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,
Springer,
1983.
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.
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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.
.
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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.
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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.
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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.
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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
Note: There is NO problem assignment due March 19.
Problems 4.2.1, 4.2.2, 4.2.3 of John's chapter 4 will be included
in the assignment due March 26. Additional
problems will be assigned March 19. There will be no problem session
March 19.
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.
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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
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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.
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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.
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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.
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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.
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Lec. 14 Apr. 30: Some applications involving
chaotic dynamics. The Lorentz system. Near-integrable Hamiltonians. Melnikov's
method.