**Lecture Schedule**

**, 08/23**Introduction, what is a dynamical system?, the importance of nonlinearity, motivating examples, Mathematica demo, 1D ODE as vector field on R, geometric and physical interpretation of an ODE

**08/25**Examples, sketching solutions, fixed points, stable and unstable fixed points, method of linear stability analysis

**08/29**Pathologies of solutions of IVP and the fundamental theorem to avoid most of them, essential applications of uniqueness of solutions (graphs of solutions can't intersect, solutions never reach fixed points in finite time).

**08/31**Potentials, Lyapunov functions, saddle-node bifurcation

**09/06**Nondimensionalization, analytic criterion for saddle-node bifurcation, normal form for saddle-node bifurcation, transcritical bifurcation, SIS transmission model

**09/08**SIS model, basic reproduction number, normal form for transcritical bifurcation, pitchfork bifurcation, motion of bead on a rotating hoop

**09/13**Analysis of singular limit of bead on a rotating hoop, ODEs on circle, simple harmonic oscillator

**09/15**Nonlinear oscillator and saddle node bifurcation, application to overdamped pendulum with constant torque, review of solving x'=Ax when A is diagonalizable matrix

**09/20**Jordan Canonical Form for 2x2 matrices, formula solution of linear systems of ODEs in R^2

**09/22**Phase plane for linear systems of ODEs in R^2 for matrices in JCF, geometric mapping properties of 2x2 matrices, phase plane for linear systems of ODEs in R^2

**09/27**Stable and unstable manifolds of fixed point, trace-determinant plane, nonlinear systems: notions of stability of a fixed point

**09/29**Exam 1

**10/04**Phase flow, linear structural stability, topological conjugacy and equivalence

**10/06**More examples of topological conjugacy and equivalence, fundamental existence and uniqueness theorem of solutions of IVP, applications, linearization of nonlinear ODE at fixed point, statement of Hartman-Grobman theorem

**10/11**Fall Break

**10/13**Applications of Hartman-Grobman theorem

**10/18**Limit cycles, homoclinic and heterclinic connections, Poincare-Bendixson theorem, pplane demo

**10/20**Stable manifold theorem, nullclines, sketch the complete phase plane diagram

**10/25**Lotka-Volterra competition model, conservation of energy for mechanical systems (pendulum, double well potential)

**10/27**Conservative systems and their properties, SIR transmission model, Hamiltonian systems

**11/01**More about Hamiltonian systems, key application of index theory, closed orbits, strategies to show existence of closed orbits (index theory, Poincare-Bendixson theorem, Lienard system)

**11/03**Ruling out closed orbits: Gradient systems, energy functions, global Lyapunov functions, local Lyapunov functions

**11/08**More about local Lyapunov functions (examples illustrating the Chetaev method and LaSalle's invariance theorem), Bendixson-Dulac criterion

**11/10**Exam 2

**11/15**Saddle node bifurcation (normal form, analytic criterion, physical example)

**11/17**Rock-scissors-paper dynamics, How to numerically solve odes (Matlab's ode45 or LOSODE), Hopf bifurcation (normal form)

**11/22**Class cancelled

**11/24**Thanksgiving

**11/29**Hopf bifurcation (analytic criterion), Introduction to Lorenz system

**12/01**Properties of Lorenz system

**12/06**Properties of Lorenz system, what is chaos?

12/14 FInal exam