CDS 140, Winter 2016

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* Center manifold theorem
* Center manifold theorem
| Perko, 2.11-2.13
| Perko, 2.11-2.13
| [ hw4-wi16.pdf] <br> Due: 3 Feb (Wed)
| [ hw4-wi16.pdf] <br> Due: 10 Feb (Wed)
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| 8 Feb <br> 10 Feb <br> 12 Feb
| 8 Feb <br> 10 Feb <br> 12 Feb
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* Periodic orbits and limit cycles
* Periodic orbits and limit cycles
| Perko, 3.1-3.3
| Perko, 3.1-3.3
| [[CDS 140a Winter 2016 Homework 5|HW 5]] <br> Due: 10 Feb (Wed)
| [ hw5-wi16.pdf] <br> Due: 17 Feb (Wed)
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|- valign=top
| 17 Feb <br> 19 Feb
| 17 Feb <br> 19 Feb

Revision as of 15:27, 8 February 2016

Introduction to Dynamics


  • Douglas MacMartin,
  • John Doyle,
  • Lectures: MWF, 1-1:55, 213 ANB
  • Office hours:
    • DGM: Mon 11:30-1pm, Wed 2-3 pm (please e-mail to confirm)

Teaching Assistants

  • Benson Christalin (CDS)
  • Contact:
  • Office hours: Monday 2-3 pm


  • Homework will be due Wednesdays in class (or by 5pm to Benson). Homework assignments are posted below.
  • Piazza If you have not received an email to sign up for Piazza, please email us!
  • Dates for recitation are shown in italics in schedule. Dates where JCD is lecturing given with asterisk
  • HW2 is posted below. Note that if you want, you may hand in the last question on HW1 with HW2 (since we haven't covered that yet), but probably easier for you to hand in with HW1.

Course Description

Analytical methods for the formulation and solution of initial value problems for ordinary differential equations. Basics in topics in dynamical systems in Euclidean space, including equilibria, stability, phase diagrams, Lyapunov functions, periodic solutions, Poincaré-Bendixon theory, Poincaré maps. Introduction to simple bifurcations, including Hopf bifurcations, invariant and center manifolds.

Lecture Schedule

Date Topic Reading Homework
4 Jan
6 Jan
8 Jan
Linear Differential Equations L1-1 Lecture notes

LinSys notes

  • Course overview and administration
  • Linear differential equations
  • Matrix exponential, diagonalization
  • Stable and unstable spaces
  • S + N decomposition, Jordan form

Perko, 1.1-1.10

Due: 13 Jan (Wed)
11 Jan
13 Jan
15 Jan
Nonlinear differential equations

Lecture notes

  • Existence and uniqueness
  • Flow of a differential equation
  • Linearization
Perko, 2.1-2.6 hw2-wi16.pdf
Due: 20 Jan (Wed)
20 Jan* 2 hrs
22 Jan
Chaos, fractals, and global analysis using SOStools
  • Chaotic systems (logistic map, Mandelbrot set)
  • SOStools for finding Lyapunov functions
25 Jan
(2 hrs)
Jan 29
Behavior of differential equations L4
  • Stable and unstable manifolds
  • Stability of equilibrium points for planar systems
Perko, 2.7-2.10 hw3-wi16.pdf
Due: 3 Feb (Wed)
1 Feb*
3 Feb*
5 Feb*
Non-hyperbolic differential equations
  • Lyapunov functions
  • Center manifold theorem
Perko, 2.11-2.13 hw4-wi16.pdf
Due: 10 Feb (Wed)
8 Feb
10 Feb
12 Feb
Global behavior
  • Limit sets and attractors
  • Krasovskii-Lasalle invariance principle (if time)
  • Periodic orbits and limit cycles
Perko, 3.1-3.3 hw5-wi16.pdf
Due: 17 Feb (Wed)
17 Feb
19 Feb
Limit cycles
  • Poincare' map
  • Bendixson criterion for limit cycles in the plane
  • Describing functions (if time)
Perko, 3.4-3.5, 3.7 HW 6
Due: 17 Feb (Wed)
22 Feb 2 hrs
26 Feb
  • Sensitivity analysis
  • Structural stability
  • Bifurcation of equilibrium points
Perko 4.1-4.2

BFS 3.2 and 5.4 (PDF)

HW 7
Due: 24 Feb (Wed)
29 Feb*
2 Mar*
4 Mar*
  • Hopf bifurcation
  • Application example
Perko 4.3-4.5 + notes HW 8
Due: 2 Mar (Wed)
7*, 9* Mar
Course review Final exam
Due: 18 Mar (Wed). Pick up from Nikki Fountleroy, 107 Steele Lab


The primary text for the course (available via the online bookstore) is

 [Perko]  L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, 2006.

The following additional texts may be useful for some students:

 [G&H]  J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, 1990.
 [H&S]  M. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Springer-Verlag, 1990.
 [J&S]  D. Jordan and P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers, Fourth Edition. Oxford University Press, 2007. (On reserve in SFL)
 [Ver]  F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edition. Springer, 2006. (On reserve in SFL)


The final grade will be based on homework and a final exam:

  • Homework (75%) - There will be 9 one-week problem sets, due in class approximately one week after they are assigned. Late homework will not be accepted without prior permission from the instructor.
  • Final exam (25%) - The final will be handed out the last day of class and is due back at the end of finals week. Open book, time limit to be decided (likely N hours over a 4-8N hour period).

The lowest homework score you receive will be dropped in computing your homework average. In addition, if your score on the final is higher than the weighted average of your homework and final, your final will be used to determine your course grade.

Collaboration Policy

Collaboration on homework assignments is encouraged. You may consult outside reference materials, other students, the TA, or the instructor. Use of solutions from previous years in the course or from other external sources is not allowed. All solutions that are handed should reflect your understanding of the subject matter at the time of writing.

You can use MATLAB, Mathematica or a similar programs, but you must show the steps that would be required to obtain your answers by hand (to make sure you understand the techniques).

No collaboration is allowed on the final exam. You will also not be allowed to use computers, but the problems should be such that extensive computation is not required.

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