0.1 Preface to Second Edition . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
2 Nonlinear Oscillators 5
2.1 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Pure Nonlinear Oscillator 19
3.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Exact period of vibration . . . . . . . . . . . . . . . . . . 22
3.2 Exact periodical solution . . . . . . . . . . . . . . . . . . . . . . . 24
3.2.1 Linear case . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Odd quadratic nonlinearity . . . . . . . . . . . . . . . . . 26
3.2.3 Cubic nonlinearity . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Adopted Lindstedt-Poincaré method . . . . . . . . . . . . . . . . 28
3.4 Modi.ed Lindstedt-Poincaré method . . . . . . . . . . . . . . . . 31
3.4.1 Comparison of the LP and MLP methods . . . . . . . . . 32
3.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Exact amplitude, period and velocity method . . . . . . . . . . . 34
3.6 Solution in the form of Jacobi elliptic function . . . . . . . . . . 35
3.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Solution in the form of a trigonometric function . . . . . . . . . . 39
3.7.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Pure nonlinear oscillator with linear damping . . . . . . . . . . . 42
3.8.1 Parameter analysis . . . . . . . . . . . . . . . . . . . 44
3.8.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.9 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Free Vibrations 49
4.1 Homotopy-perturbation technique . . . . . . . . . . . . . . . . . 51
4.1.1 Duffing oscillator with a quadratic term . . . . . . . . . . 54
4.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Averaging solution procedure . . . . . . . . . . . . . . . . . . . . 57
4.2.1 Solution in the form of an Ateb function . . . . . . . . . . 57
4.2.2 Solution in the form of the Jacobi elliptic function . . . . 64
4.2.3 Solution in the form of a trigonometric function . . . . . . 70
4.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Hamiltonian Approach solution procedure . . . . . . . . . . . . . 75
4.3.1 Approximate frequency of vibration . . . . . . . . . . . . 75
4.3.2 Error estimation . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.3 Comparison between approximate and exact solutions . . 79
4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Oscillator with linear damping . . . . . . . . . . . . . . . . . . . 86
4.4.1 Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . 88
4.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Oscillators with odd and even quadratic nonlinearity . . . . . . . 93
4.5.1 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . 95
4.5.2 Exact solution for the asymmetric oscillator . . . . . . . . 97
4.5.3 Solution for the symmetric oscillator . . . . . . . . . . . . 99
4.5.4 Oscillations in an optomechanical system . . . . . . . . . 104
4.5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Oscillators with the time variable parameters 115
5.1 Oscillators with slow time variable parameters . . . . . . . . . . . 116
5.2 Solution in the form of the Ateb function . . . . . . . . . . . . . 116
5.2.1 Oscillator with linear time variable parameter . . . . . . . 119
5.3 Solution in the form of a trigonometric function . . . . . . . . . . 121
5.3.1 Linear oscillator with time variable parameters . . . . . . 122
5.3.2 Non-integer order nonlinear oscillator . . . . . . . . . . . 123
5.3.3 Levi-Civita oscillator with a small damping . . . . . . . . 124
5.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4 Solution in the form of a Jacobi elliptic function . . . . . . . . . 128
5.4.1 Van der Pol oscillator with time variable mass . . . . . . 130
5.4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.5 Parametrically excited strong nonlinear oscillator . . . . . . . . . 137
5.5.1 Solution procedure . . . . . . . . . . . . . . . . . . . . . . 139
5.5.2 Numerical simulation . . . . . . . . . . . . . . . . . . . . 146
5.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Forced Vibrations 151
6.1 Oscillator with constant excitation force . . . . . . . . . . . . . . 152<
6.1.1 Solution of the odd-integer order oscillator . . . . . . . . . 154
6.1.2 The oscillator with additional small nonlinearity . . . . . 158
6.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2 Harmonically excited pure nonlinear oscillator . . . . . . . . . . . 163
6.2.1 Pure odd-order nonlinear oscillator . . . . . . . . . . . . . 163
6.2.2 Bifurcation in the oscillator . . . . . . . . . . . . . . . . . 166
6.2.3 Harmonically forced pure cubic oscillator . . . . . . . . . 169
6.2.4 Numerical simulation and discussion . . . . . . . . . . . . 173
6.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.3 Forced vibrations of the pure nonlinear oscillator . . . . . . . . . 179
6.3.1 Design of excitation and derivation of amplitude-frequency
equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.3.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7 Two-Degree-of-Freedom Oscillator 185
7.1 System with nonlinear viscoelastic connection . . . . . . . . . . . 186
7.1.1 Model with strong nonlinear viscoelastic connection . . . 187
7.1.2 Solution procedure . . . . . . . . . . . . . . . . . . . . . . 188
7.1.3 Pure nonlinear viscoelastic connection . . . . . . . . . . . 191
7.1.4 Special case . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.1.5 .Steady-state.solution . . . . . . . . . . . . . . . . . . . . 195
7.1.6 Mechanical vibration of the vocal cord . . . . . . . . . . . 198
7.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.2 System with nonlinear elastic connection . . . . . . . . . . . . . . 203
7.2.1 Two-degree-of-freedom Van der Pol oscillator . . . . . . . 205
7.2.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.3 Complex-valued di¤erential equation . . . . . . . . . . . . . . . . 213
7.3.1 Adopted Krylov-Bogolubov method . . . . . . . . . . 214
7.3.2 Method based on the first integrals . . . . . . . . . . . . . 216
7.3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.4 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8 Chaos in Oscillators 231
8.1 Chaos in ideal oscillator . . . . . . . . . . . . . . . . . . . . . . . 232
8.1.1 Homoclinic orbits in the unperturbed system . . . . . . . 233
8.1.2 Melnikov.s criteria for chaos . . . . . . . . . . . . . . . . . 235
8.1.3 Numerical simulation . . . . . . . . . . . . . . . . . . . . 238
8.1.4 Lyapunov exponents and bifurcation diagrams . . . . . . 241
8.1.5 Control of chaos . . . . . . . . . . . . . . . . . . . . . . . 242
8.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.2 Chaos in non-ideal oscillator . . . . . . . . . . . . . . . . . . . . . 245
8.2.1 Modeling of the system . . . . . . . . . . . . . . . . . . . 246
8.2.2 Asymptotic solving method . . . . . . . . . . . . . . . . . 247
8.2.3 Stability and Sommerfeld e¤ect . . . . . . . . . . . . . . . 248
8.2.4 Numerical simulation and chaotic behavior . . . . . . . . 253
8.2.5 Control of chaos . . . . . . . . . . . . . . . . . . . . . . . 257
8.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.3 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9 Vibration of the Axially Purely Nonlinear Rod 263
9.1 Model of the axially vibrating rod . . . . . . . . . . . . . . . . . 263
9.2 Solving procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.2.1 Solving of the equation with displacement function . . . . 266
9.2.2 Solving of the equation with time function . . . . . . . . . 269
9.3 Frequency of axial vibration . . . . . . . . . . . . . . . . . . . . . 270
9.4 Solution illustration and simulation . . . . . . . . . . . . . . . . . 272
9.5 Period and frequency of vibration of a muscle . . . . . . . . . . . 274
9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.7 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
A Periodical Ateb functions 279
B Fourier series of the ca Ateb function 283
C Averaging of Ateb functions 287
D Jacobi elliptic functions 291
E Euler's integrals of the first and second kind 293
F Inverse incomplete Beta function 295
