• Syllabus
• Dynamical Systems software links (Not all are fully functional. Caution: Some links lead to commercial sites, none of which is endorsed by the instructor. :) )
• Desmos Cobweb Diagrams
• Geogebra Cobweb Diagrams
• Youtube for creating Matlab Cobweb Diagram
• Wolfram Alpha (Mathematica)
• Older software that might be available and useful if you can use java:
  • Boston University Dynamics Website: Install Google Chrome Browser; install the Chrome extension: Chrome Extension for Java. It takes a few minutes to get started as the extension recompiles all of the Java to JavaScript (so that the browser can run it natively), but it seems to work smoothly after that.
  • 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 on Canvas

Homework Assignments

Set 1, due FRI Aug. 30

1. Strogatz: Read Prefaces and Chapter 1: Overview.
2. Devaney: Read Preface and Ch's 1-4
3. Devaney Ch 3: 1,3,4,5,6,7abh,8,11,12,13,14
4. 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.
5. Do the following tasks with the help of any software which allows cobweb diagrams (see software suggestions above).
1. Experiment by typing in other functions with parameters. (Nothing to hand in.)
2. 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.

1. Devaney Ch 4: 1bcg,2ab,7 (Optional for 7: Use cobweb diagram software to experiment and to check your answers to problem 7.)

1. Read Devaney: Chapters 5-6.
2. >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.)
3. Complete the following chart for periods n=1,2,...,10 for the quadratic map with c=-2 (Q_c(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? You will need to think of the properties of the graph of the nth iterate for higher iterates.

1. Read Devaney: Chapters 6-7.
2. 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.)
3. 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 here.
4. Do the lab tasks on tangent and period-doubling bifurcations. Handout in class or download here.

1. Read Devaney: Chapters 7-10.
2. 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.
3. 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.
4. Lab Related Work:
1. 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
2. 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:Q_c(x)=x²+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.
• 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.

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

1. 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).
2. 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.
3. Extra Credit: Ch. 10: 20 Prove directly only the property that periodic points are dense for the doubling function.

1. Read Strogatz Ch's1 - 2
2. Sec. 2.1: 1,2
3. 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.
4. 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.)

1. Read Strogatz Ch 3. Focus on Sections 3.1, 3.2, 3.4. Just skim the other sections.
2. 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.
3. 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 "useful" functions.
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.
4. Change of variables:
1. 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.
2. Logistic map: x_n+1=ax_n - bx_n². Rescale by y=Ax. Choose A so that the "new" map is y_n+1=ay_n - ay_n².

1. Read Strogatz Ch 5, sections 5.0, 5.1, 5.2
2. 5.1: 3,5,7*; [Extra Credit 9, 10ace]
   *5.1Notes:
• 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.
3. 5.2: 1,3*,6*,7*,8*, 13ab*; [Extra Credit: 11*, 13c*]
   *5.2 Notes:
• 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 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 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.

1. Read Strogatz Ch 6: 6.0-6.4, 6.7
2. 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).
3. 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).
4. 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.
5. Read 12.2 (the Henon map)
6. 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.
7. In 6.3: EXTRA CREDIT 16
8. In 6.4: 2
9. In 6.7: 1 DO A PHASE PLANE SKETCH ONLY FOR B=1.
10. 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?
11. 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.
12. 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.

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

1. Read Devaney Ch 17.
2. 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.

1. (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)
   
   
2. (30pts) Analyze the dynamics and bifurcations of the family:
   
   x_n+1=a(x_n-x_n³/3). Both x and a are real.
   
   Hints:
   1. 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.
   2. 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.
   3. 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₊.)
   
   
3. (15pts) Number of period-n attracting intervals and decorations. (Use of software like Mathematica is encouraged.)
   1. 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.
   2. 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.
   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?
   
   
4. (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 sought-after points.
   
   (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.
   (iii) 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 at p₊. 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.
   (iii) 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 z^2-1.
   
   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.
   
   
5. (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?
   
   
6. (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. (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.)
   
   

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