Introduction to Nonlinear Oscillations

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.

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
 

• By (author) Vladimir I. Nekorkin