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Unformatted text preview: 12.006J/18.353J Nonlinear Dynamics I: Chaos Daniel H. Rothman Massachusetts Institute of Technology Contents Acknowledgements and references 5 1 Pendulum 7 1.1 Free oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Global view of dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Energy in the plane pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Stability of solutions to ODEs 14 2.1 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Conservation of volume in phase space 19 4 Damped oscillators and dissipative systems 22 4.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 Phase portrait of damped pendulum . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5 Forced oscillators and limit cycles 28 5.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Van der Pol equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 Energy balance for small α . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.4 Limit cycle for α large . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.5 A final note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Parametric oscillator 38 6.1 Mathieu equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.2 Elements of Floquet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.3 Stability of the parametric pendulum . . . . . . . . . . . . . . . . . . . . . . 41 6.4 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.5 Further physical insight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 Fourier transforms 47 7.1 Continuous Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . 47 7.2 Discrete Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.3 Inverse DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.4 Autocorrelations, power spectra, and the Wiener-Khintchine theorem . . . . 52 7.5 Power spectrum of a periodic signal . . . . . . . . . . . . . . . . . . . . . . . 54 7.5.1 Sinusoidal signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7.5.2 Non-sinusoidal signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7.5.3 t max /T = integer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.6 Quasiperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 7.7 Aperiodic signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 1 8 Poincar´ e sections 68 8.1 Construction of Poincar´...
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