دانلود کتاب An Introduction to Nonlinear Oscillations

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Contents
Preface
1 Nonlinear Physical Systems
1.1 Introduction
1.2 Examples of Nonlinear Physical Systems
1.3 Dimensionless Form of Differential Equations
1.4 Exact Solution for Period of a Pendulum
1.5 Exact Solution of d2y/dt2 + y + ey3 = 0
Problems
References
2 The Perturbation Method
2.1 Introduction
2.2 Secular Terms
2.3 Lindstedt-Poincare Method
2.4 Worked Examples
2.5 Shohat Expansion
2.6 Existence of a Periodic Solution
Problems
References
3 Method of Slowly Varying Amplitude and Phase
3.1 Introduction
3.2 First Approximation of Krylov and Bogoliubov
3.3 Worked Examples Using the Method of Krylov and Bogoliubov
3.4 Method of Krylov-Bogoliubov-Mitropolsky
3.5 Worked Examples Using the Method of Krylov-Bogoliubov-Mitropolsky
3.6 Stationary Amplitudes and Their Stability
3.7 Equivalent Linearization
3.8 Nonlinear Oscillations with Finite Damping
Problems
References
4 Multi-Time Expansions
4.1 Introduction
4.2 Two-Time Expansion
4.3 Worked Examples Using the Two-Time Expansion
4.4 Derivative Expansion Procedure
4.5 Worked Examples Using the Derivative Expansion Procedure
Problems
References
5 Forced Oscillations
5.1 Introduction
5.2 Forced Oscillations of Linear Systems
5.3 Combination Tones
5.4 Subharmonic Oscillations
5.5 Iteration Methods for Harmonic Oscillations without Damping
5.6 Perturbation Theory Applied to Forced Oscillations
5.7 Worked Examples Using the Perturbation Method
5.8 Duffing Equation: Resonance Curves and Jump Phenomena
Problems
References
6 Advantages and Disadvantages of Various Techniques
6.1 Introduction
6.2 Perturbation Method
6.3 Method of Slowly Varying Amplitude and Phase
6.4 Multi-Time Expansion
6.5 Procedures for Solving Nonlinear Problems
Appendix A: Mathematical Relations
A.1 Trigonometric Functions
A.2 Factors and Expansions
A.3 Solution of Quadratic Equations
A.4 Solution of Cubic Equations
A.5 Differentiation of a Definite Integral with Respect to a Parameter
References
Appendix B: Series Expansions
B.1 Uniform Convergence
B.2 Weierstrass M Test for Uniform Convergence
B.3 Properties of Uniformly Convergent Series
B.4 Power Series
B.5 Taylor Series of a Function of a Single Variable
B.6 Taylor Series of a Function of Two Variables
References
Appendix C: Fourier Series
C.1 Definition of Fourier Series
C.2 Convergence of Fourier Series
C.3 Expansion of F(A cos x, - A sin x) in a Fourier Series
References
Appendix D: Asymptotic Expansions
D.1 Gauge Functions and Order Symbols
D.2 Asymptotic Expansions
D.3 Uniform Expansion
D.4 Elementary Operations on Asymptotic Expansions
D.5 Examples
References
Appendix E: Basic Theorems of the Theory of Second-Order Differential Equations
E.1 Introduction
E.2 Existence and Uniqueness of the Solution
E.3 Dependence of the Solution on Initial Conditions
E.4 Dependence of the Solution on a Parameter
References
Appendix F: Linear Second-Order Differential Equations
F.1 Basic Existence Theorem
F.2 Homogeneous Linear Differential Equations
F.3 Nonhomogeneous Linear Differential Equations
F.4 Linear Second-Order Homogeneous Differential Equations with Constant Coefficients
F.5 Linear Second-Order Nonhomogeneous Differential Equations with Constant Coefficients
References
Appendix G: Existence of Periodic Solutions of Certain Second-Order Differential Equations
G.1 Limit Cycles
G.2 Lienard-Levinson-Smith Theorem
G.3 Levinson-Smith Theorem
G.4 Cartwright-Littlewood Theorem
G.5 Levinson Theorem
References
Appendix H: Stability of Limit Cycles
H.1 Introduction
H.2 Stability Condition
References
Appendix I: Numerical Examples
I.1 Introduction
I.2 Simple Pendulum
I.3 Nonlinear Spring
I.4 Rayleigh Equation
Bibliography
Index