**Hale, Jack, Ordinary Differential Equations, 2nd
Ed., Krieger, 1980.**

**Verhulst, F., Nonlinear Differential Equations and
Dynamical Systems, Springer, 1990.**

**Guckenheimer and Holmes, Nonlinear Oscillations,
Dynamical Syetems,a and Bifurcations of Vector Fields,**

**Lec. 2 Jan. 29: Local existence for the 1st
order by Picard iteration. Extension to nth-order systems. Relation**
**to a fixed-point theorem. Examples. Reading: Chapter 1 of F. John's
notes.**
**Download odehwk2.tex.**
**Download odehwk2.pdf.**

**Lec. 3 Feb. 5: Global existence of solutions.
Euler polygon construction of an approximate global solution.**
**Reading: pp. 33-42 of F. John's notes. pp. 371-377 of C & L.
.**
**Download odehwk3.tex.**
**Download odehwk3.pdf.**

**Lec. 4 Feb. 12: . Classification of critical
points of autonomous systems in 2D. Stability and**
**asymptotic stability of critical points. General topological considerations
in n dimensions. Periodic solutions.**
**Poincaré-Bendixon theory for existence of periodic solutions
in 2D.**
**Examples of limit cycles. Reading: Chapter 3 of F. John's notes.
pp. 391-398 of C & L.**
**Download odehwk4.tex.**
**Download odehwk4.pdf.**
**Correction: Question 5(c) was wrongly worded for this problem and
may be disregarded. The P-B theorem**
**says nothing about establishing non-existence of a periodic orbit.**

**Lec. 5 Feb. 19: . Outline of proof of the Poincaré-Bendixon
theorem. Stability of periodic solutions.**
**Continuous dependence of solutions on parameters, with application
to perturbation theory.**
**Evolution of volume in phase space. Volume preserving flows and
the analogy with incompressible fluid motion.**
** Reading: pp. 43-49 of F. John's notes.**
**Download odehwk5.tex.**
**Download odehwk5.pdf.**
**Correction: In problem (5) the term yz in the second equation should
be xz. In problem (3c) "extremum" is used**
**to mean "critical point".**

**Lec. 6 Feb. 26: . Volume conservation and recurrence.
Autonomous Hamiltonian systems with Hamiltonian. Principle**
**of least action and the Lagrangian. Definition of generalized momenta
from the Lagrangian. Example: the double**
**pendulum. Linear ODE's. The theory for equations of second order.
Linearly independent solutions. Variation of parameters.**
**Solving the inhomogeneous problem.**
**Reading: consult any elementary text to review solutions of 2nd
order linear ODE's. See Tabor, chapter 2,**
**for an introduction to Hamiltonian dynamics.**
**Download odehwk6.tex.**
**Download odehwk6.pdf.**
**Correction to hwk6: In problem (1), in the expression for H, the
parenthesis after m_2 should be removed. In problem 4(b),**
**the equation should be y"+2xy'+2y=0.**

**Lec. 7 Mar. 5: Linear systems of n first-order
linear equations, and the scalar nth-order linear equation.**
**Linear independence, Wronskian, solution of the inhomogeneous equation
by quadrature, reduction of order**
**to n-1 equations. Classical second-order linear equations: Bessel
functions, Legrendre functions.**
**Examples of solutions. Reading: Ch. 4 of John's notes. A useful
reference for special functions is**
**Abramowitz and Stegun, Handbook of Mathematical Functions, Dover
(1970).**

l

**Lec. 8 Mar. 19: Linear systems with constant
coefficients. The fundamental solution matrix. Direct calculation from
the Jordan**
**Normal form. The function exp(Ax) as a fundamental solution. Analytic
functions of matrices. exp(Ax) by residue theory.**
**Readiang: Chapter 5 of John's notes to page 118.**
**Download odehwk7.tex.**
**Download odehwk7.pdf.**

**Lec. 9 Mar. 26: Linear systems with periodic
coefficients. Floquet multipliers. Boundedness of solutions.m Hill's
and Mathieu's**
**equations. The representation Y(t)=P(t)exp(tC).**
**Readiang: Chapter 6 of John's notes, pp. 129-138. See some
remarks on pertubative solution of Mathieu's**
**equation fora periodic solutions: Mathieu.pdf.Mathieu.tex**
**Download odehwk8.tex.**
**Download odehwk8.pdf.**

**Note: Homeworks 8 and 9 will be due together April
9. There will be no homework session April 2. The last problem session**
**of the course will be April 16, covering Homework
10. I will be assigning review problems April 16 and 23rd which will form**
**a take-home final, due in my office (717 WWH) on or
before Thursday, May 2.**

**Lec. 10 Apr. 2: Stability theory for linear
equations. iPrevious results. Liapunov's theorem, with examples. The method
applied to the linear case with constant coefficients.**
**Application of the theory to nonlinear problems using the real Jordan
Normal Form.**
**Reading: Chapter 5 of John's notes, pp. 118-128.**
**Download odehwk9.tex.**
**Download odehwk9.pdf.**
**Corrections: In problem 1 z^(2) is periodic with period 2pi,
not pi. In problem 3(a) the "...origin of the (x,eta) plane."
In problem 4, the nonlinear problem is**
**stable but not asymptotically so according to the Liapunov theorem.
However, what happens at points where the Liapunov function vnaishes?**

**Lec. 11 Apr. 9: Two point boundary-value problems.
Formulation, solution alternatives, the adjoint operator, Green's
function. Sturm-Liouville**
**eigenvalue problems. Reformulation as an integral
equation, eigenfunction expansion, completeness.**
**Reading: John Ch. 8, C & L ch. 7.**
**Download odehwk10.tex.**
**Download odehwk10.pdf.**

**Lec. 12 Apr. 16: Bifurcation theory for autonomous
systems and boundary-value problems. Examples of the**
**pitchfork bifurcation: Buckling of the pinned-pinned
beam under compression.**
**Other bifurcation of first-order scalar equations.
General formulation for ODE systems. Hopf bifurcation.**
**Reading: Drazin 3-22, Verhulst ch. 13, Guckenheimer&Holmes,
ch. 3, especially section 3.4.**
**Download odehwk11.tex.**
**Download odehwk11.pdf.**

**Lec. 13 Apr. 23: Multiscale perturbation methods.
The method of averaging. Two-timing methods. Adiabatic invariants.**
**Singular perturbations. Relaxation oscillations
in the Van der Pol oscillator. Boundary layer methods.**
**Reading: Drazin, Verhulst, Hale have sections
relating to this material.**
**Download odehwk12.tex.**
**Download odehwk12.pdf.**

**Lec. 14 Apr. 30: Some applications involving
chaotic dynamics. The Lorentz system. Near-integrable Hamiltonians. Melnikov's**
**method.**