Prof. Bruce Peckham, Department of Mathematics and Statistics, University of Minnesota, Duluth

- Syllabus
- Dynamical Systems software links (Not all are fully functional. Caution: Some links lead to commercial sites, none of whichendorsed by the instructor. :) )
- Boston University Dynamics Website
- Prof. Luis Alseda (Barcelona, Spain) links to dynamical systems web software
- ODE Factory
- Rice University Direction Field and Phase Plane software: http://math.rice.edu/~dfield/dfpp.html
- Winplot. Free downloadable plotting software with graphical iteration (cobweb) capabilities. For Windows only.
- Filled Julia Set software for quadratic maps and singular maps
- Fraqtive - free Mandelbrot-Julia set software to install
- Check grades recorded to date.

**Set
1, due FRI Sept. 1 **

- Strogatz: Read Preface and
Chapter 1: Overview.
- Devaney: Read Preface and Ch's
1-4
- Devaney Ch 3:
1,3,4,5,6,7abh,8,11,12,13,14
- Reproduce the table on p. 23 using a spreadsheet (such
as EXCEL). All your numbers might not match exactly. Why? (Thought
question - not to hand in.) If your printout is more than one page,
provide only the first page.
- Do the following tasks with the help of the BU Website
by linking to "Java Applets for chaos and fractals" and then
"Linear Web" or "Nonlinear Web" (from any computer
with web access and JRE (Java Runtime Environment) installed:
- Experiment to see how to operate the Linear Web and Nonlinear Web software. (Nothing to hand in.)
- Do Experiment 3.6 on
pages 25-6 of the text, including the essay and Notes and Questions 1 and
2 (but not 3). The book suggests using 10 initial conditions for each of
the three functions, but you may use just 5. You
may use either a spreadsheet or Nonlinear Web (or both). All the
functions are available via a menu in Nonlinear Web. You can type the
functions in yourself in Excel. Hints: You can use the mod function to
write the doubling function. Also for the doubling function, you might
want to experiment with the number of decimal places printed. Make sure
you iterate at least 70-100 times for each seed. You may use printouts to
support your writeup, but you are not required
to do so.

- Devaney Ch 4: 1bcg,2ab,7 (Optional: Use BU website ->; Java Applets for Chaos and Fractals ->; Linear Web to experiment and to check your answers to problem 7.)

- Read Devaney: Chapters 5-6.
- >Ch 5: 1bcj, 2abc, 3, 4ad,
5,7,9 (for f'(x_0)=+1 only; OK to assume the
result of 8)

Extra Credit: State and prove a neutral fixed point theorem analogous to problem 9, except assuming F'(x_0)=-1. (Big hint: Expand F in a power series around the fixed point, compute F(F(x)), and apply problem 9.) - Complete the following chart for periods n=1,2,...,10 for the quadratic map with c=-2 (Q
_{c}(x)=x^{2}+c) using the following column headings:

n, # period-n pts, # least period-n points, # least period-n orbits. Hint: You can use Nonlinear web for n<=6, but you will need to think of the properties of the graph of the nth iterate for higher iterates.

- Read Devaney: Chapters 6-7.
- Ch 6: 1aef (Suggestion: use software for part f), 3,4,5,6,7,8,9 (Sketch the bifurcation diagram only; plot attracting fixed points with solid lines, repelling fixed points with dashed lines.)
- Experiment 6.4 We will do this experiment together as a class. Each person should record the eventual behavior of four seeds on the class graph. Seeds for each person will be distributed in class; a chart on which to plot your results will also be provided. Assigned parameter values are provided here.
- Do the lab tasks on tangent and period-doubling
bifurcations. Handout in class or download here.

- Read Devaney: Chapters 7-10.
- Ch 7: 1,2,3,9,10,11,12,13 (OK to assume result of 9 to prove 12 and 13, so this is not a formal proof.) Extra Credit: 8.
- Show that for any given real number r>0 there exists
a c such that rx(1-x) is conjugate to y
^{2}+c. Hint: Try a conjugacy of the form h(x) = Ax + B. Solve for A,B,c in terms of r. - Lab Related Work:
- Ch. 8: Do Experiment 8.3: Windows in the Orbit Diagram. Do Notes and Questions 1,3,4. Extra credit: Question 5. Also extra credit: Ch. 8: 16,17
- Attracting parameter
intervals for the quadratic map. (This is an extension of the lab work
from HW Set 3, # 3.) Use the Nonlinear Web/Orbit Diagram software at the
BU web site and/or spread sheets and/or Mathematica to locate ALL
parameter intervals of attracting least-period-n orbits for n=1,2,3,4,5
for the quadratic map:Q
_{c}(x)=x^{2}+c. Restrict the parameter c to -2<c<1/4. Explain briefly how you obtained your answers.

Hints: - Use the graph of the nth iterate to look for approximate parameter values where period-n orbits are born. The slider on the BU Nonlinear Web software is especially useful for changing parameter values. Or you can use the Manipulate command in Mathematica.
- Use the approximate parameter values obtained in the previous step to magnify the appropriate region in the Orbit Diagram software. Locate the appropriate saddle-node and/or period-doubling bifurcations which correspond to the "birth/death/change of stability" of the attracting periodic orbit you are investigating. The disadvantage of this software is that you cannot read the mouse position off directly, and you cannot "back out" one step at a time from your magnifications. The Orbit Diagram software has the big advantage that you can choose a magnification region either with the mouse or by typing in window ranges. You are also allowed to change the number of transient iterates to ``hide'', and the number beyond the hidden iterates to ``display.''
- You could find the endpoints of the attracting intervals by using only Nonlinear Web software, and doing graphical iteration to see whether orbits are drawn toward a specific periodic orbit or not. This, however, tends to be slow, and it is difficult to locate endpoints accurately.

- Reread Ch 9.
- Ch. 9: 1,2,5,7,8,9

- Read Devaney Ch. 10. Know the three properties for a dynamical system to be chaotic. Skim Ch's 11 and 12. Read carefully The Period 3 Theorem (p. 133), >Sarkovskii's Theorem (p. 137, including the Sarkovskii ordering), and the negative Schwarzian derivative theorem (p. 158).
- Extend your "Charts" from HWs 3, 4 and 5 to
include the following. For n=1 to 10, fill in columns: n, # slns to Q
_{-2}^{n}(x)=x, # least per-n pts for c=-2, # least per-n orbits for c=-2, # of c-intervals corresponding to attr. least per-n orbits (as c decreases), # of least period-n orbits born (as c decreases) in period-doublings (from period n/2), # of least period-n orbits born in saddle-node bifurcations, # of least period-n windows. - Extra Credit: Ch. 10: 20 Prove directly only the
property that periodic points are dense for the doubling function.

d

- Read Strogatz Ch's1 - 2
- Sec. 2.1: 1,2
- Sec. 2.2: 3*,7*, 8, 9, 13c

*For 3 and 7, determine equilibrium points (fixed points) (exact location for 3, approximate location for 7) and their stability (attracting, repelling or neither), sketch phase lines, and sketch corresponding solution curves. - Other sections in Ch. 2: 2.4.4, 2.4.7, 2.5.2, 2.6.1, 2.7.1, 2.8.2c (For 2.8.2c you may use software to print the slope field, or sketch a slope field by hand. The Rice University software plots slope fields for 1D differential equations. It also allows selecting initial conditions and plotting the corresponding solution curve, but I want you to do this part by hand. Mathematica, of course, allows everything, but you need to figure out how to use it. Try the VectorPlot command, for example. Sketch several solution curves by hand on the slope field. Check with StreamPlot command.)

- Read Strogatz Ch 3
- Ch. 3: 3.1.1, 3.2.1, 3.4.3. In all three problems,
replace "sketch vector fields" with "sketch phase lines".
- Locate and identify all bifurcations in the family: dx/dt = x (r - (1 - x
^{2})) (r - (2x^{3}- 2x)). It is OK to use software to plot any "useful" functions. - Change of variables:
- Logistic D.E.: dx/dt = ax-bx
^{2}. Rescale by y=Ax. (A is a constant number.) Choose A so that the "new" differential equation (in y) is dy/dt = ay - ay^{2}. - Logistic map: x
_{n+1=}ax_{n}- bx_{n}^{2}. Rescale by y=Ax. Choose A so that the "new" map is y_{n+1=}ay_{n}- ay_{n}^{2}.

- Read Strogatz Ch 5, sections 5.0, 5.1, 5.2
- 5.1: 3,5,7*; EXTRA CREDIT 10ace, Extra Credit: 9

*5.1Notes: - Do 5.1 #7 by hand. Include both nullclines (dx/>dt=0 and dy/dt=0) and any real eigenvectors in your sketch.
- 5.2: 1,3*,6*,7*,10*, 13ab*;
EXTRA CREDIT: 11*, Extra Credit: 13c.

*5.2 Notes: - In 5.2: 3,6,7,10: You may use software (Rice University phase plane software, for example) to obtain the phase portrait. Include several phase curves. Add arrows by hand to indicate the direction of travel as time increases. Add and label nullclines and any real eigenspaces in all phase portraits.
- In 5.2 #11: Do not "solve" the system analytically. Do show the one-dimensional eigenspace and the phase portrait (including several direction vectors and several phase curves).
- In 5.2 # 13: You may use software (Rice University phase space software, for example) for your phase portraits. Sketch any equilibria, nullclines and real eigenspaces on the printout. If there are two eigenvalues with the same sign, indicate the "stronger" one with double arrows. (Or do it all by hand.) Hint: Think of the different regions of the "trace-determinant space"; you should have 2 main (not borderline) cases, depending on the parameters. Assume that the mass m>0 and the spring constant k>0. ADDITIONAL HINT: YOU SHOULD HAVE TWO PHASE PORTRAITS, ONE WHERE THE EIGENVALUES ARE COMPLEX, AND ONE WHERE THEY ARE REAL AND DISTINCT. YOU NEED NOT SKETCH A PHASE PORTRAIT FOR THE BORDERLINE CASE WHERE THERE ARE REPEATED REAL EIGENVALUES. PICK EXAMPLE PARAMETER VALUES FOR M,B,K, ONE SET FOR EACH CASE.

- Read Strogatz Ch 6: 6.0-6.4, 6.7
- Read Strogatz 7: 7.0, Example 7.1.1, 7.3 first paragraph
(Poincare-Bendixon Theorem statement) and last
paragraph (No Chaos in the Plane, p.210).
- Read Strogatz Ch 8: 8.0, 8.1 up to beginning of Example 8.1.1 and
the paragraph on Transcritical and Pitchfork
bifurcations (p 246).
- Read Strogatz Ch 9: (the Lorenz equations) 9.0, 9.2 (p. 311 only),
9.3 (p. 317-319 only) and Color Plate 2.
- Read 12.2 (the Henon map)
- In 6.1: 5, 8. For 5, you need not sketch the vector field; instead, include the direction field at least along the nullclines. For both 5 and 8, use the phase portrait to describe the fate of ALL orbits.
- In 6.3: EXTRA CREDIT 16
- In 6.4: 2
- In 6.7: 1 DO A PHASE PLANE SKETCH ONLY FOR B=1.
- In 8.1: 2 Change the directions to: Plot a bifurcation diagram showing the x-value of the equilibria as a function of the parameter mu. Make the curve solid where the equilibrium is attracting, and dashed if it is unstable (either a saddle or repelling). Sketch three representative phase portraits, one each for mu>0, mu=0, and mu<0. Describe the fate of all orbits for all three cases. What type of bifurcation is this?
- In 8.2: 5 Use software (Rice University) to plot three phase portraits: mu>0, mu=0, mu<0. Describe the fate of all orbits in all three cases. >
- For any phase plane in this assignment, include computation of linearizations around all equilibrium points to classify as sink/saddle/source, with sinks and sources subclassified as nodes or spirals, stable/unstable manifolds of all saddle points, andnullclines.

- Read Devaney Chapters 16 and 17
- Devaney, Ch 15: 1d, 2e, 3e, 5ac, 6 (hint: translate), 8ac, 9, 11
- Devaney, Ch 16: 1,3,5abc (for 5b, restrict c to be real),6a,9bc

Test 2, Tues. Nov. 21, 5-6:30 Room TBA. Covers Strogatz material and Devaney Chs 15, 16 and part of 17. Select here for Midterm 2 topic list. Note that this list expands the topics list for Midterm 1, but topics that are primarily for one-dimensional maps will not be asked again on Midterm 2. Select here for a practice test and here for an older practice test with slightly different topics, but with solutions. The second practice test is from 2007 before we used the Strogatz text, but the topics covered are still similar.

- Read Devaney Ch 17.
- Do Lab Experiment 17.5 in Devaney. Indicate the location of the eight c values on the copy of the Mandelbrot set which was handed out in lab. Answer Notes and Questions #2. #3 may be done for extra credit.