12 The Van der Pol Method. Self-Sustained Oscillations and Truncated Systems
• 13 Forced Oscillations of a Linear Oscillator
• 14 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom
• 15 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force
• 16 Parametric Oscillations
• 17 Answers to Selected Exercises
• Bibliography
• Index
• End User License Agreement