MATH 345 Jan-April 2013 Lecture Summary


We
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Date Lec-
ture
Contents
1 1231


0102 L01
outline
overview; examples of the geometric approach
0104 L02 PART I. One dimensional flows
§2.1-2.3   one dimensional flow, fixed points, their stability, logistic equation
2 0107 L03 §2.4   stability analysis: by graph and by derivative
0109 L04 demonstration of the software XPPAUT
§2.5   the theorem on unique existence of solutions
0111 L05 §2.5   impossibility of oscillation
§2.6   potential
§3.1   saddle-node bifurcation
3 0114 L06 §3.1   bifurcation diagram, normal form
0116 L07 §3.2   transcritical bifurcation
0118 L08 §3.4   pitchfork bifurcation: supercritical and subcritical
4 0121 L09 §3.4   subcritical pitchfork bifurcation with higher order damping term, jump and hysteresis
0123 L10 §3.5   over-damped bead on a rotating vertical hoop
0125
L11 §3.6   imperfect bifurcation
5 0128
L12 §3.6   imperfect bifurcation continued
§3.7   insect outbreak
0130
L13 §3.7   insect outbreak continued
PART II. Two dimensional flows
Chapter 5   2D linear flows
0201
L14 §5.2   classification of 2D linear flows
6 0204
L15 Chapter 6   2D nonlinear flows
§6.3   fixed points and linearization
0206
L16
§6.3   examples
0208
L17 §6.4   competition models
7 0211

Family Day
0213

midterm exam
0215
L18 §6.5   conservative systems, example: Newton's equation
0218-0222
midterm break
8 0225
L19 §6.5   example: Hamiltonian systems
§6.6   Reversible systems
0227
L20 §6.6   examples
0301
L21 Chapter 7   Limit cycles
§7.1   limit cycles, examples
§7.2   nonexistence: gradient flows
9 0304
L22 §7.2   nonexistence: Lyapunov functions, Dulac's criterion
0306
L23 §7.2   examples of Dulac's criterion
§7.3   Poincare-Bendixson theorem
0308
L24 §7.3   examples of Poincare-Bendixson theorem
10 0311
L25 §7.4   Lienard equation, Poincare map
0313
L26 §8.1   SN, TC and PF bifurcations for 2D systems
§8.2   Introduction to Hopf bifurcation
0315
L27 §8.2   Hopf bifurcation
11 0318
L28 §8.2   one more example
§8.3   Belousov-Zhabotinsky reaction: oscillating chemical reactions
          links: wikipedia, youtube video 1, youtube video 2
0320
L29 §8.3   continued
§8.7   Poincare map
0322
L30 §8.7   continued
12 0325
L31 §8.6   invariant tori, (optional: an example of an invariant torus)
§9.2   Lorenz equations
0327
L32 §9.2   Lorenz equations continued
0329

Good Friday
13
0401

Easter Monday
0403
L33
§9.3   chaos and strange attractor
§9.5   exploring the parameter space
0405

finish §9.5 and review