Ordinary Differential Equations, Spring 2002 (G63.2470)

Lecture: Tuesday, 5:10-7:00 p.m., Room 1013 WWH. First class January 22.

Problem session: 4:00-5:00, Room 1013 WWH. First session February 5. (TA:  Elisha Kobre.)
Homework is due the Tuesday following. First assignment will handed out January 22, due January 29.

Books on reserve:

Arnold, V.I. Ordinary Differential Equations, MIT Press ,1987.

Coddington & Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.

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

John, Fritz, Ordinary Differential Equations, CIMS Lecture Notes, 1965.

Tabor, Michael, Chaos and integrability in nonlinear dynamics, John Wiley, 1989.

Drazin, P.G. ,  Nonlinear Systems, Cambridge, 1992.

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.

Office hours: 3:00-4:00 Tuesday, Room 717 WWH, or by appointment (childress@cims.nyu.edu, 998-3135)

Revised Syllabus:

1. Overview.
2. Local existence and uniqueness.
3. Global existence. Approximation of global solutions by Euler construction. Two-dimensional systems
4. Autonomous equations in the plane continued. Poincaré- Bendixon theory.
5. P-B theory concluded, continuous dependence on parameters, volume in phase space.
6. Hamiltonian systems. Algebra of second-order linear equations. Solution techniques.
7. Classical second-order equations.  Formulation for the general 1st-order system.
8. Constant coefficient linear equations. Jordan blocks, fundamental solutions,  analytic structure.
9.  Periodic coefficients.
10. Stability- Liapunov methods.
11. Two point boundary-value problems.
12. Bifurcation and Perturbation I.
13. Bifurcation and perturbation II.
14. Chaotic systems.

Homework and notes:

Lec. 1 Jan. 22: Overview: Scope of the course. General first-order scalar ode.  Linear versus nonlinear equations.
Local and global issues of existence and uniqueness. The simple pendulum equation as an example of qualitative
theory of a second-order equation. The phase plane. Autonomous versus non-autonomous equations.
Stability of solutions.
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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.
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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.
Download odehwk4.tex.
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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.
Download odehwk5.tex.
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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.
Download odehwk6.tex.
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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).
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
Download  odehwk8.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.
Download  odehwk9.tex.
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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