Math 110.421, Dynamical Systems

Spring 2011 Course Syllabus

http://www.mathematics.jhu.edu/brown/courses/s11/421.htm

Text:  B. Hasselblat and A. Katok, A First Course in Dynamics, 1^st edition, Cambridge University Press (2003), ISBN 0 521 58750 6 (paperback).

Course Material: The core of the course will center on the text material, and will cover most of the book.

Grade Policy:    There will be homework sets and a (possibly take home) exam or two.  The schedule of homework and the exams will be given below in time. 

Homework:    Homework based on the current week’s lectures will be posted here on the course web site sometime on Wednesday after lecture. These assignments will be due the following Wednesday. You are strongly encouraged to do your homework in groups. You are required, however, to write up your homework on your own. Homework is essential educational part of this course. You will be graded mostly on your ability to work through problems completely and concisely.  This effort will be evaluated on the exam, which cannot be done in any collaborative way.

Course Policy: You are responsible for lecture notes, any course material handed out, and attendance in class.  While I will not formally record your attendance, I will easily get to know you and your rate of presence over time.  Since I will be following the book, the lectures will be conducted as if you have already read the material and attempted some homework problems.  In this manner, you can focus mainly on those parts of the lectures that cover the areas of your reading you found difficult to understand. My teaching style is that of interactive discussion and I will rely on your input in developing the material.  Active participation in the classroom is a great way to generate the discussion necessary to fully grasp the material.

Help Room:      213 Kreiger Hall.  The hours are 9am – 9pm on Monday through Thursday, and 9am – 5pm on Friday.  This free service is a very valuable way to get one-on-one help on the current material of a class from other students outside the course.  It is staffed by graduate students and advanced undergraduates.

Ethics Statement:  The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful.  Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion. Offenses may be reported to medical, law, or other professional or graduate schools when a cheater applies.

Violations can include cheating on exams, plagiarism, reuse of assignments without permission, improper use of the Internet and electronic devices unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition. Ignorance of these rules is not an excuse.

In this course, as in many math courses, working in groups to study particular problems and discuss theory is strongly encouraged.  Your ability to talk mathematics is of particular importance to your general understanding of mathematics.

You should collaborate with other students in this course on the general construction of homework assignment problems.  However, you must write up the solutions to these homework problems individually and separately.  If there is any question as to what this statement means, please see the professor or the grader.

For more information, see the guide on "Academic Ethics for Undergraduates" and the Ethics Board web site (http://ethics.jhu.edu).

Students with Disabilities:  Students with documented disabilities or other special needs that require accommodation must register with the Office of Academic Advising. After that, remind me of your needs at least 5 days prior to each exam; we will need to have received confirmation from Academic Advising.

Math 110.421, Dynamical Systems

Spring 2011 Tentative Schedule

The details of this material may be updated and reformed as the semester progresses.

Problem Notes:

1. Use the ratio $\displaystyle\frac{x_{n+1}}{x_n}$ to construct a map on the interval $\left[ 2,4\right]$ which defines the dynamical system.  Interestingly, the fixed point of this map will be the asymptotic growth rate for the rabbit population.
2. You may want to review Proposition 2.6.7 on page 63, showing that the metric defined on the circle is actually a metric.
3. This is actually easier than you may think.  With $c=7.1$, you know the value of $\alpha$.   Just get it close to an integer.

Extra Problems:

1. Approximate $\sqrt{110}$ using Heron’s method for calculating square roots, to an accuracy of .001.  Try this using starting points of 10 and 11 for your rectangle sides, and 1 and 110, noting the convergence properties.
2. Do the following for the ODE $\displaystyle \frac{dp}{dt} = \frac{p}{2}-450$ :
   1. Solve the ODE by separating variables.  Justify why the absolute value sign is not necessary when stating a single expression for the general solution.
   2. Calculate the equivalent discrete dynamical system (on the real line) which takes each point of the real line to its position along the solution curve of the ODE one time unit later.  (Hint:  As in class, the function which defines the discrete dynamical system will be linear, but this case will include a specific, non-zero constant term).
   3. Discuss the simple dynamics of the discrete dynamical system.      
3. Show $f(x)=\sqrt{x}$ is a $\displaystyle\frac{1}{2}$-contraction on $\left.\left[ 1,\infty\right.\right)$ without using any derivative material (that is, without using Propositions 2.2.3 or 2.2.5).
4. Show, without using the Contraction Principle, that a contraction cannot have 2 or more fixed points.
5. (Do this one on a separate sheet of paper)  Prove that the area $A$ of any triangle in the plane is given by half the length of any side (the base $b$) times the length of the line segment starting at the line through the base, perpendicular to this line, and ending at the vertex of the triangle opposite the base (this is the height $h$).  So show $\displaystyle A=\frac{1}{2}bh$.
6. Find all periodic points (to an accuracy of $\displaystyle\frac{1}{1000}$) of the discrete dynamical system given by the map $f(x)=\log(x-1)+5$ on the interval $[2,\infty)$.
7. Show that an invertible map on $I=[a,b]$, where $b>a$, must satisfy all of the following (Hint:  All of these can be shown by assuming the property does not hold and then finding a contradiction):
   • $f$ is injective (one-to-one).
   • $f$ is surjective (onto;  the range must be ALL of $I$).
   • $f$ must satisfy either $f(a)=a$ and $f(b)=b$, or $f(a)=b$ and $f(b)=a$.
8. Figure 2.2.4 of the book provides a visual example of an invertible map on the unit circle (called $S^1$) that contains something called homoclinic points.  Construct an example (similar to the one in the figure or different) with an explicit expression for $f(x)$.  Note:  Any continuous map on the unit interval with both endpoints fixed can be viewed as a map on $S^1$.  But can you construct one that does not fix the endpoints?  Can you construct one that is also differentiable on all of $S^1$?  Can you graph this function? 
9. Solve the first order, autonomous non-linear ODE system in cylindrical coordinates below and show that there exists an asymptotically stable limit cycle (Hint:  Since the system is completely uncoupled, you can solve each ODE separately).  What are the eigenvalues of the $2\pi$-periodic point at $\vec{p}=(1,0,0)$?  \begin{align*}  \dot{r}&=\frac{1}{2}r(1-r)\\  \dot{\theta} &= 1\\ \dot{z} &= -z.\end{align*}
10. Show that the logistic map $f(x)=\lambda x(1-x)$ on the interval $[0,1]$ does not induce a discrete dynamical system for $\lambda\not\in [0,4]$.
11. Show that for $\lambda\in(1,3)$, there is an attracting fixed point of the logistic map at $x=1-\frac{1}{\lambda}$.
12. Find the change of variables which takes $f(x)=(r_0-ax)x$ to $f(y)=ky(1-y)$.
13. Show that the annulus $A$ below (in gray with its boundary circles) is homeomorphic to the cylinder $C$ of height $h>0$ and radius $s>0$, centered on the $z$-axis and resting on the $z=0$ plane.  of course, $C$ has no top or bottom, but the side does have the circle edges on top and bottom.  Do this by explicitly constructing the map $f:A\to C$, and its inverse $g:C\to A$.  Hint:  A good way to do this is to construct $f$ from (part of) the plane into three space using polar coordinates in the plane and cylindrical coordinates in three space.  Then construct $g$ in a similar manner.  When restricted to $A$ and $C$, the maps should be continuous bijections.  Verify this. 
14. Show that you cannot have a continuous surjective contraction on $S^1$.  However, construct a continuous, non-trivial contraction on the circle $S^1$.  Hint:  A continuous map cannot break teh circle in its image (or it would not be continuous at the break).  But it can fold the circle.
15. Show that the function $g(x,y)=\left( 2\cos 2x, 4y^2-y^4\right)$, from the plane to itself, can be made into a function on the standard infinite cylinder $C=S^1\times\mathbb R$.  Show also that by limiting the domain appropriately, one can use $g$ to construct a function on the torus $T=S^1\times S^1$.  

For the next two problems, let $f:\vec{v}\mapsto B\vec{v}$, where $B=\left[\begin{array}{cc} \lambda&0\\ 0&\mu\end{array}\right]$.

16. If $0<\lambda<\mu< 1$ (no absolute values here:  both eigenvalues are positive), find the 2-dimensional system of linear, homogeneous, first-order ODEs whose time-1 map is $f$.
17. If $0<\left| \lambda\right|<\left| \mu\right|< 1$ (I want absolute values here), then the equation of the curves of motion where the solutions live are given by $\left| y\right| = C\left| x\right|^\alpha$, where $\alpha=\frac{\log\left| \mu\right|}{\log\left| \lambda\right|}$.  Derive this equation.
18. Do the following:
    1. For the hyperbolic map $f:\mathbb R^2\to\mathbb R^2$, given by $f(\vec{x})=A\vec{x}$, where $A=\left[\begin{array}{cc}0&1\\ 1&1\end{array}\right]$, find a nontrivial initial vector $\vec{v}$ so that the orbit of $\vec{v}$ goes to the origin.  That is, $\mathcal O_{\vec{v}}\longrightarrow\vec{0}$.
    2. Draw a phase portrait of $f$, giving some representative invariant curves (use a range of $-15$ to $15$ for the axes).  Show the orbit of $\vec{v}$ given your answer in Part a..  Also show the orbit for $\vec{v}=\left[\begin{array}{c} 1\\ 1\end{array}\right]$.
    3. Recast the Lemmings Problem as a first order vector recursion in the plane.  Find the starting vector.
    4. Use Proposition 3.1.13 to find the functional form for the Yearly population of lemmings.  Use this to determine the total population of lemmings this spring, if the first two lemmings were one year old in 1990.
19. Show the following:
    1. For any rational rotation of the circle, all orbits are periodic of the same period.
    2. For any irrational circle rotation, no orbits are periodic.
20. In class, the circle rotation $R_\alpha:S^1\to S^1$ was discussed, where $\alpha=\frac{1}{3+\frac{1}{5+\frac{1}{c}}}$, $c>1$, where $c$ is not rational.  It was determined that $1-3\alpha=\delta$ was the closest return of the orbit of 0 to 0 of the first few iterates ($\delta$ is the distance between the third iterate of 0 and 0 itself).  It was also stated that it was smaller in value than the distance between the fourth iterate of 0 to 0 (this was $\beta=4\alpha-1$).  Show this using the continued fraction form of $\alpha$.
21. Show that a linear map on the real line of the form $f(x)=kx$ corresponds to a continuous map on $S^1$ iff $k$ is an integer.  Graph the circle map on the unit interval with the endpoints identified.  Note:  Remember that two points in the real line correspond to the same point on the circle if their difference is an integer.   This is the parameterization that we will always refer to by default.
22. Let $h:\mathbb R\to\mathbb R$ be a linear map so that $h(\vec{x})=A\vec{x}$, where $A=\left[\begin{array}{cc} a & b\\ c & d\end{array}\right]$ and $a,b,c,d\in\mathbb Z$.  Do the following:
    1. Show $h$ induces a map on the standard two torus $\mathbb T = S^1\times S^1$.  Hint:  Two vectors in the plane are in the same equivalence class on the torus (correspond to the same point on the torus), if they differ by a vector with integer entries.
    2. What can you say about the orbits of points on $\mathbb T$ given by
       1. $A=\left[\begin{array}{rr} 0& 1\\ -1& 0\end{array}\right]$.
       2. $A=\left[\begin{array}{cc} 1& 1\\ 0& 1\end{array}\right]$.
       3. $A=\left[\begin{array}{cc} 2& 1\\ 1& 1\end{array}\right]$.
23. In the proof of Proposition 4.2.8 on page 113 of the text, it is claimed that the first return map on the waist curve under the constant flow induced by $\gamma$ which is not rational, is a rotation by $\gamma$.  Show this.
24. Draw the Lissajous figure corresponding to the $(x_2,x_4)$-planar projection of the toral flow given in Section 4.2.4 when $\omega_1 = 2$ and $\omega_2 = 5$.  For these same values of $\vec{\omega}$, draw the orbit of the origin on the torus using the unit square representation of the torus in the plane with the sides identified.  If you like, also try to draw the orbit on the "surface of a doughnut" relaization of the torus.  This is not necessary, however.

25. Find the rotation number for the following invertible circle map:

$f(x) = \left\{ \begin{array}{cl} \frac{x^2+1}{3} & 0\le x<\frac{1}{3}\\ 6x-\frac{44}{27} & \frac{1}{3}\le x< \frac{10}{27} \\ 2x-\frac{4}{27} & \frac{10}{27}\le x< \frac{1}{2}\\ \frac{8}{5}\left( x-\frac{1}{2}\right)+\frac{23}{27} & \frac{1}{2}\le x < \frac{21}{27}\\ \frac{1}{6}\left( x-\frac{21}{27}\right)+\frac{35}{27}& \frac{21}{27}\le x< 1\end{array}\right.$.

26. Show that for a circle map $f: S^1\to S^1$ with a lift $F:\mathbb R\to\mathbb R$ that the function $F(x)-x(\textrm{deg} f)$ is periodic.

27. Find a suitable lift $F:\mathbb R\to\mathbb R$ for the rotation map $R_\alpha: S^1\to S^1$ where $\alpha = 2\pi$.  Graph both $F$ and $R_\alpha$. 

28. Do the following:

    1. Show that the constant flow on the 2-torus is an isometry (Hint:  Build the proper metric on the torus.)

    2. Show that the map $f(x,y)=(x+y,y)$ on the unit cylinder $S^1\times\left[0,1\right]$ preserves volume but is not an isometry.

29. For the normalized undamped pendulum $\ddot{x}+\sin(2\pi x)=0$, carefully draw either the phase plane or the phase cylinder.  Re-derive the potential energy and the total energy of the system and show that that total energy is a constant of motion.  Label on your drawing the energy levels of the equilibria and describe what is happening on the orbits that comprise the energy level that includes the saddle.  Answer thoroughly this question:  Describe in detail the differences between the first return map on the open interval of the vertical axis that runs from the top separatrix to the bottom separatrix, and any time-$t$ map within the region bounded by the two separatrices.

30. For the circular billiard within the unit circle, derive the expression for the caustic as a function of the incidence angle for the orbit.

31. Derive an expression for the twist map on the state space cylinder, which is the billiard map for the circular table of radius $r>0$.

32. prove that the generating function of the circular billiard table is $H(s,s^\prime)=-2\sin\frac{1}{2}\left(s^\prime-s\right)$, and describe the critical set in the $ss^\prime$-plane.

33. For the linear map of the circle given by $f(z) = z^2$, locate all period-2,3, and 4 points.

34. Show that Proposition 7.1.3 holds for $m=-2$.

35. For the hyperbolic linear map of the torus given by the matrix $\left[\begin{array}{cc}3&2\\ 1&1\end{array}\right]$, draw the torus with its two canonical loops that correspond to the edges of the unit square in $\mathbb R^2$, viewed as a fundamental domain.  Then carefully draw the images of these two curves under the toral map.  You may want to draw the images of the edges of the fundamental domain in $\mathbb R^2$ first.

36. For the hyperbolic linear map of the torus given by the matrix $\left[\begin{array}{cc}2&1\\ 1&1\end{array}\right]$, finding and counting the fixed points of the $n$th iterate of the map involved finding the preimages of $\left[\begin{array}{c}0\\ 0\end{array}\right]$ under the related map $G_n = F_L^n-I_2$ (See the proof of Proposition 7.1.10).  These were the integer vectors lying inside the parallelogram $\left(L^n-I_2\right)\left([0,1)\times[0,1)\right)$, where $L$ is the linear map of $\mathbb R^2$given by the same matrix.  Do the following:

    1. Use the construction of $G_3$ to calculate the number of fixed points of the third iterate of the toral map.

    2. Draw the parallelogram for $G_3$ and mark all of the integer vectors in this image of the half-open square.

    3. Use $G_3$, as well as $G_1$, and $G_2$ to identify the points in the original fundamental domain that correspond to integer vectors in the parallelogram (these are the actual toral points fixed by the third iterate). 

37. The book uses the fact that expanding maps of $S^1$ are topologically mixing to show that they are chaotic.  Without using topological mixing, show that expanding maps of $S^1$ are chaotic.

38. Show that the cylindrical twist map from EP28b has a sensitive dependence on initial conditions but is not chaotic.

39. Show that an isometry cannot exhibit a sensitive dependence on initial conditions.

40. Calculate the $f$-capacity and hence the box dimension of the Cantor Set formed by successfully removing the middle half of each subinterval of the unit interval at each stage.

41. Show that for a map $f:X\to X$ on a metric space, the $n$th orbit segment metric $d_n^f$ defined in class is actually a metric.  Recall the definition:  for $d$ a metric on $X$, we have $\displaystyle d_n^f(x,y)=\max_{0\le i\le n-1} d\left(f^i(x),f^i(y)\right).$