CDS 140, Winter 2015
Differential Equations and Dynamical Systems
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.
| 5 Jan
| Linear Differential Equations
| HW 1 |
Due: 14 Jan (Wed)
| 12 Jan
| Nonlinear differential equations
||Perko, 2.1-2.6|| HW 2 |
Due: 21 Jan (Wed)
| 21 Jan
| Behavior of differential equations
||Perko, 2.7-2.10|| HW 3 |
Due: 28 Jan (Wed)
| 26 Jan
| Non-hyperbolic differential equations
||Perko, 2.11-2.13|| HW 4 |
Due: 4 Feb (Wed)
| 4 Feb
| Global behavior
||Perko, 3.1-3.3|| HW 5 |
Due: 11 Feb (Wed)
| 9 Feb
| Limit cycles
||Perko, 3.4-3.5, 3.7|| HW 6 |
Due: 18 Feb (Wed)
| 18 Feb
|| Perko 4.1-4.2 + notes
|| HW 7 |
Due: 25 Feb (Wed)
| 23 Feb
||Perko 4.3-4.5 + notes|| HW 8 |
Due: 4 Mar (Wed)
| 2 Mar
| Other classes of dynamical systems
||TBD|| HW 9 |
Due: 13 Mar (Fri)
| 11 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 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.