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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
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.
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Seitenzahl: 395
Veröffentlichungsjahr: 2015
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Contents
Cover
Related Titles
Title Page
Copyright
Preface
1 Introduction to the Theory of Oscillations
1.1 General Features of the Theory of Oscillations
1.2 Dynamical Systems
1.3 Attractors
1.4 Structural Stability of Dynamical Systems
1.5 Control Questions and Exercises
2 One-Dimensional Dynamics
2.1 Qualitative Approach
2.2 Rough Equilibria
2.3 Bifurcations of Equilibria
2.4 Systems on the Circle
2.5 Control Questions and Exercises
3 Stability of Equilibria. A Classification of Equilibria of Two-Dimensional Linear Systems
3.1 Definition of the Stability of Equilibria
3.2 Classification of Equilibria of Linear Systems on the Plane
3.3 Control Questions and Exercises
4 Analysis of the Stability of Equilibria of Multidimensional Nonlinear Systems
4.1 Linearization Method
4.2 The Routh–Hurwitz Stability Criterion
4.3 The Second Lyapunov Method
4.4 Hyperbolic Equilibria of Three-Dimensional Systems
4.5 Control Questions and Exercises
5 Linear and Nonlinear Oscillators
5.1 The Dynamics of a Linear Oscillator
5.2 Dynamics of a Nonlinear Oscillator
5.3 Control Questions and Exercises
6 Basic Properties of Maps
6.1 Point Maps as Models of Discrete Systems
6.2 Poincaré Map
6.3 Fixed Points
6.4 One-Dimensional Linear Maps
6.5 Two-Dimensional Linear Maps
6.6 One-Dimensional Nonlinear Maps: Some Notions and Examples
6.7 Control Questions and Exercises
7 Limit Cycles
7.1 Isolated and Nonisolated Periodic Trajectories. Definition of a Limit Cycle
7.2 Orbital Stability. Stable and Unstable Limit Cycles
7.3 Rotational and Librational Limit Cycles
7.4 Rough Limit Cycles in Three-Dimensional Space
7.5 The Bendixson–Dulac Criterion
7.6 Control Questions and Exercises
8 Basic Bifurcations of Equilibria in the Plane
8.1 Bifurcation Conditions
8.2 Saddle-Node Bifurcation
8.3 The Andronov–Hopf Bifurcation
8.4 Stability Loss Delay for the Dynamic Andronov–Hopf Bifurcation
8.5 Control Questions and Exercises
9 Bifurcations of Limit Cycles. Saddle Homoclinic Bifurcation
9.1 Saddle-node Bifurcation of Limit Cycles
9.2 Saddle Homoclinic Bifurcation
9.3 Control Questions and Exercises
10 The Saddle-Node Homoclinic Bifurcation. Dynamics of Slow–Fast Systems in the Plane
10.1 Homoclinic Trajectory
10.2 Final Remarks on Bifurcations of Systems in the Plane
10.3 Dynamics of a Slow-Fast System
10.4 Control Questions and Exercises
11 Dynamics of a Superconducting Josephson Junction
11.1 Stationary and Nonstationary Effects
11.2 Equivalent Circuit of the Junction
11.3 Dynamics of the Model
11.4 Control Questions and Exercises
12 The Van der Pol Method. Self-Sustained Oscillations and Truncated Systems
12.1 The Notion of Asymptotic Methods
12.2 Self-Sustained Oscillations and Self-Oscillatory Systems
12.3 Control Questions and Exercises
13 Forced Oscillations of a Linear Oscillator
13.1 Dynamics of the System and the Global Poincaré Map
13.2 Resonance Curve
13.3 Control Questions and Exercises
14 Forced Oscillations in Weakly Nonlinear Systems with One Degree of Freedom
14.1 Reduction of a System to the Standard Form
14.2 Resonance in a Nonlinear Oscillator
14.3 Forced Oscillation Regime
14.4 Control Questions and Exercises
15 Forced Synchronization of a Self-Oscillatory System with a Periodic External Force
15.1 Dynamics of a Truncated System
15.2 The Poincaré Map and Synchronous Regime
15.3 Amplitude-Frequency Characteristic
15.4 Control Questions and Exercises
16 Parametric Oscillations
16.1 The Floquet Theory
16.2 Basic Regimes of Linear Parametric Systems
16.3 Pendulum Dynamics with a Vibrating Suspension Point
16.4 Oscillations of a Linear Oscillator with Slowly Variable Frequency
17 Answers to Selected Exercises
Bibliography
Index
End User License Agreement
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Guide
Cover
Contents
Begin Reading
List of Illustrations
Figure 1.1
Figure 1.2
Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7
Figure 2.8
Figure 2.9
Figure 2.10
Figure 2.11
Figure 3.1
Figure 3.2
Figure 3.3
Figure 3.4
Figure 3.5
Figure 3.6
Figure 3.7
Figure 3.8
Figure 3.9
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 4.6
Figure 4.7
Figure 4.8
Figure 4.9
Figure 4.10
Figure 5.1
Figure 5.2
Figure 5.3
Figure 5.4
Figure 5.5
Figure 5.6
Figure 5.7
Figure 5.8
Figure 5.9
Figure 5.10
Figure 5.11
Figure 5.12
Figure 6.1
Figure 6.2
Figure 6.3
Figure 6.4
Figure 6.5
Figure 6.6
Figure 7.1
Figure 7.2
Figure 7.3
Figure 7.4
Figure 7.5
Figure 7.6
Figure 7.7
Figure 7.8
Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5
Figure 8.6
Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4
Figure 9.5
Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9
Figure 10.1
Figure 10.2
Figure 10.3
Figure 10.4
Figure 10.5
Figure 10.6
Figure 10.7
Figure 10.8
Figure 10.9
Figure 11.1
Figure 11.2
Figure 11.3
Figure 11.4
Figure 11.5
Figure 11.6
Figure 11.7
Figure 11.8
Figure 11.9
Figure 11.10
Figure 11.11
Figure 12.1
Figure 12.2
Figure 12.3
Figure 12.4
Figure 12.5
Figure 12.6
Figure 13.1
Figure 13.2
Figure 13.3
Figure 13.4
Figure 13.5
Figure 14.1
Figure 14.2
Figure 14.3
Figure 14.4
Figure 15.1
Figure 15.2
Figure 15.3
Figure 15.4
Figure 15.5
Figure 15.6
Figure 16.1
Figure 16.2
Figure 16.3
Figure 16.4
Figure 16.6
Figure 16.7
Figure 16.8
List of Tables
Table 3.1
Table 10.1
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Vladimir I. Nekorkin
Introduction to Nonlinear Oscillations
Author
Vladimir I. Nekorkin
Institute of Applied Physics of the Russian Academy of Sciences
46 Uljanov str.
603950 Nizhny Novgorod
Russia
Cover
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Preface
At the foundation of this course material are lectures on a general course in the theory of oscillations, which were taught by the author for more than 20 years at the Faculty of Radiophysics at Nizhny Novgorod State University (NNSU).
The aim of the course was not only to express fundamental ideas and methods of the theory of oscillations as a science of evolutionary processes, but also to teach the audience the methods and techniques of solving specific (practical) problems.
The key role in forming this lecture course is played by qualitative methods of the theory of dynamical systems and methods of the theory of bifurcations, which follow the tradition of Nizhny Novgorod school of nonlinear oscillations. These methods are even used when solving simple problems, where, in principle, their use is not necessary. Such a way of presenting the following material allows us, first of all, to reveal the essence and fundamental principles of the methods, and, secondly, for the reader to develop the skills necessary to put them to use, which appears to be important for the transition to studying more complex problems.
The book is constructed in the form of lectures in accordance with the syllabus of the course “Theory of Oscillations” for the Faculty of Radiophysics at NNSU. Yet, the content of nearly every lecture in this book is expanded further than it is usually presented during the reading of a formal lecture. This makes it possible for the reader to gain additional knowledge on the subject. At the end of each lecture, there are test questions and problems for revision and independent study.
This text could also prove useful to undergraduate and graduate students specializing in the field of nonlinear dynamics, information systems, control theory, biophysics, and so on.
The author is grateful to the colleagues at the department of “Theory of Oscillations and Automated Control” for many useful discussions on the topics of this text and to the colleagues from the department of Nonlinear Dynamics at the Institute of Applied Physics of the Russian Academy of Sciences.
Vladimir I. Nekorkin Nizhny Novgorod October 2014
