Math 5260 Dynamical Systems
Fall 2019 Course Homepage
Prof. Bruce Peckham, Department of Mathematics and
Statistics, University of Minnesota, Duluth
- Dynamical Systems software links (Not all are fully
functional. Caution: Some links lead to commercial sites, none of which is endorsed by the instructor. :) )
- Older software that might be available and useful if you can use java:
- Check grades on Canvas
Note: Both the Devaney and Strogatz textbooks are available in electronic format through the course page on Canvas.
Solutions for Strogatz are also on Canvas. Solutions for Devaney are at
1, due FRI Aug. 30
- Strogatz: Read Prefaces and
Chapter 1: Overview.
- Devaney: Read Preface and Ch's
- Devaney Ch 3:
- Reproduce the table on p. 23 using a spreadsheet (such
as GOOGLE SHEETS or 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 any software which allows cobweb diagrams (see software suggestions above).
- Experiment by typing in other functions with parameters. (Nothing to hand
- 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 any software which allows you to view cobweb diagrams (or both).
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.
Set 2, Due Friday Sept 6
- Devaney Ch 4: 1bcg,2ab,7
(Optional for 7: Use cobweb diagram software to experiment and to check your answers to problem 7.)
Set 3, Due Friday Sept 13.
- 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 (Qc(x)=x2+c)
using the following column headings:
n, # period-n pts, # least period-n points, # least period-n orbits. Hint:
You can graph iterates of the quadratic map for low values of n, but beyond n=6 this becomes less useful. Why?
will need to think of the properties of the graph of the nth iterate for
Set 4, Due Friday Sept 20.
- 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 the 6-7 assigned individually.
Do this by iterating a fixed number of iterates (say 100), throwing them out, and plotting or recording the next fixed number (say 100). The second 100 are an approximation to the "fate" of the orbit.
All orbits should start at x=0.
You can do this with a spreadsheet and/or graphical iteration software (Desmos or Boston University).
A chart on which to plot your results will
be provided. Assigned parameter values are provided
- Do the lab tasks on tangent and period-doubling
bifurcations. Handout in class or download here.
Software references below here may need to be updated.
Set 5 Due Fri. Sept. 27.
- 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 y2+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 Desmos software and/or Boston University Orbit Diagram software
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:Qc(x)=x2+c.
Restrict the parameter c to -2 < c < 1/4. Explain briefly how you
obtained your answers.
- Use the graph of the
nth iterate to look for approximate parameter values where period-n
orbits are born.
- 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.''
- For small values of n, the (2n)th iterate might be useful to determine an endpoint of the c-interval for an
attracting period-n orbit.
- You could find the
endpoints of the attracting intervals by using only 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.
Set 6 Due Wed. Oct. 2
- Reread Ch 9.
- Ch. 9: 1,2,5,7,8,9
Test 1: Monday Oct 7. 5-6:30 Room: SCC 130. Chapters 1-9 and selected parts of Chapters 10 - 12 of
Devaney. Select here Midterm 1 topic list.
See a practice test here and solutions here.
Set 7 Due Fri. Oct. 11
- 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-2n(x)=x,
# least per-n pts for c=-2, # least per-n orbits for c=-2,
# 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 (as c decreases),
# of c-intervals corresponding to attr. least per-n
orbits (as c decreases),
# of least period-n windows.
- Extra Credit: Ch. 10: 20 Prove directly only the
property that periodic points are dense for the doubling function.
Set 8 Due Fri Oct. 18.
- 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
- Other sections in Ch. 2: 2.4.4, 2.4.7, 2.5.2, 2.6.1 (extra credit),
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.)
Set 9 Due Wed. Oct. 23.
- Read Strogatz Ch 3. Focus on Sections 3.1, 3.2, 3.4. Just skim the other sections.
- Ch. 3: 3.1.1, 3.2.1, 3.4.3. In all three problems, sketch the bifurcation diagram and
sketch several "representative" phase lines on the bifurcation diagram.
- Sketch the bifurcation diagram and locate and identify all bifurcations in the family: dx/dt = x (r - (1 - x2)) (r - (2x3
- 2x)). Sketch several representative phase lines on the bifurcation diagram. It is OK to use software to plot any
Plotting help: in Desmos, plotting a function f(x) is the same as plotting (t, f(t)) for a < t < b. To switch the roles of x and y, you could plot (f(t), t) instead. This idea might be useful for this problem. For those of you using Mathematica, you could do a similar thing with ParametricPlot.
- Change of variables:
- Logistic D.E.: dx/dt = ax-bx2. Rescale by y=Ax. (A is a
constant number.) Choose A so that the "new" differential
equation (in y) is dy/dt
= ay - ay2.
- Logistic map: xn+1=axn - bxn2. Rescale
by y=Ax. Choose A so that the "new" map is yn+1=ayn - ayn2.
Set 10 Due Fri. Nov. 1
- Read Strogatz Ch 5, sections 5.0, 5.1, 5.2
- 5.1: 3,5,7*; [Extra Credit 9, 10ace]
5.2: 1,3*,6*,7*,8*, 13ab*;
[Extra Credit: 11*, 13c*]
- Do 5.1 #7 by hand.
Include and label both nullclines (dx/dt=0 and dy/dt=0)
and any real eigenspaces in your sketch.
Include some vectors on the nullclines, some on the real eigenspaces, and some elsewhere.
- In 5.2: 3,6,7,8: 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
- 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 for your
phase portraits. Sketch and label 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.
Set 11 Due Fri. Nov. 8
- 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. 212 (p.210 in first edition).
- 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. 249 (p 246 in first edition).
- Read Strogatz Ch 9: (the Lorenz equations) 9.0, 9.2, p. 319 (p. 311 in first edition),
9.3 ,p. 325-327 only (pp. 317-319 in first edition) 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
- 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, and nullclines.
Set 12 Due Fri. Nov. 15
- Read Devaney Chapters 15 and 16
- 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
2, Mon. Nov. 18, 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.
Set 13 Due Mon. Nov. 25.
- 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 handed out in class. Answer Notes and
Questions #2. You can the Fraqtive software to locate the appropriate c-values, and either use a spreadsheet or BU software: Java Applets -> The Quadratic Map Applet, to determine the "period" associated with each "bulb." Recall that we have a theorem that guarantees that if you start iterating from z=0 you will be attracted to any existing periodic orbit.
For those of you using the online version of the book, here are the 8 Julia sets.
#3 may be done for extra credit.
Final Problem Set Due Wednesday, Dec. 11 at
Do problems 1-7.
100 points total.
Ground rules for the final problem set.
Unilike for HW sets during the semester, you are not allowed
to collaborate for the final problem set.
You may use software, computer programs, books, internet,
but any use of these references must be fully documented!!!!! Fill out the reference sheet provided.
You may ask me questions, but no other person.
It may be useful to look at the course summary.
Do All problems 1-7. Problems 5-7 have choices.
100 points total.
(http://www.d.umn.edu/~bpeckham/www) is maintained by Bruce Peckham
(firstname.lastname@example.org) and was last
modified on Tuesday, 26-Nov-2019 15:43:47 CST
- (15pts) Analyze the dynamics and bifurcations of the following family. (x,y and a are real.) Restrict the parameter a to [.6, 1].
Assume x and y are both non-negative.
dx/dt = x - x2 - xy/(x+.25)
dy/dt = -.5y + axy/(x+.25)
- (30pts) Analyze the dynamics and bifurcations of the family:
xn+1=a(xn-xn3/3). Both x and a are real.
- Note that Boston University `Nonlinear Web' software includes this family. If you don't have this software on your computer, you can use the Windows PCs in SCC 118. Check out a key in SCC 140. Or you can find or write software to compute graphical iteration and/or orbit diagrams for this cubic family.
In addition to the other standard behaviors you might look for in analyzing
the dynamics of any family of maps,
determine which orbits stay bounded, which
bounded orbits stay positive, and which bounded orbits stay negative, and
determine various invariant - or noninvariant - intervals.
Remember the goal of analyzing a family of dynamical systems is to describe the behavior all
orbits, and how that description changes as parameter(s) are varied.
Part of your description can be to compare/contrast the dynamics of this family
to that of a "known" family, such as x^2+c.
For example, you might observe that, in the cubic family above, there is a similar sequence of bifurcations
as a varies (in some specific parameter interval) and for x values in some specific interval(s) to the
bifurcations that occur for x^2+c for c between .25 and -2 and x is restricted to be between -p+ and p+.)
- (15pts) Number of period-n attracting intervals and decorations.
(Use of software like Mathematica is encouraged.)
Hint: A periodic orbit can be superattracting only if the orbit includes x=0 (z=0).
What equation can you write down that guarantees that the origin is a period-k point.
How many real/complex solutions does this equation have?
- Determine the number of
superattracting least period-n orbits in the family
x->x2+c for n=1,2,3,4,5,6. For n=1,2,3,4,5, locate and label
the superattracting period-n point having x coordinate zero
on the orbit diagram printout. Explain how you obtained your answers.
- Determine the number of superattracting period-n orbits in the family
z->z2+c, for n=1,2,3,4,5,6. For n=1,2,3,4,5, locate and label
the parameter value of each superattracting period-n orbit
on the Mandelbrot set printout. Explain how you obtained your answers.
- (20pts) Fixed, prefixed, and preperiod-2 points in x->x^2+c and z->z^2 + c.
In x2+c, both x and c are real; in z2+c, both z and c are complex.
Justify all answers. Using software (Desmos, BU programs, Mathematica, ...) both for graphical and computational purposes might be useful.
If you cannot compute exact answers, find approximations to
(A) Consider the map Q(x) = x^2-1. (x is real.)
(i) Compute the two fixed points p+ and p-.
Both should be repelling. Verify this. Show that there is also a superattracting orbit formed by x=0 and x=-1.
(ii) Compute all points (excluding p+) that are eventually fixed
at p+. Hint: there is only one! You may compute it
exactly or compute a decimal approximation.
Using graphical iteration, locate any 5 points that are eventually
fixed at p-.
You need not compute their exact values.
Choose them so they are not ALL on the same orbit.
Hint: there is an infinity of them!
(iv) Using graphical iteration, locate any 3 points (excluding x=0 and x=-1) that are eventually period-two.
Recall that eventually period-two points must land exactly ON either x=0 or x=-1,
not just be attracted toward the period-two orbit.
(v) Label all nine of your preperiodic points from parts (ii), (iii),
and (iv) as a1, b1, ..., b5, c1, c2, c3, respectively,
on a graphical iteration diagram.
(B) Consider the map Q(z) = z^2-1. (z is complex.)
(i) Compute the two fixed points p+ and p-.
Both should be repelling. Verify this.
Show that there is also a superattracting orbit between z=0 and z=-1.
(ii) Compute any one NONREAL point that is eventually fixed
Hint: there is an infinity of them in general, but no NONREAL points which land
on p+ in fewer than two iterates! You may compute it
exactly or compute a decimal approximation, or use the Mandelbrot/Julia Set Applet to determine
its location on the picture of the z-plane.
Compute any one NONREAL point that is eventually
fixed at p-.
A decimal approximation is adequate.
Hint: there is an infinity of them, but no NONREAL points which land on
p- in fewer than three iterates.
(iv) Compute any one NONREAL point that is eventually period-two.
(v) Label all nine of your REAL preperiodic points from parts A(ii), A(iii),
and A(iv) as a1, b1, ..., b5, and
c1, c2, c3, respectively,
on the Julia Set diagram provided for z^2-1. Label the three preperiodic
points from parts B(ii), B(iii), and B(iv) as A, B, and C, respectively, on
the Julia Set diagram provided in class.
(vi) Describe briefly the fate of all initial conditions for the map
C. (Extra Credit: 5pts) Find analagous preperiodic points on the Julia set
for Douady's "rabbit" on the diagram provided.
You may compute them or use iteration software to locate them.
In either case, locate them on the graph of the rabbit Julia set.
- (6 pts) Attend class Mon. Dec. 2 (Lorenz system and attractor) OR
For the 3D Lorenz system, fix sigma at 10, b at 8/3. Let r vary.
Construct a bifurcation diagram in x and r indicating the values of equilibria for r in (0,30). Compute the stability of the equilibria: sink saddle or source. Indicate stable by solid lines, saddle or source by dashed lines.
Are any equilibria attracting at the classic parameter value of r=28?
- (7 pts) Attend class Wed. Dec. 4 (Henon map and attractor, horseshoes) OR
Consider the following version of the Henon map: (x,y)->(a-by-x^2, x).
Show that the map is invertible (compute the inverse). Find all fixed points and linearize around the fixed points. What are the eigenvalues of the linearizations?
- (7 pts) Attend class Fri. Dec. 6 (Newton's method, singular perturbations, and related student projects) OR
Derive the formula for Newton's method for the complex function F:
N(z)=z-F(z)/F'(z), where z is the guess for a root of F, and N(z) is the
new "improved" guess by taking the linear approximation (tangent line if real)
at (z, F(z)). Show that zeroes of F are superattracting fixed points of N.
Show N for F(z)=z2+1 is conjugate to Q(z)=z2. (See problem 2, Chapter 18.5 in Devaney for more details.)