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. INDICE: INTRODUCTION TO THE THEORY OF OSCILLATIONS .General Features of the Theory of Oscillations .Dynamic Systems.Attractors .Structural Stability of Dynamic Systems.Control Questions and Exercises..ONE DIMENSIONAL DYNAMICS.Qualitative Approach. Rough Equilibria.Bifurcations of Equilibria .Systems on the Circle .Control Questions and Exercises..STABILITY OF EQUILIBRIA. A CLASSIFICATION OF EQUILIBRIA OF TWO–DIMENSIONAL LINEAR SYSTEMS.Definition of the Stability of Equilibria .Classification of Equilibria of Linear Systems on the Plane .Control Questions and Exercises ..ANALYSIS OF THE STABILITY OF EQUILIBRIA OF MULTIDIMENSIONAL NONLINEAR SYSTEMS .Linearization Method .The Routh–Hurwitz Stability Criterion .The Second Lyapunov Method .Hyperbolic Equilibria of Three–Dimensional Systems .Control Questions and Exercises ..LINEAR AND NONLINEAR OSCILLATORS.The Dynamics of a Linear Oscillator .Dynamics of a Nonlinear Oscillator .Control Questions and Exercises ..BASIC PROPERTIES OF MAPS .Point Maps as Models of Discrete Systems.Poincare Map .Fixed Points.One–Dimensional Linear Maps .Two–Dimensional Linear Maps .One–Dimensional Nonlinear Maps: Some Notions and Examples .Control Questions and Exercises..LIMIT CYCLES .Isolated and Nonisolated Periodic Trajectories. Definition of a Limit Cycle.Orbital Stability. Stable and Unstable Limit Cycles.Rotational and Librational Limit Cycles .Rough Limit Cycles in Three–Dimensional Space.The Bendixson–Dulac Criterion .Control Questions and Exercises..BASIC BIFURCATIONS OF EQUILIBRIA IN THE PLANE .Bifurcation Conditions .Saddle–Node Bifurcation .The Andronov–Hopf Bifurcation .Stability Loss Delay for the Dynamic Andronov–Hopf Bifurcation .Control Questions and Exercises ..BIFURCATIONS OF LIMIT CYCLES. SADDLE HOMOCLINIC BIFURCATION.Tangent Bifurcation of Limit Cycles .Saddle Homoclinic Bifurcation .Control Questions and Exercises ..THE SADDLE–NODE HOMOCLINIC BIFURCATION. DYNAMICS OF SLOW–FAST SYSTEMS IN THE PLANE.Homoclinic Trajectory .Final Remarks on Bifurcations of Systems in the Plane .Dynamics of a Slow–Fast System .Control Questions and Exercises ..DYNAMICS OF A SUPERCONDUCTING JOSEPHSON JUNCTION.Stationary and Non–Stationary Effects .Equivalent Circuit of the Junction .Dynamics of the Model .Control Questions and Exercises..THE VAN DER POL METHOD. SELF–SUSTAINED OSCILLATIONS AND TRUNCATED SYSTEMS.The Notion of Asymptotic Methods .Self–Sustained Oscillations and Self–Oscillatory Systems.Control Questions and Exercises ..FORCED OSCILLATIONS OF A LINEAR OSCILLATOR .Dynamics of the System and the Global Poincaré.Map .Resonance Curve .Control Questions and Exercises ..FORCED OSCILLATIONS IN WEAKLY NONLINEAR SYSTEMS WITH ONE DEGREE OF FREEDOM .Reduction of a System to the Standard Form.Resonance in a Nonlinear Oscillator .Forced Oscillation Regime.Control Questions and Exercises..FORCED SYNCHRONIZATION OF A SELF–OSCILLATORY SYSTEM WITH A PERIODIC EXTERNAL FORCE .Dynamics of a Truncated System .The Poincaré.Map and Synchronous Regime.Amplitude–frequency Characteristic of a Self–Oscillatory System .Control Questions and Exercises..PARAMETRIC OSCILLATIONS .The Floquet Theory.Basic Regimes on Linear Parametric Systems.Pendululm Dynamics with a Vibrating Suspension Point.Oscillations of a Linear Oscillator with Slowly Variable Frequency.Control Questions and Exercises..
- ISBN: 978-3-527-41330-0
- Editorial: Wiley VCH
- Encuadernacion: Rústica
- Páginas: 290
- Fecha Publicación: 08/04/2015
- Nº Volúmenes: 1
- Idioma: Inglés