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 R2 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 R2 II SC2.4 – 2.5 drawing phase portraits lecture3 lecture3
3 IV: Linear Systems on Rd SC2.3, P1.9 Jordan normal form, stability theory: decomposition of phase space Rd = Es + Ec + Eu 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 Rd SCB.3 differential structure of embedded submanifolds of Rd 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
9 XVI: Normal Forms Theory P 2.13 Poincare–Dulac normal forms
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
2021-05-08, Ho Cheung Pang