Contents -- Preface -- List of Figures -- List of Tables -- 1. Background and General Comments -- 1.1 Truly Nonlinear Functions -- 1.2 Truly Nonlinear Oscillators -- 1.3 General Remarks -- 1.4 Scaling and Dimensionless Form of Differential Equations -- 1.4.1 Linear Damped Oscillator -- 1.4.2 Nonlinear Oscillator -- 1.4.3 ẍ + axp = 0 -- 1.4.4 ẍ + ax + bx1/3 = 0 -- 1.5 Exactly Solvable TNL Oscillators -- 1.5.1 Antisymmetric, Constant Force Oscillator -- 1.5.2 Particle-in-a-Box -- 1.5.3 Restricted Duffing Equation -- 1.5.4 Quadratic Oscillator -- 1.6 Overview of TNL Oscillator Methods -- 1.6.1 Harmonic Balance -- 1.6.2 Parameter Expansion -- 1.6.3 Averaging Methods -- 1.6.4 Iteration Techniques -- 1.7 Discussion -- Problems -- References -- 2. Establishing Periodicity -- 2.1 Phase-Space -- 2.1.1 System Equations -- 2.1.2 Fixed-Points -- 2.1.3 ODE for Phase-Space Trajectories -- 2.1.4 Null-clines -- 2.1.5 Symmetry Transformations -- 2.1.6 Closed Phase-Space Trajectories -- 2.1.7 First-Integrals -- 2.2 Application of Phase-Space Methods -- 2.2.1 Linear Harmonic Oscillator -- 2.2.2 Several TNL Oscillator Equations -- 2.3 Dissipative Systems: Energy Methods -- 2.3.1 Damped Linear Oscillator -- 2.3.2 Damped TNL Oscillator -- 2.3.3 Mixed-Damped TNL Oscillator -- 2.4 Resume -- Problems -- References -- 3. Harmonic Balance -- 3.1 Direct Harmonic Balance: Methodology -- 3.2 Worked Examples -- 3.2.1 ẍ + x3 = 0 -- 3.2.2 ẍ + x-1 = 0 -- 3.2.3 ẍ + x2sgn(x) = 0 -- 3.2.4 ẍ + x1/3 = 0 -- 3.2.5 ẍ + x-1/3 = 0 -- 3.3 Rational Approximations -- 3.3.1 Fourier Expansion -- 3.3.2 Properties of ak -- 3.3.3 Calculation of x -- 3.4 Worked Examples -- 3.4.1 ẍ + x3 = 0 -- 3.4.2 ẍ + x2sgn(x) = 0 -- 3.4.3 ẍ + x-1 = 0 -- 3.5 Third-Order Equations -- 3.5.1 Castor Model -- 3.5.2 TNL Castor Models -- 3.6 Resume -- 3.6.1 Advantages -- 3.6.2 Disadvantages -- Problems.

References -- 4. Parameter Expansions -- 4.1 Introduction -- 4.2 Worked Examples -- 4.2.1 ẍ + x3 = 0 -- 4.2.2 ẍ + x-1 = 0 -- 4.2.3 ẍ + x3/(1 + x2) = 0 -- 4.2.4 ẍ + x1/3 = 0 -- 4.2.5 ẍ + x3 = o(1 - x2) x -- 4.2.6 ẍ + sgn(x) = 0 -- 4.3 Discussion -- 4.3.1 Advantages -- 4.3.2 Difficulties -- Problems -- References -- 5. Iteration Methods -- 5.1 General Methodology -- 5.1.1 Direct Iteration -- 5.1.2 Extended Iteration -- 5.2 Worked Examples: Direct Iteration -- 5.2.1 ˙ x + x3 = 0 -- 5.2.2 ẍ + x3/(1 + x2) = 0 -- 5.2.3 ẍ + x-1 = 0 -- 5.2.4 ẍ + sgn(x) = 0 -- 5.2.5 ẍ + x1/3 = 0 -- 5.2.6 ẍ + x-1/3 = 0 -- 5.2.7 ẍ + x + x1/3 = 0 -- 5.3 Worked Examples: Extended Iteration -- 5.3.1 ẍ + x3 = 0 -- 5.3.2 ẍ + x-1 = 0 -- 5.4 Discussion -- 5.4.1 Advantages of Iteration Methods -- 5.4.2 Disadvantages of Iteration Methods -- Problems -- References -- 6. Averaging Methods -- 6.1 Elementary TNL Averaging Methods -- 6.1.1 Mickens-Oyedeji Procedure -- 6.1.2 Combined Linearization and Averaging Method -- 6.2 Worked Examples -- 6.2.1 ẍ + x3 = -2 ˙ x -- 6.2.2 ẍ + x3 = - x3 -- 6.2.3 ẍ + x3 = (1 - x2) x -- 6.2.4 ẍ + x1/3 = 2 x -- 6.2.5 ẍ + x1/3 = (1 x2) x -- 6.2.6 ẍ + x = 2( ˙ x)1/3 -- 6.2.7 General Comments -- 6.3 Cveticanin's Averaging Method -- 6.3.1 Exact Period -- 6.3.2 Averaging Method -- 6.3.3 Summary -- 6.4 Worked Examples -- 6.4.1 ẍ + α.1 = 2 x -- 6.4.2 ẍ + α1 = 2( ˙ x)3 -- 6.4.3 ẍ + α.1 = (1 x2) x -- 6.5 Chronology of Averaging Methods -- 6.6 Comments -- Problems -- References -- 7. Comparative Analysis -- 7.1 Purpose -- 7.2 ẍ + x3 = 0 -- 7.2.1 Harmonic Balance -- 7.2.2 Parameter Expansion -- 7.2.3 Iteration -- 7.2.4 Comments -- 7.3 ẍ + x1/3 = 0 -- 7.3.1 Harmonic Balance -- 7.3.2 Parameter Expansion -- 7.3.3 Iteration -- 7.3.4 Comment -- 7.4 ẍ + x3 = -2 x -- 7.4.1 Mickens-Oyedeji -- 7.4.2 Combined-Linearization-Averaging.

7.4.3 Cveticanin's Method -- 7.4.4 Discussion -- 7.5 ẍ + x1/3 = 2o x -- 7.5.1 Combined-Linearization-Averaging -- 7.5.2 Cveticanin's Method -- 7.5.3 Discussion -- 7.6 ẍ + x3 = (1 - x2) x -- 7.6.1 Mickens-Oyedeji -- 7.6.2 Cveticanin's Method -- 7.6.3 Discussion -- 7.7 ẍ + x1/3 = (1 x2) ˙ x -- 7.8 General Comments and Calculation Strategies -- 7.8.1 General Comments -- 7.8.2 Calculation Strategies -- 7.9 Research Problems -- References -- Appendix A Mathematical Relations -- A.1 Trigonometric Relations -- A.1.1 Exponential Definitions of Trigonometric Functions -- A.1.2 Functions of Sums of Angles -- A.1.3 Powers of Trigonometric Functions -- A.1.4 Other Trigonometric Relations -- A.1.5 Derivatives and Integrals of Trigonometric Functions -- A.2 Factors and Expansions -- A.3 Quadratic Equations -- A.4 Cubic Equations -- A.5 Differentiation of a Definite Integral with Respect to a Parameter -- A.6 Eigenvalues of a 2 × 2 Matrix -- References -- Appendix B Gamma and Beta Functions -- B.1 Gamma Function -- B.2 The Beta Function -- B.3 Two Useful Integrals -- Appendix C Fourier Series -- C.1 Definition of Fourier Series -- C.2 Convergence of Fourier Series -- C.2.1 Examples -- C.2.2 Convergence Theorem -- C.3 Bounds on Fourier Coefficients -- C.4 Expansion of F(a cos x,-a sin x) in a Fourier Series -- C.5 Fourier Series for (cos θ)α and (sin θ)α -- References -- Appendix D Basic Theorems of the Theory of Second-Order Differential Equations -- D.1 Introduction -- D.2 Existence and Uniqueness of the Solution -- D.3 Dependence of the Solution on Initial Conditions -- D.4 Dependence of the Solution on a Parameter -- References -- Appendix E Linear Second-Order Differential Equations -- E.1 Basic Existence Theorem -- E.2 Homogeneous Linear Differential Equations -- E.2.1 Linear Combination -- E.2.2 Linear Dependent and Linear Independent Functions.

E.2.3 Theorems on Linear Second-Order Homogeneous Differential Equations -- E.3 Inhomogeneous Linear Differential Equations -- E.3.1 Principle of Superposition -- E.3.2 Solutions of Linear Inhomogeneous Differential Equations -- E.4 Linear Second-Order Homogeneous Differential Equations with Constant Coefficients -- E.5 Linear Second-Order Inhomogeneous Differential Equations with Constant Coefficients -- E.6 Secular Terms -- References -- Appendix F Lindstedt-Poincaré Perturbation Method -- References -- Appendix G A Standard Averaging Method -- References -- Appendix H Discrete Models of Two TNL Oscillators -- H.1 NSFD Rules -- H.2 Discrete Energy Function -- H.3 Cube-Root Equation -- H.4 Cube-Root/van der Pol Equation -- References -- Bibliography -- Index.