# Lecture plan

Full set of notes, some corrections incorporated: lectures. (Please do not distribute without written permission.)

SC: Schaeffer and Cain, P: Perko

Week | Lecture | References | Topics | Notes | Panopto link |
---|---|---|---|---|---|

2 | I: Introduction | SC2.2 – 2.3 | phase space/flows, exponentiation of matrices | lecture1 | lecture1 |

2 | II: Linear Systems on R^{2} I | SC2.4 – 2.5 | autonomous systems of two linear equations, phase portraits, examples galore — nodes, centres, foci, saddles | lecture2 | lecture2 |

3 | III: Linear Systems on R^{2} II | SC2.4 – 2.5 | drawing phase portraits | lecture3 | lecture3 |

3 | IV: Linear Systems on R^{d} | SC2.3, P1.9 | Jordan normal form, stability theory: decomposition of phase space R^{d} = E^{s} + E^{c} + E^{u} | lecture4 | lecture4 |

4 | V: Local Well-posedness I | SC3.2 - 3.3 | Lipschitz condition, Picard's local existence and uniqueness theorem and proof | lecture5 | lecture5 |

4 | VI: Local Well-posedness II | SC4.2, 4.5 | Gronwall's Lemma with proof, continuous dependence on initial data, finite time blow-up | lecture6 | lecture6 |

5 | VII: Hyperbolic Critical Points | SC6.1, 1.4, 1.6 | linearization of 2X2 systems, examples: van der Pol, Duffing, Lotka–Volterra, activator-inhibitor | lecture7 | lecture7 |

5 | VIII: Embedded Submanifolds of R^{d} | SCB.3 | differential structure of embedded submanifolds of R^{d} | lecture8 | lecture8 |

6 | IX: Stable Manifold and Hartman-Grobman Theorems | SC6.6, 6.9, 6.10, P2.7, 2.8 | topological equivalence and conjugacy, Stable Manifold Theorem, Hartman–Grobman Theorem, applications | lecture9 | lecture9 (in-person) |

6 | X: The Method of Lyapunov | SC6.2, 6.5, P2.9 | Lyapunov functions and stability, proof, examples and applications | lecture10 | lecture10 |

7 | XI: Gradient and Hamiltonian Systems | SC6.5, SC6.8, P2.14 | conservation of energy, nondegenerate critical points, examples and applications | lecture11 | lecture11 |

7 | XII: Critical Points of Planar Systems I | P2.10 | topological saddles, spirals, and centres | lecture12 | lecture12 |

8 | XIII: Critical Points of Planar Systems II | P2.11 | non-hyperbolic critical points, examples | lecture13 | lecture13 |

8 | XIV: Critical Points of Planar Systems II — continuation | P2.11 | sectoring behaviour | lecture14 | lecture14 |

9 | XV: Centre Manifold Theory | SC6.9, P2.12 | statement of the Local Centre Manifold Theorem | lecture15 | lecture15 |

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9 | XVI: Limit Sets | P3.2, 3.3 | trajectories, limit sets, attractors | lecture16 | lecture16 |

10 | XVII: Limit Sets — continuation | P3.2, 3.3, SC 6.6.4 | periodic orbits, limit cycles, examples including the Lorenz system | lecture17 | lecture17 |

10 | XVIII: Poincare Map and Stability I | SC7.3, 7.10.1, P3.4, 3.5 | the Poincare map, stable manifold theorem for periodic orbits, some Floquet theory | lecture18 | lecture18 |

11 | XIX: Poincare Map and Stability II | SC7.3, 7.10.1, P3.4, 3.5 | more on the Poincare map, examples | lecture19 | lecture19 |

11 | XX: Poincare–Bendixson Theorem | SC7.2 | Poincare–Bendixson Theorem and proof | lecture20 | |

12 | XXI: Index Theory I | P3.12 | Notion of index, applications | lecture21 | lecture21 |

12 | XXII: Index Theory II | P3.12 | Bendixson's index formula and proof, index at infinity | lecture22 | lecture22 |

14 | XXIII: One-Dimensional Local Bifurcations I | SC8.1 – 8.4 | degeneracy/transversality/symmetries leading to one-dimensional bifurcations — saddle-node, transcritical, and pitchfork; examples | lecture23 | lecture23 |

15 | XXIV: One-Dimensional Local Bifurcations II | SC8.7 – 8.8 | continuation of XXIII, Hopf bifurcation; examples | lecture24 | lecture24 |

15 | XXV: One-Dimensional Global Bifurcations I | SC9.1 – 9.2 | bifurcations of limit cycles, examples | lecture25 | lecture25 |

16 | XXVI: One-Dimensional Global Bifurcations II | SC9.6 - 9.7 | continuation of XXV, Neimark–Sacker bifurcation, period-doubling bifurcation, examples | lecture26 | lecture26 |

16 | Revision Session I | revision1 | |||

17 | Revision Session II | revision2 |