# Calendar

LEC # | TOPICS |
---|---|

1 | Introduction |

2 | PendulumFree Oscillator Global View of Dynamics Energy in the Plane Pendulum Stability of Solutions to ODEsLinear Systems Nonlinear Systems |

3 | Conservation of Volume in Phase SpaceDamped Oscillators and Dissipative SystemsGeneral Remarks Phase Portrait of Damped Pendulum Summary Forced Oscillators and Limit CyclesGeneral Remarks Van der Pol Equation |

4 | Forced Oscillators and Limit Cycles (cont.)Energy Balance for Small ε Limit Cycle for ε Large A Final Note |

5 | Parametric OscillatorMathieu Equation Elements of Floquet Theory Stability of the Parametric Pendulum Damping Further Physical Insight |

6 | Fourier TransformsContinuous Fourier Transform Discrete Fourier Transform Inverse DFT Autocorrelations, Power Spectra, and the Wiener-Khintchine Theorem |

7 | Fourier Transforms (cont.)Power Spectrum of a Periodic Signal - Sinusoidal Signal - Non-sinusoidal Signal - t _{max}/T ≠ Integer- Conclusion |

8 | Fourier Transforms (cont.)Quasiperiodic Signals Aperiodic Signals Poincaré SectionsConstruction of Poincaré Sections |

9 | Poincaré Sections (cont.)Types of Poincaré Sections - Periodic - Quasiperiodic Flows - Aperiodic Flows First-return Maps 1-D Flows Relation of Flows to Maps - Example 1: The Van der Pol Equation |

10 | Poincaré Sections (cont.)Relation of Flows to Maps (cont.) - Example 2: Rössler Attractor - Example 3: Reconstruction of Phase Space from Experimental Data Fluid Dynamics and Rayleigh-Bénard ConvectionThe Concept of a Continuum Mass Conservation |

11 | Fluid Dynamics and Rayleigh-Bénard Convection (cont.)Momentum Conservation - Substantial Derivative - Forces on Fluid Particle Nondimensionalization of Navier-Stokes Equations Rayleigh-Bénard Convection |

12 | Fluid Dynamics and Rayleigh-Bénard Convection (cont.)Rayleigh-Bénard Equations - Dimensional Form - Dimensionless Equations - Bifurcation Diagram - Pattern Formation - Convection in the Earth |

13 | Midterm |

14 | Introduction to Strange AttractorsDissipation and AttractionAttractors with d = 2 Aperiodic Attractors Example: Rössler Attractor Conclusion |

15 | Lorenz EquationsPhysical Problem and ParameterizationEquations of Motion - Momentum Equation - Temperature Equation Dimensionless Equations |

16 | Lorenz Equations (cont.)Stability Dissipation Numerical Equations Conclusion |

17 | Hénon AttractorThe Hénon MapDissipation Numerical Simulations Experimental AttractorsRayleigh-Bénard ConvectionBelousov-Zhabotinsky Reaction DefinitionFractals |

18 | Fractals (cont.)Examples Correlation Dimension ν - Definition - Computation Relationship of ν to D |

19 | Lyaponov ExponentsDiverging Trajectories Example 1: M Independent of Time Example 2: Time-dependent Eigenvalues Numerical Evaluation Lyaponov Exponents and Attractors in 3-D Smale's Horseshoe Attractor |

20 | Period Doubling Route to ChaosInstability of a Limit Cycle Logistic Map Fixed Points and Stability |

21 | Period Doubling Route to Chaos (cont.)Period Doubling Bifurcations Scaling and Universality |

22 | Guest Lecture by Lorenz |

23 | Period Doubling Route to Chaos (cont.)Universal Limit of Iterated Rescaled ƒ's Doubling Operator Computation of α |

24 | Period Doubling Route to Chaos (cont.)Linearized Doubling Operator Computation of δ Comparison to Experiments |

25 | Intermittency (and Quasiperiodicity)General Characteristics of Intermittency One-dimensional Map Average Duration of Laminar Phase Lyaponov Number |

26 | Intermittency (and Quasiperiodicity) (cont.)Quasiperiodicity Special Topic |