Math 134: Dynamical Systems


Instructor: Katy Craig, MS 7354, katy•craig•math at gmail • com

Teaching Assistant: William Oakley, MS 2344, wgoakley at ucla • edu

Lecture/Section/Office Hours: Times and Locations

Textbook: S. Strogatz, Nonlinear Dynamics and Chaos, second edition.

Other Recommended References:

S. Strogatz, Lectures on Nonlinear Dynamics and Chaos. (YouTube videos from when our textbook’s author taught a similar course at Cornell)

J. D. Crawford, Introduction to Bifurcation Theory. (Free, if you download from the campus network)

  1. S.Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos.

Exams: There will be two midterms and one final exam. The examinations will be closed book and closed note. There will be no retaking or rescheduling of exams under any circumstances, as the grading scheme allows you to drop your lowest midterm score.

First Midterm: Wednesday, February 4th, 9:00-9:50AM

Second Midterm: Wednesday, March 4th, 9:00-9:50AM

Final Exam: Wednesday, March 18th, 8:00-11:00AM

Grading Policy:

Your grade will be calculated based on whichever of the following two schemes results in a higher grade.

Scheme 1:                                            Scheme 2:

    Homework: 10%                                    Homework: 10%

    First Midterm: 25%                                 Highest Midterm Score: 40%

    Second Midterm: 25%                            Final: 50%

    Final: 40%

If you have questions about the grading of any assignment or exam, you have one week after it is handed back to request a regrade.

Prerequisites: Math 33B




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  1. Homework assignments will be posted on this website and collected Wednesday during lecture.

  2. Only the problems marked with an asterisk should be submitted for grading.

  3. At least one problem on each of the exams will be chosen from the non-asterisked homework problems.

  4. No late homework will be accepted. (Talk to me if you transfer into the course partway through the quarter, and we’ll work something out.)

  5. The lowest two homework grades will be dropped and will not count toward the final grade.

Homework 0 (due Wednesday, January 7th): HW0.pdf

Homework 1 (due Wednesday, January 14th): HW1.pdf (for Q6, the y-axis should be labeled “x(t)” and the x-axis should be labeled “t”)

Homework 2 (due Wednesday, January 21st): HW2.pdf (for Q5, part a, the equation should be "av(t) = A h'(t)" not "av(t) = A · h(t)")

Homework 3 (due Wednesday, January 28th): HW3.pdf (for Q8, you should sketch V(x) as a function of x, not r -- this is a type-o)

Homework 3.5 (extra practice for midterm, not to be turned in): HW3p5.pdf (for Q2, f(x)<0 should be f’(x)<0 and -x1/2 should be -|x|1/2)

Your midterm will be of similar difficulty to the homework problems I have assigned, and some problems will be identical to the non-asterisked homework problems.

Homework 4 (due Wednesday, February 11th): HW4.pdf (for Q2, use a computer; for Q3 assume z(0)=0; Q5(b) should read -dV/d\theta = 1)

Homework 5 (due Wednesday, February 18th): HW5.pdf

Homework 6 (due Wednesday, February 25th): HW6.pdf(for Q7, remove the last bit of bullet 2:``but the effect is more severe for the aliens.”)

Homework 6.5 (extra practice for midterm, not to be turned in): HW6p5.pdf (for Q3, it should be \gamma = r \omega^2/g, not r\omega^2 g)

Homework 7 (due Wednesday, March 11th): HW7.pdf (for Q3, the fixed point should be (0,2), not (0,0).)

Homework 7.5 (extra practice for final, not to be turned in): HW7p5.pdf (for Q2, you may assume there are no FPs in the trapping region)

read by today

Strogatz 1.0-1.3

Strogatz 2.0-2.3

Strogatz 2.4-2.6

Strogatz 2.7-2.8

Strogatz 3.0-3.1

Strogatz 3.2-3.3


Strogatz 3.4-3.5



Strogatz 3.6-3.7

Strogatz 4.0-4.3

Strogatz 4.4-4.5


Strogatz 5.0-5.1


Strogatz 5.2-5.3

Strogatz 6.0-6.2


Strogatz 6.3-6.4

Strogatz 6.5-6.6

Strogatz 6.7

Strogatz 6.8


Strogatz 7.0-7.1


Strogatz 7.2

Strogatz 7.3

Strogatz 8.0-8.1

Strogatz 8.2-8.3

topics to be covered

Dynamical systems and ordinary differential equations

1D flows, linear vs. nonlinear, fixed points, stability, population dynamics

Linear stability analysis, existence and uniqueness, impossibility of oscillations

Potentials, introduction to numerical methods

Intro to bifurcations, saddle-node bifurcation, bifurcation diagrams

Normal forms, transcritical bifurcation

Happy MLK Day!

Pitchfork bifurcation

Pitchfork bifurcation (ctd.), application to overdamped bead on rotating hoop

Application to overdamped bead on rotating hoop (ctd.), dimensional analysis

Imperfect bifurcations, insect outbreak model

Flows on the circle, beating, nonuniform oscillators, ghosts, bottlenecks

Oscillator examples, applications to overdamped pendulum, fireflies

Midterm 1, covering lectures 1-12

2D linear systems, phase portraits


Classification of linear systems, characteristic equation, types of fixed points

2D nonlinear systems, existence/uniqueness, trajectories cannot cross

Happy President’s Day!

Stability, fixed points, linearization/effect of nonlinear terms, hyperbolicity, Hartman-Grobman theorem

Special nonlinear systems, conservative/reversible systems, heteroclinic/homoclinic orbits

Application of nonlinear phase plane analysis to classic pendulum problem

Index theory, local vs global methods

Index theory continued

Limit cycles

Midterm 2, covering lectures 13-24

Ruling out limit cycles, gradient systems, Liapunov functions

Existence of closed orbits, Poincare-Bendixson thm, trapping regions, impossibility of chaos

Bifurcations in higher dimensions, saddle-node, transcritical, pitchfork bifurcations

Hopf bifurcation, supercritical/subcritical/degenerate types

Final Exam, 8-11AM, covering lectures 1-30

Outline of Course:

Part 1: One dimensional flows

  1. fixed points and linear stability analysis

  2. bifurcation theory

  3. -saddle-node, transcritical, pitchfork

  4. flows on the circle

Part 2: Two dimensional flows

  1. linear vs. nonlinear systems

  2. phase plane analysis

  3. limit cycles

  4. bifurcation theory

  5. -saddle-node, transcritical, pitchfork, Hopf