## MATH 345 Jan-April 2013 Lecture Summary

 We ek 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