Nekorkin - Introduction to Nonlinear Oscillations

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Author

Vladimir I. Nekorkin

Institute of Applied Physics of the Russian Academy of Sciences

46 Uljanov str.

603950 Nizhny Novgorod

Russia

Cover

iStock 5401380, AndrzejStajer

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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

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.