# Math 415 Applied Dynamical Systems, Chaos and Modeling: Fall 2017

 Lecture Room: B239 Van Vleck Lecture Time: 9:30–10:45 TR Lecturer: Jean-Luc Thiffeault Office: 503 Van Vleck Email: Office Hours: Tuesdays 10:45–11:45; Thursdays 1:30–2:20

## Final exam

The final exam is on Thursday December 21 at 2:45pm, in the regular classroom (B239). It will last 2 hours. The exam consists of 4 questions. You are responsible for all the material in the class, except as noted below, but the exam will lean heavily on the post-midterm parts.

Here is a not-completely-exhaustive list of what you're expected to know:

• Fixed points, nullclines, linearization, stability, drawing a phase portrait.
• Existence, uniqueness, consequences.
• Limit cycles: ruling out their existence. Gradient systems. Lyapunov functions: know how and why they work, and how to derive them for simple examples. Using index theory to rule out limit cycles. Poincaré–Bendixson Theorem.
• Conserved quantities: show a quantity is conserved, find energy for simple systems such as [conserved + potential] systems. Plot phase portrait. Find the equation for an orbit linking multiple fixed points (heteroclinic orbit). Pendulum.
• NOT COVERED: maps, chaos, Lorenz equations. Basically anything that came after homework 5 (up to and including 7.3).

## Syllabus

See the official syllabus.

## Textbook

The textbook for the class is Nonlinear Dynamics and Chaos by Steven Strogatz (Second Edition).

## Prerequisites

Math 319 or 320, or consent of instructor.

## Homework

Every two weeks or so I will assign homework from the textbook and post it here.

HW 1 (Due Sept 21 26): page 36: 2.1: 1–4; 2.2: 1–4, 7, 10, 13; 2.3: 1, 4; 2.4: 1, 3, 6, 9; 2.5: 1, 2, 4, 6; 2.7: 1, 3. Solutions

HW 2 (Due Oct 10 12): page 80: 3.1: 1, 2; 3.2: 1, 2, 4; 3.3: 1 [except (d)]; 3.4: 1, 2, 4, 5, 7, 10, 11, 15; 3.5: 2, 3, 6 [except (e)]; 3.6: 2, 4; 3.7: 1–4. Solutions

HW 3 (Due Oct 26): page 115: 4.1: 1–3; 4.3: 2, 3; 4.4: 4; 4.5: 1, 3; page 140: 5.1: 3–6, 9, 10; 5.2: 1–4, 7, 10, 11, 13. Solutions

HW 4 (Due Nov 9 Nov 14): page 181: 6.1: 1, 2, 4, 6; 6.2: 2; 6.3: 1, 2, 4, 5, 9 (except (e)), 11, 13; 6.4: 3, 4. [6.3.2, 6.3.5: OK in this case to assume centers persist.] Solutions

HW 5 (Due Nov 28): page 187: 6.5: 1, 2, 3, 11, 19; 6.7: 2 [except the 'reversible' question in (c)]; 6.8: 1, 3, 4, 5, 6, 8; page 230: 7.1: 1, 4, 8; 7.2: 2, 6, 10, 14; 7.3: 1, 3, 4. Solutions

## Piazza

We'll use Piazza Q&A for discussions about the class and related topics. Feel free to post questions and answers there about homeworks and exams, logistics, or relevant interesting things you found on the web. Note that we will only use Piazza for the Q&A feature, not for posting the actual homeworks.

## Schedule of Topics

 lecture date sections topic 1 09/07 1 Introduction 2 09/12 2.1–2.4 One-dimensional ODEs 3 09/15 2.5–2.8 Existence and uniqueness 4 09/19 3.1 Saddle-node bifurcations 5 09/21 3.2–3.4 Transcritical and pitchfork bifurcations 6 09/26 3.5 Overdamped bead 7 09/28 3.6 Imperfect bifurcations; catastrophes 8 10/03 3.7 Insect outbreak 9 10/05 4 Flows on the circle 10 10/10 4.5; 5.1 Fireflies; Two-dimensional linear flows 11 10/15 5.2 Two-dimensional linear flows (cont'd) 12 10/17 5.2; 6.1 Two-dimensional linear flows (finish); Phase plane 13 10/19 6.1–6.3 Phase plane (cont'd) 14 10/24 6.4 Lotka–Volterra model 15 10/26 – Modeling plants in lake Wingra (notes) – 10/31 – Homework Q&A – 11/02 – MIDTERM (first three homeworks) 16 11/07 6.5 Convervative systems 17 11/09 6.7, 6.8 Pendulum; Index theory 18 11/14 6.8, 7.1 Index theory (cont'd); Limit cycles 19 11/16 7.1–7.2 Limit cycles (cont'd) 20 11/21 7.3 Poincaré–Bendixson Theorem 21 11/28 9.2 The Lorenz equations (Lorenz's paper) 22 11/30 9.2–9.3 The Lorenz equations: numerical demonstration (Matlab files) 23 12/05 10.0–10.1 One-dimensional maps 24 12/07 10.2–10.4 Periodic orbits of maps; Chaos (Matlab files) – 12/12 – Discussion of final

There will be a midterm exam in class and a final exam. The final grade will be computed according to:

 Homework 35% Midterm exam 30% Final exam 35%

## Exam Dates

 Midterm exam Thursday November 2, 2017 at 9:30–10:45 (in class) Final exam Thursday December 21, 2017 at 14:45–16:45 (room B239)

## Midterm Results

 # of scores 53 mean score 65.4 standard deviation 19.5 median score 64

Midterm solutions