Bruce B. Peckham - Fall 2001 Dynamical Systems (Math 5260)

Math 5260 Dynamical Systems

Fall 2001
Prof. Bruce Peckham

Syllabus

Set 1 Due Friday 9/14/2001:
- Read Ch 1
- Read Ch 2
- 2.1: 1,2
- 2.2: 3,7,8,9,13c
- 2.3: 4
- 2.4: 7
- 2.5: 2
- 2.6: 1
- 2.7: 1
- 2.8: 2c (You may use software to draw the slope field. Differential Systems is available on the Mac Lab Server, or in CCtr 118 on the Mac G3's.)
Lab tasks for Friday 9/14/2001.
- For Differential Equations:
  1. Find the software Differential Systems. (Lab Software -> Programming and Math) We will be using only one equation at a time (see Equations -> Build Your Own), and therefore only two variables. Make sure you are looking at the correct two (under Plot).
  2. For the following differential equations, Plot -> Field Marks (we called it the slope field in class), select the indicated initial conditions via the mouse, and you should see the computer's numerical solution plotted on the screen. In the process, you should be able to reproduce Figures 2.1.2, 2.1.3, 2.3.4, 2.8.2, 2.8.3 from Strogatz.
    1. dx/dt = -.5 * x, x(0)=2.
    2. dy/ds = -.5 * y, y(0)=2
    3. dx/dt = sin(x), x(0)=pi/4
    4. dx/dt = sin(t) (why can there be oscillations???)
    5. dx/dt=x(1-x), x(0)=.5
    6. Do homework problem 2.8.2c
- For Maps:
  1. Intro to "cobweb" diagrams for map iteration. (I will give a short presentation.)
  2. Find the software Chaos and Dynamics. (Course Software -> Math -> Math 5695)
  3. Open Lab 2 - Plot and Analyze. We will work with the "logistic function": rx(1-x).
  4. Add a graph and change the parameter to 0.5. Double click on the screen to give an initial condition. Use several initial conditions, or just visualize what the cobweb diagram would look like for different initial conditions. It is sometimes useful to click on Transients to eliminate the first part of the cobweb and make the eventual behavior more obvious. Locate any fixed points. Are they attracting or repelling? Explain (to me verbally) the eventual fate of ALL orbits, and how the fate depends on the initial conditions.
  5. Repeat these tasks for parameter values of 1.5, 2.5, 3.1, 3.5, 3.8, 3.83, 4.0
Set 2 Due Wed. 9/26/2001:
- Read Sections 10.1 and 10.2 in Strogatz
- Do Section 10.1 problems:
  1. 1 (also graph f², f³)
  2. 2
  3. 9 (including finding all fixed points and their stability)
  4. 11,12,13
- Extra problems
  1. Analyze the behavior of orbits under iteration of F(x)=x+1/2 sin(x)
  2. (Taken from A First Course in Chaotic Dynamical Systems by Robert L. Devaney) Consider the "doubling function": D(x)=2x mod 1, 0<=x<1.
    - Discuss the behavior of the orbits for the following initial conditions: .3, .7, 1/8, 1/16, 1/7, 1/14, 1/11, 1/22
    - Sketch the graph of D² and D³. Describe what the graph of Dⁿ would look like. How many fixed points are there for D, D², D³, Dⁿ?
    - Extra Credit. The computer may lie. Using the Lab 2 software from Chaos and Dynamics, investigate the doubling function for several different initial conditions. Include initial conditions you can compute explicitly, such as 0.2. Does the computer orbit match the orbit you compute by hand? Explain why not. :)
  3. Analyze the eventual behavior of orbits of x_n+1=f_(a,b)(x_n) for f_(a,b)(x)=ax+b. Consider all initial points and all possible real values of a and b. In particular, you should include the existence of fixed points and their stability for all paramter values.
Lab tasks for Friday 9/21/2001.
- For Maps:
  1. Find the software Chaos and Dynamics. (Course Software -> Math -> Math 5695)
  2. Open Lab 2 - Plot and Analyze. We will work with the "logistic function": rx(1-x).
  3. Add a graph and change the parameter to 0.5. Double click on the screen to give an initial condition. Use several initial conditions, or just visualize what the cobweb diagram would look like for different initial conditions. It is sometimes useful to click on Transients to eliminate the first part of the cobweb and make the eventual behavior more obvious. Locate any fixed points. Are they attracting or repelling? Explain (to me verbally) the eventual fate of ALL orbits, and how the fate depends on the initial conditions.
  4. Repeat these tasks for parameter values of 1.5, 2.5, 3.1, 3.5, 3.8, 3.83, 4.0
  5. Use Devaney Lab software (Lab 2) to plot the eventual behavior of the logistic map for "your" parameter values on the class graph.
  6. Use Lab 2 software to start locating the parameter intervals of attracting period-n orbits for n=1,2,3,4,5 for the logistic map:f_r(x)=rx(1-x) . Restrict the parameter r to 0<r<4.
Set 3 Due Wed. 10/10/2001:
- Section 10.2 in Strogatz: problems 1 and 2 (interpret "globally attracting" as attracting for all x in (0,1))
- Doubling map continued. Let D(x)=2x mod 1, 0<=x<1. Characterize all starting points which eventually land on 0 (and therefore are "eventually fixed").
- Leftover task from 9/21/2001 Lab: Use Lab 2 software to start locating the parameter intervals of attracting period-n orbits for n=1,2,3,4,5 for the logistic map:f_r(x)=rx(1-x). Restrict the parameter r to 0<r<4. Explain how you obtained your answers. Some representative graphs would be useful, but you do not need to provide graphs for each case.
- Chain rule. Let p be a period-3 point for x_n+1=f(x_n). Show that the slope of the third iterate of f is the same at the three points p, f(p) and f²(p).
- Back to Differential equations. Read Sections 0,1,2,4 in Chapter 3.
- Section 3.1: 1
- Section 3.2: 1
- Section 3.4: 12
- Locate and identify all bifurcations in the family: dx/dt = x (r - (1 - x²)) (r - (2x³ - 2x)).
Set 4 Due Friday October 19, 2001 (move to Monday Oct. 22)
- Read in second reference: "A First Course in Chaotic Dynamical Systems" by Robert L., Devaney. There is a copy on reserve for our class in the Library, and I have a copy that you can borrow as well. Skim Chapters 1, 2, 3 and 4. Read Chapter 5. Almost all of the material has already been covered either in Strogatz or in my lectures.
- Read Strogatz Section 10.3. Do problems 1,2,4,5,7abc.
- Complete the chart for periods n=1,2,...,10 for the logistic map with r=4 (f₄(x)=4x(1-x)) using the following column headings: n, # slns to fⁿ(x)=x, # prime period-n points, # prime period-n orbits.
- Complete the chart for periods n=2,3,...,10 for the logistic map with (f_r(x)=rx(1-x)) as r increases from 0 to 4, using the following column headings: n, # period-n saddle-node bifurcations, # period-n/2 period-doubling bifurcations, # period-n windows, # r intervals with attracting period-n orbit, total number of period-n orbits born via both saddle-node and period-doubling bifurcations. Hint: Our software only graphs up through the 6th iterate, but if you assume that period-n orbits are only born through period-doublings or saddle-nodes, and that every attracting orbit eventually loses its stability due to a period-doubling, then you can determine the rest of the chart entries. Note that an r interval with an attracting period-n orbit is not necessarily in a period-n window. For example, the attracting period-6 orbit which appears when a period-3 orbit loses stablility via period-doubling is still in the period-3 window. You may want to investigate the orbit diagram with the Lab 3 software (or the Java version "Orbit Diagram" at http://math.bu.edu/DYSYS/applets/). Here the use of the term "window" becomes more obvious.
- The invariant Cantor set for the logistic map with r>4. Carefully sketch or print out by computer a graph of the logistic function: f_r(x)=rx(1-x) using any r>4 (r=4.1, for example). Define the following "exit" sets for n=1,2,... :
  E_n = {x in [0,1] s. t. f^j(x) is in [0,1] for j=0,1,...,n-1, but not for j=n}.
  Let I₀=[0,1/2]\E₁, and I₁=[1/2,1]\E₁. Define the following "n+1-year surviving" sets:
  I_{i₀,...,i_n}={x in [0,1] s. t. f^j(x) is in I_{i_j}.
  Sketch and label E₁, E₂, E₃, and any one subinterval in E₄. On the same graph label I₀₀, I₀₁, I₁₀, I₁₁, I₁₀₁, and I₁₀₀.
Midterm Wednesday Oct. 24 (moved to Friday Oct. 26)
HW Set 5 Due Wednesday 11/7/2001
1. Read in Strogatz: Sec. 10.4, Sec. 10.6 pp 369-372 only
2. Sec. 10.4: problems 3,6
3. Read Devaney: Chapters 6,7,8, 9; Section 10.1
4. Change of variables:
  1. Logistic D.E.: dx/dt = ax-bx². Rescale by y=Ax. Choose A so that the "new" differential equation (in y) is dy/dt = ay - ay².
  2. Logistic map: x_n+1=ax - bx². Rescale by y=Ax. Choose A so that the "new" map is y_n+1=ay - ay².
5. Consider the shift map sigma on the symbol sequence space Sigma; assume the metric we defined in class:
  d(s,t)=Sum_i=1^infinity |s_i-t_i|/ 2ⁱ.
  How many digits must mathch in two sequences in order for them to be within 0.01 of each other?
6. Conjugacies. Assume the maps f and g are conjugate via the homeomorphism h. That is, g o h = h o f (the circle is composition). Show
  1. x is a fixed point for f implies h(x) is a fixed point for g.
  2. y is a fixed point for g implies h^-1(x) is a fixed point for f.
  3. x is a period-q point for f implies h(x) is a period-q point for g.
  4. Extra Credit. fⁿ(x) -> p implies gⁿ(h(x)) -> h(p)
7. Period-6 windows for the logistic map. Locate all period-6 windows. Use Lab 2 software to identify approximate parameter values, and then Lab 3 software to see the window. For each period-6 window, record the parameter range for the window, the parameter range corresponding to an attracting period-6 orbit, and the parameter range corresponding to an attracting period-12 orbit.
8. Superattracting orbits. Locate all approximate parameter values where the logistic family has a superattracting orbit of period 1,2,3,4. Use what you know to locate the parameter intervals where attracting orbits are found. Then use the chain rule to locate the parameter value (using Lab 3 software) where the critical point is on the appropriate orbit. Verify for periods 1 and 2 analytically (also using the fact that the critical point must be on a superattracting orbit).
9. Conjugacy between the logistic map: x_n+1=rx - rx² and the quadratic map: y_n+1=y² + c. For homework, you already found out that for any given r, the logistic map is "conjugate" to the quadratic map with c=r/2 - r²/4 via the conjugacy: x -> y= h(x) = -r(x-1/2). Locate the following points on the orbit diagram for the logistic map. Then compute and locate the corresponding points on the orbit diagram of the quadratic map. Describe any corresponding features (eg, a bifurcation point).
  - x=1/2 (Why is this point special?)
  - (x,r)=(1/2,2)
  - (x,r)=(0,1)
  - (x,r)=(2/3,3)
  - (x,r)=(.51437,3.82843)
  - (x,r)=(.4864,3.84148)
10. Extra Credit: A. Show (using an epsilon-delta proof) that the shift map: sigma: Sigma -> Sigma is continuous.
11. Extra Credit. We know that the period-3 window "opens" because of a period-3 saddle-node bifurcation. Describe the condition for the "closing" of the period-3 window. Hint: Find some "subintervals" which are invariant under f³ just after the saddle-node bifurcation. What happens as r is increased? What happens when a subinterval ceases to remain invariant?

Set 6 Due Wednesday 11/21/2001:

Read Sections 5.1 and 5.2 in Strogatz
5.1: 3,5,7,9,10ace
5.2: 1,3,5,7,9,11,13
Notes:
- Do 5.1 #7 by hand. Include both nullclines (dx/dt=0 and dy/dt=0) and any real eigenvectors in your sketch.
- In 5.2: 3,5,7,9: You may use software to obtain the phase portrait. Include nullclines and any real eigenspaces in all phase portraits.
- In 5.2 #11: Do not "solve" the system. Do show the one-dimensional eigenspace and the phase portrait.

Set 7 Due Monday 12/3/2001: (later if more problems are added)

Read Sections 6.0 - 6.3 in Strogatz.
6.1: #6 Find equilibria and their eigenvalues. Sketch nulclines, some vectors in the vector field (direction only). Include some vectors on the nullclines. Then sketch the phase portrait. Identify the stable and unstable manifolds of any saddles. Describe the fate of all orbits and how that fate depends on initial conditions. Include in this description the basin of attraction of any attracting equilibria.
6.1 (cont.) 7, 8 (describe the fate of all orbits), 12, 13
6.2: 1 Use dx/dt=-x, dy/dt=-2y as an example.
6.3: Continue with problem 6.1.6: Linearize around each equilibrium point. Sketch the phase portrait of each linearization. Comment briefly on how the phase portraits of the linearizations compare to the nonlinear phase portrait.
Analyze the following system: dx/dt=x(3-x-2y); dy/dt=y(2-x-y).
Perform the same tasks as for problem 6.1.6 above, including the tasks listed under 6.1 and 6.3. This system is used to model two different species competing for the same resources. It leads to the "principle of competitive exclusion." Explain using your phase portrait.

Set 8 Due Monday 12/10/2001: (later if more problems are added)

Read Sections 7.1 - 7.3, 8.1, 8.2, 8.4 in Strogatz.
7.1 1,3,5
7.2: CANCEL. 1 Sketch some level curves of V along with some trajectories of solutions. Are the trajectories perpendicular to the level sets?
7.3: CANCEL. 1, 2. For part 1e, assume there are no fixed points other than the origin. For extra credit, show this is true.
8.1: 1, 6. For part 6a, sketch nullclines for the three cases: before the bifurcation, at the bifurcation, and after the bifurcation.
8.2: 1,2,3 Hint for 1: First find fixed points. Linearize. Look for parameter values for which the fixed points have pure imaginary eigenvalues.
8.4: 3 (extra credit)

Final Problem Set Due Tuesday 12/18/2001

(10pts)Do a bifurcation analysis of the family:

dx/dt = ax - xy
dy/dt = -y + x^2.
(10pts)Find a two-dimensional family of differential equations, depending on the parameter a in (-epsilon, epsilon), for some epsilon > 0, and exhibiting the following properties: for a<0, the differential equations have a single attracting limit cycle; for a>0, the differential equations have a repelling limit cycle in between two attracting limit cycles. Write the differential equation first in polar coordinates; then convert to rectangular coordinates.
(20pts) Do a bifurcation analysis of the family

x_n+1=a(x_n-x_n^3/3).

Rather than describing all dynamics and bifurcations from scratch, oncentrate on similarities and differences from the bifurcations of the logistic family. You may assume that the dynamics and bifurcations of the logistic family is "known." Recall that you had a homework problem where you studied this family for a=3. Note also that the Devaney Lab 2 and Lab 3 software includes this family.

Midterm 2 Tuesday Dec. 18 10-12am in Cina 308.

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