Remus Radu

Institute for Mathematical Science
Stony Brook University

office: Math Tower 4-103
phone: (631) 632-8266

MAT 351: Differential Equations:
Dynamics & Chaos
Spring 2016
Schedule & Homework


The PDF version of the schedule is available for print here.

Note: Z=Zhang, S=Strogatz, DHS=Devaney,Hirsch,Smale
Date Topic Notes Assignments
Jan 26 Introduction: differential equations & dynamical systems
Jan 28 First order autonomous equations
Differential equations in dimension one: equilibrium & stability
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
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
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
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)
Apr 21 A model for the Lorenz attractor
Poincaré map
DHS Ch. 14
Apr 26 Chaotic attractor
Reading (see Figures 6, 7): A new twist in knot theory
Animation several trajectories (Video)
DHS Ch. 14
HW7 (due May 5)
Apr 28 Discrete dynamical systems
DHS Ch. 15
May 3 Discrete dynamical systems; Examples S10
DHS Ch. 15
May 5 Fractals and dimension
Three-dimensional ODEs - Open Problems
May 16 Projects -- due at 5:30pm in Math Tower 4-103

Last updated May 2016