Institute for Mathematical Science
Stony Brook University
office: Math Tower 4-103
phone: (631) 632-8266
e-mail: remus.radu@stonybrook.edu
The PDF version of the schedule is available for print here.
| Date | Topic | Notes | Assignments |
| Jan 26 | Introduction: differential equations & dynamical systems | ||
| Jan 28 | First order autonomous equations Differential equations in dimension one: equilibrium & stability |
Z3.1-3.2 S2.1-2.4 |
|
| Feb 2 | Stability, Lyapunov function & examples | S2.4-2.7 Notes Bb |
HW1 (due Feb 11) |
| Feb 4 | Existence & uniqueness of solutions Bifurcations, normal forms |
Z3.2, S2.5 | |
| Feb 9 | Bifurcations: saddle-node, transcritical & examples | Z3.3, S3.1-3.2 Notes Bb |
HW2 (due Feb 18) |
| Feb 11 | Bifurcations: transcritical, pitchfork, hysteresis | S3.3-3.4 | |
| Feb 16 | Dimension two: Linear systems | Z5, S5.1-5.2 | |
| Feb 18 | Classification of linear systems | S5.2, 6.1-6.2 | HW3 (due Feb 25) |
| Feb 23 | Nonlinear systems: sinks, saddles, sources, stability, hyperbolicity Hartman-Grobman theorem; Examples |
S6.3-6.5 | |
| Feb 25 | Stable/unstable manifolds, closed orbits, limit cycles An example of Hopf bifurcation |
S7.1, 8.2 | HW4 (due Mar 8) |
| Mar 1 | Conservative systems, energy and nonlinear centers | S6.5 | |
| Mar 3 | Gradient systems, Lyapunov functions and examples | S7.2, Z6.2 | |
| Mar 8 | Dulac's criterion, Bendixon's negative criterion | S7.1-7.3 Z6.3-6.4 |
HW5 (due Mar 24) |
| Mar 10 | Poincaré-Bendixon theorem | Z6.4-6.5 | |
| Mar 15 | Spring break (no class) | ||
| Mar 17 | Spring break (no class) | ||
| Mar 22 | Applications of Poincaré-Bendixon theorem | S7.3 | |
| Mar 24 | Bifurcations in two-dimensional systems | S8.1-8.2 | Practice problems |
| Mar 29 | Hopf bifurcations Review |
S8.2-8.3 | Project Topics |
| Mar 31 | Midterm (1:00-2:20pm, in class) -- Midterm | ||
| Apr 5 | Hopf bifurcations; Examples | Notes Bb DHS Ch. 8 |
|
| Apr 7 | Homoclinic bifurcations; Lorenz system | S8.4, S9.2 | |
| Apr 12 | Lorenz system & properties | S9.2, Notes Bb | |
| Apr 14 | Dissipative systems, attractors, examples | S9.3, Notes Bb | HW6 (due Apr 21) |
| Apr 19 | Lorenz attractor
Stable manifold of the origin: (Video & Lorenz System Example by Alex Vladimirsky) |
S9.3 | |
| Apr 21 | A model for the Lorenz attractor Poincaré map |
DHS Ch. 14 Pictures |
|
| Apr 26 | Chaotic attractor Reading (see Figures 6, 7): A new twist in knot theory Animation several trajectories (Video) |
DHS Ch. 14 Pictures |
HW7 (due May 5) |
| Apr 28 | Discrete dynamical systems Chaos |
S10 DHS Ch. 15 |
|
| May 3 | Discrete dynamical systems; Examples | S10 DHS Ch. 15 |
|
| May 5 | Fractals and dimension Three-dimensional ODEs - Open Problems |
S11 | |
| May 16 | Projects -- due at 5:30pm in Math Tower 4-103 | ||