This is a course on the modern, qualitative theory of ordinary differential equations. The emphasis is not on producing elaborate formulae for `exact solutions' but on understanding the structural features of such solutions: do they settle to equilibrium? do they oscillate? what happens if the parameters are changed? and so on. The course is fastpaced and demanding.
Prerequisites: Math 230 or 231 and 250 or 251 are prerequisites for this class.
Meeting Times: The class meets twice a week, on Tuesdays and Thursdays at 9.4511.00 a.m. in 109 Osmond. (Occasionally, we may meet in another room for experiments or technology demonstrations; these meetings will be notified in the previous class and by email.)
Textbook: Steven H. Strogatz, Nonlinear Dynamics and Chaos, AddisonWesley.
Office hours will take place Tuesday 23 p.m. and Wednesday, 10.3011.30 a.m. Students are strongly encouraged to make use of available office hours to discuss any questions or problems that they may have about the course or about mathematics more generally.
Calculators will not be necessary for the course, and are not permitted on the tests or on the final exam. You will need to use the computer programs dfield and pplane for plotting the solutions to differential equations. These are available from this web site in Java versions which should run in any Javaenabled web browser.
Academic Integrity Statement All Penn State policies regarding ethics and honorable behavior apply to this course. Academic integrity is the pursuit of scholarly activity free from fraud and deception and is an educational objective of this institution. Academic dishonesty includes, but is not limited to, cheating, plagiarizing, fabricating of information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. For any material or ideas obtained from other sources, such as the text or things you see on the web, in the library, etc., a source reference must be given. Direct quotes from any source must be identified as such. All exam answers must be your own, and you must not provide any assistance to other students during exams. Any instances of academic dishonesty will be pursued under the University and Eberly College of Science regulations concerning academic integrity.
Grading Your grades for this course will be computed on the basis of weekly homework assignments, an inclass midterm, and a final exam. Homework assignments will be posted on this web site and handed out in class. They will be due on following Thursdays: 1/20, 2/3, 2/17, 3/17, 3/31, 4/14, 4/28, and I will aim to return graded homework on the following Tuesdays. Each assignment will contain five questions.
Grades will be calculated as follows:
 Each homework question will be graded out of 25 points. The best six of your homework assignments will contribute to your total score, up to a maximum of 150 points. The midterm will be graded out of 50 points, and the final exam out of 100 points.
 Grades will be assigned on the basis of your total score. I anticipate that around 245 points will suffice for an A grade, but this may be changed as the course progresses.
Final Exam. The final exam will take place on Tuesday May 3rd at 4:40 pm in 109 Osmond. It will be a comprehensive exam.
January


Tue

Thu


4.

6.


11.

13.
First order differential equations. Phase space and the picture of an ODE as giving a 'flow' in phase space. Various tricks for reducing higherorder, or nonautonomous, differential equations to systems of firstorder autonomous equations. 

18.
Fixed points of differential equations. Local analysis using Taylor's Theorem. Classification of fixed points into stable, unstable, and 'halfstable'. 
20.
Homework 1 due. Existence and uniqueness questions. Finite time blowup for nonlinear equations. 

25.
Numerical methods for solving ODE; Euler's method. Taylor series and the estimation of error in Euler's method. Using Euler's method to prove the existence and uniqueness theorem for ODE. 
27.
The definition of a bifurcation ("change in the qualitative behavior of a system as a parameter is varied"). Example: the bead on a rotating wire. 
February


Tue

Thu


1.
Examples and definitions of various kinds of bifurcations: saddlenode, transcritical, pitchfork. 
3.
Homework 2 due. Taylor's theorem and normal forms for bifurcations. The criterion for recognizing a saddlenode bifurcation. 

8.
More examples of Taylor's theorem and normal forms. Imperfect bifurcations and the cusp catastrophe. 
10.
Oscillatory phenomena (Chapter 4 Strogatz). The nonuniform oscillator and associated bifurcations; frequency locking in fireflies. 

15.
Begin study of linear systems (mostly 2dimensional) as described in Chapter 5 of Strogatz. Eigenvalues, eigenvectors, and their application to solving linear systems. Detailed discussion of the case where the eigenvalues are real and distinct. 
17.
Homework 3 due. 2dimensional linear systems: the case of complex eigenvalues. Phase portraits of various kinds of 2dimensional linear systems: saddles, centers, spirals, nodes. 

22.
Practice Problems 5.2.4, 5.2.6, 5.2.8 will be discussed today. Review Homework 3. Brief introduction to the Jordan Normal Form. Higherdimensional linear systems. 
24.
Class was canceled because I was ill. Here is what I had intended to tell you. 

March


Tue

Thu


1.
Review Session. 
3.
Midterm Exam. 

8.
NO CLASS  SPRING BREAK 
10.
NO CLASS  SPRING BREAK 

15.
Showed computer examples of various secondorder nonlinear systems and the qualitative features of their solutions including: attractive fixed points, saddles, nonlinear centers, limit cycles. Defined the notions of 'attractive' and 'Liapunov stable' for fixed points. Introduced Liapunov functions which give a sufficient condition for Liapunov stability. 
17.
Homework 4 due today. Discussed linearization near fixed points of 2D systems (using Taylor's theorem). Sketched proof that there is a Liapunov function near to a fixed point where the linearization has eigenvalues with negative real part; this is an example of a general principle that when the eigenvalues have nonzero real part (hyperbolic case) the linearization governs the local behavior of the nonlinear system. Discussed one competitive exclusion example (book p.155) in detail. 

22.
Worked practice problems on sketching phase portraits: 6.3.1, 6.3.3. Reviewed theorems relating Liapunov functions to stability, and gave detailed proof of one such theorem. Defined the notion of a conservative system, and observed that such a system can have no attracting fixed points. 
24.
Homework 5 assigned. Defined hyperbolicity for fixed points. State the Stable Manifold Theorem and the classical terminology ('separatrix', 'basin of attraction') for the 2D case. More details an examples of conservative systems. Homoclinic and heteroclinic orbits. 

29.
Defined the (Poincare) index of a closed path with respect to a vector field. Discussed its properties: homotopy invariant, obtained by adding up contributions from interior fixed points. Applications to the (non) existence of closed orbits.

31. Homework 5 due. Held pop quiz. Review examples related to the Poincare index. Discussed Dulac's criterion for ruling out closed orbits, and its relationship to the Divergence Theorem. 
April


Tue

Thu


5.
Returned homework 5. Reviewed it in detail. 
7.
PoincareBendixson theorem. Trapping regions. Closed orbits and limit cycles. 

12.
Lienards theorem. The Lienard plane. Proof of Lienard's theorem (simplest case); construction of a trapping region. Example of the van der Pol oscillator. The strongly nonlinear limit; relaxation oscillations. 
14.
The saddlenode bifurcation of fixed points in two dimensions. "Bifurcations occur where hyperbolicity breaks down." Significance of tangent nullclines. 

19.
Transcritical and pitchfork bifurcations of fixed points in two dimensions. 
21.
The Hopf bifurcation: simple example using polar coordinates. Supercritical and subcritical Hopf bifurcations. 

26.
Chemical oscillators: the BelusovZhabotinsky reaction. 
28.
Review session. Assigned additional review problems (as requested): 7.1.1, 7.1.7, 7.1.8, 7.2.7, 7.2.12, 7.2.13, 7.3.5, 8.1.1, 8.1.6, 8.1.7, 8.2.2, 8.2.3, 8.2.4 