Math 443, Section C1; Fall Semester 1998
Ordinary Differential Equations
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Professor:
Eugene Lerman
334 Illini Hall
244-9510
lerman@math.uiuc.edu, URL http://www.math.uiuc.edu/~lerman/
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Class Time/Place:
MWF 10 343 Altgeld
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Prerequisites:
Undergraduate analysis or a smattering
of topology of metric spaces.
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Text:
No text required. Recommended text:
P. Glendinning Stability, Instability and Chaos.
An Introduction to the Theory of Nonlinear Differential Equations.
Syllabus:
- 1.
- Introduction:
- phase spaces, vector fields and flows;
- existence and uniqueness theorems
- limit sets
- 2.
- Stability: various notions of stability, Liapunov functions
- 3.
- Linear differential equations
- exponentiation of operators
- real and complex Jordan normal forms of matrices, autonomous
linear ODEs.
- Floquet theory
- 4.
- Linearization and hyperbolicity
- linearization
- stable manifold theorem
- Hartman-Grobman theorem
- 5.
- Dynamics in two dimensions
- Poincaré index
- Bendixon and Dulac criteria
- Poincaré-Bendixon theorem
- 6.
- Periodic orbits and Poincaré maps
- Poincaré maps
- structural stability, genericity, transversality
- 7.
- Bifurcation theory
- center manifold theorem
- the saddle node bifurcation
- the pitchfork bifurcation
- the Poincaré-Andronov-Hopf bifurcation
last changed 5/8/1998