Math 119B: Ordinary Differential Equations, Spring, 2017

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Announcements

Here is a list of possible topics for the take-home final. You are also free to propose any other dynamical systems topic related to your own interests (mathematical, numerical, or an application).

• Limit cycles in planer, polynomial systems
• Floquet theory and stability of periodic orbits
• Proof of the Hopf bifurcation theorem
• Bogdanov-Takens bifurcation at a double-zero eigenvalue
• Periodic Hopf bifurcation
• Ergodic theory of dynamical systems
• Liapounov exponents
• The Melnikov method
• Classic papers in chaos (e.g. Lorenz, May, or Ruelle-Takens)
• The Feigenbaum constant
• Fractals and Hausdorff dimension
• Routes to chaos
• Chaos in Hamiltonian systems (e.g. the Henon-Heiles Hamiltonian)
• The Arnold cat map
• The Henon map

Text

Nonlinear Dynamics and Chaos, 1st ed., Steven H. Strogatz.

The ODEs from population dynamics that we discussed in class were taken from some course notes on Modeling Population Dynamics by André de Roos.

Syllabus

The course will cover material from Chapters 7 – 12 of the text: limit cycles, bifurcation theory and chaotic dynamical systems.

Important Dates

• First class: Mon, Apr 3
• Last day to add: Tue, Apr 18
• Last day to drop: Fri, Apr 28
• Last class: Wed, Jun 7
• Academic holidays: Mon , May 29

Exams

There will be one Midterm and a Final

• Midterm: Wed, May 10 (in class)
• Final: Will consist of a take-home project, due Wed, June 14.

Midterm

Here are solutions to the midterm.

The midterm will be in class on Wed, May 12. It will cover material from Chapters 7 and 8:

• 7.0 – 7.1 Limit cycles
• 7.2 Criteria to rule out limit cycles. Gradient flows. Liapounov functions. Dulac's criterion.
• 7.3 Poincare-Bendixson theorem
• 7.5 Relaxation oscillations. Van der Pol equation.
• 7.6 Weakly nonlinear oscillators. Poincare-Lindstedt method. Method of multiple scales.
• 8.0 Review of bifurcations in one-dimensional systems.
• 8.1 Saddle-node, transcritical, and pitchfork bifurcations of equilibria in planar systems.
• 8.2 Hopf bifurcations.
• 8.4 Homoclinic bifurcations.

Here are some sample midterm problems. Here are sample midterm solutions.

Homework

Homework problems will be assigned each week and collected in class.

Grade

The course grade will be based on (weights in parentheses):

• Homework (20%)
• Midterm (30%)
• Final (50%)

Other resources

If you want to type good-looking mathematics, the standard tool is LaTeX, or one of its many variants. See here to get started.

MATLAB is a useful platform for the numerical exploration of dynamical systems (and many other things). MATLAB for Students is available to UCD students at no charge for Spring Quarter. Go to the https://software.ucdavis.edu website and select MatLab for Students to access the software. This license will expire on June 30th.

Homework Sets

Problem numbers refer to the exercises in the text.

Set 1 (Friday, Apr 14)

• Sec. 7.1, p.227: 7.1.4, 7.1.6, 7.1.8, 7.1.9
• Sec. 7.5, p.233: 7.5.4
• Sec. 7.6, p.234: 7.6.2, 7.6.13, 7.6.16, 7.6.19

Set 2 (Friday, Apr 21)

• Sec 7.2, p. 229: 7.2.1, 7.2.2, 7.2.7, 7.2.15, 7.2.17
• Sec 7.3, p. 231: 7.3.1, 7.3.4, 7.3.6, 7.3.9, 7.3.10

Set 3 (Friday, Apr 28):

• Sec. 8.1, p. 284: 8.1.1, 8.1.2, 8.1.3, 8.1.4
• Additional problems

Set 4 (Friday, May 5):

• Sec 8.1, p. 284: 8.1.6, 8.1.10, 8.1.13
• Sec 8.2, p. 287: 8.2.1, 8.2.2, 8.2.3

Set 5 (Friday, May 19):

• Sec 8.2, p. 287: 8.2.12
• Sec 8.4, p. 291: 8.4.3 (optional)
• Sec. 10.1, p. 388: 10.1.10, 10.1.11