MATH 118R: Dynamical Systems

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The final project is due May 5th.
The submission will be by email to shani@g.harvard.edu.
Subject line: Math 118 Final Project
File name: your full name
I will print the projects. Please make sure the format in which you submit is easy to print, and that everything can be read after printing as well. (For example, avoid ridiculously small fonts / handwriting.)
I will reply to the email to acknowledge receipt. If you have not received a reply, the project is not submitted, and you should reach out to make sure it is received.

Office hours:

Monday (5/1) 2:00pm-3:30pm, SC 333i (Assaf)
Tuesday (5/2) 2:00pm-3:30pm, SC 333i (Assaf)
Wednesday (5/3) 11:30am-1:00pm, SC 333i (Assaf)

You are encouraged, and always welcome, to come to office hours.

Class meetings:
Tuesdays and Thursdays, 9:00-10:15 am.
Location: SC 507
Attendance is expected.

Material:
Here is a link to an overleaf project with typed notes, kindly provided by Alex and Calvin. Note that there might be mistakes or missed days and information, so please do not rely entirely on this resource.

Tuesday 1/24	Introduction and overview: Dynamical system (X,T), orbits, fixed and periodic points. The quadratic family $LaTeX: \lambda x(1-x)$ . Bifurcations at $LaTeX: \lambda=1,2,3,4$ . Circle rotations. Irrational rotations are dense. Complete analysis of dynamics for $LaTeX: \lambda x(1-x)$ when $LaTeX: 0<\lambda<1$ .
Thursday 1/26	Some more introduction and overview: interval maps vs circle maps. Intermediate Value Theorem. Statement of Sharkovski's theorem. Symbolic dynamics: space of sequences and the shift map ([D, Section 1.6]). Metric spaces, continuous maps. (Topological) conjugacy of dynamical systems ([D, Def 7.4]).
Tuesday 1/31	Attracting and repelling periodic points. Stable set of a periodic point (See [D, section 1.4], Prop 4.4). Attracting / repelling fixed points for $LaTeX: \lambda x(1-x)$ . Complete analysis of dynamics for $LaTeX: \lambda x(1-x)$ when $LaTeX: 1<\lambda<2$ (see [D, section 1.5], Prop 5.3).
Thursday 2/2	Complete analysis of $LaTeX: \lambda x(1-x)$ when $LaTeX: 2<\lambda<3$ (see [D, section 1.5], Prop 5.3). Devaney's definition of chaos ([D, Section 1.8]). The circle map $LaTeX: \theta\mapsto 2\theta$ is chaotic. Circle rotations are not chaotic.
Tuesday 2/7	is chaotic. Semi-conjugacy (see [D, Example 8.9]). Shift map $LaTeX: \sigma$ is chaotic. Studying $LaTeX: \lambda x(1-x)$ for large enough $LaTeX: \lambda$ ( $LaTeX: \lambda>2+\sqrt{5}$ ). The set $LaTeX: \Lambda$ of non-escaping points. The itinerary map to the sequence space. We proved that the itinerary map is one-to-one. (See [D, Section 1.5]).
Thursday 2/9	The itinerary map is onto, sending $LaTeX: I_{\omega_0,\dots,\omega_{n-1}}\cap \Lambda$ to $LaTeX: B_{\omega_0,\dots,\omega_{n-1}}$ . Devaney's definition of chaos: third condition (sensitivity) is redundant for continuous maps (see [HK, 7.2.12]). Proof under a simplifying assumption.
Tuesday 2/14	Finished proof about sensitivity. Removed the simplifying assumption. The dynamical systems $LaTeX: \lambda_1 x(1-x)$ and $LaTeX: \lambda_2 x(1-x)$ are topologically conjugate when $LaTeX: 0<\lambda_1, \lambda_2 < 1$ . Fundamental domain argument (see [D, Section 1.9]).
Thursday 2/16	-distance and structural stability ([D, Section 1.9]). Structural stability for $LaTeX: \lambda x(1-x)$ , $LaTeX: 0<\lambda<1$ . For continuous interval maps: $LaTeX: T^3(x) \leq x < T(x) < T^2(x)$ implies all periods exist. Example: if has period 3. (see [D, Theorem 10.1]).
Tuesday 2/21	For continuous interval maps: transitivity implies chaos (see [R, Proposition 2.15]). Started proof of Sharkovski's theorem (see [D, Theorem 10.2]).
Thursday 2/23	Continued proof of Sharkovski's theorem. Case of odd period where is the smallest odd period. Justified figures 10.3 and 10.4 in [D, p. 65].
Tuesday 2/28	Final steps for Sharkovski's theorem.
Thursday 3/2	Miscellaneous: Optimality of Sharkovski's theorem (see [D, p. 66-68]). Clarifications regarding Question 16 in [D, p. 60]). Approximating a continuous system by discrete dynamics: $LaTeX: \frac{dx}{dt}=kx(L-x)$ vs $LaTeX: F(x)=\lambda x(1-x)$ . The circle as a quotient (or as ) (see [HK, 2.6.2]). [Here is a quick review of equivalence relations and quotients.] Using circle rotations to study trajectories on the torus and billiards in the square.
Tuesday 3/7	Lifts of circle maps. Degree of circle maps. (See [HK, Proposition 4.3.1]). For injective circle maps the degree is $LaTeX: \pm1$ .
Thursday 3/9	Topologically conjugate circle maps have the same degree. Rotation number for order preserving homeomorphisms. (See [HK, pages 125-126])
Tuesday 3/21	Orientation preserving homeomorphisms with rational rotation number. (See [HK, Proposition 4.3.5 and Proposition 4.3.8]
Thursday 3/23	Poincare classification theorem. ([HK, Theorem 4.3.20]). Corollaries: A transitive homeomorphism of the circle is topologically conjugate to an irrational rotation (and the conjugating homeomorphism is orientation preserving). If a homeomorphism of the circle has a dense orbit then all orbits are dense.
3/28 3/30	- Rotation numbers under topological conjugacy. Proposition 4.3.9 in [HK, p. 128]. - Denjoy's example: an orientation preserving homeomorphism of the circle with irrational rotation number and a wandering interval. Example 14.9 in [D, p. 108-9] - Schwarzian derivative. Critical points in stable sets of attracting periodic points. [D, p. 69-74].
Tuesday 4/4	Denjoy's theorem: Circle diffeomorphism with continuous second derivative and no fixed points is topologically conjugate to an irrational rotation. See [HK+, p. 401]. See also J. Milnor notes.
Thursday 4/6	Finished the proof of Denjoy's theorem. Small perturbations of circle homeomorphisms and rotation numbers (continuity of $LaTeX: \rho$ ). [HK, 4.4.5].
Tuesday 4/11	Instability of irrational rotation number. Any -diffeomorphism of the circle with irrational rotation number can be approximated arbitrarily close (by the -distance) by diffeomorphisms with periodic points. [D, Section 1.15].
Thursday 4/13	Period 3 for $LaTeX: \lambda x(1-x)$ , $LaTeX: 3<\lambda<4$ . [D, Section 1.13]. An illuminating example for: Sharkowski's theorem; the theorem regarding critical points in attracting orbits; and subsystems of the shift space.
Tuesday 4/18	Finished proof that the itinerary map is one-to-one. Subshifts of finite type [D, Section 1.13]. Definition(s) of Thue-Morse sequence [G. 15.1.1].
Thursday 4/20	Morse sequence ([G, p. 272-273]). Odometer (minimal) systems. Almost periodic sequences. Minimal subsystems of the shift space. ([G, Theorem 18.3.8])

References:
[D] Robert Devaney - An introduction to chaotic dynamical systems (Online access).
[HK] B. Hasselblatt and A. Katok - A first course in dynamics (Online access).
[G] Geoffrey R. Goodson - Chaotic Dynamics (Online access).
[R] Sylvie Ruette - Chaos on the interval (graduate text) (Online access)
[LM] D. Lind and B. Marcus - An introduction to symbolic dynamics and coding (Online access).
[HK+] B. Hasselblatt and A. Katok - Introduction to the modern theory of dynamical systems (graduate text) (Online access).

I will make references to the book in the follow forms: "Theorem X in [D]" or "[D, Theorem X]", "Exercise Y in [D, Section X]", "[HK, Exercise x.y.z]", and so on.

Course description:
Our main object of study will be an iterative dynamical system. That is, a space LaTeX: X , and a transition function $LaTeX: T\colon X\rightarrow X$ . The space will often be the space of real numbers $LaTeX: \mathbb{R}$ , or the unit circle. The general goal is to study the orbits:

$LaTeX: \begin{equation} \hspace{9mm}x_0, T(x_0), T(T(x_0)), T(T(T(x_0))), \dots \end{equation}$

where LaTeX: x_0 is some initial value.

A central theme in our study will be predictability vs. chaos. That is, when we can predict the long-term behavior of such orbit, or when the behavior is chaotic and unpredictable.

Our focus will be on the abstract mathematical theory. For example, among the results we will cover will be:

Finding and analyzing periodic points. Attracting and repelling periodic points.
Sharkovskii's theorem about periodic points for iterative dynamical systems on the real line $LaTeX: \mathbb{R}$ .
Analyzing the dynamical behavior for the functions $LaTeX: T_\lambda(x)=\lambda x(1-x)$ , depending on the value of $LaTeX: \lambda$ . (In particular, determining when it is predictable and when it is chaotic, and studying its bifurcations.)
Dynamics on the circle: rotation numbers, doubling maps. We will try to understand and classify the possible dynamical behavior for functions from the circle to itself.
Symbolic dynamics: "the shift system" on the space of sequences. The shift system, and its subsystems, are important examples of dynamical systems. Furthermore, they provide an essential tool to study the dynamical behavior of other dynamical systems, such as the ones mentioned above.
Structural stability. Suppose are two transition maps which are "very close". Must their dynamical behavior be similar? For example, If $LaTeX: \lambda_1, \lambda_2$ are very very close, do $LaTeX: T_{\lambda_1}, T_{\lambda_2}$ have similar dynamical behavior?

Prereqs:
Familiarity with calculus and linear algebra, Math 21, 22, 25, or 55.
Basic properties of the real numbers; exposure to formal mathematics, such as the definition of a limit and the $LaTeX: \varepsilon - \delta$ definition of a continuous function; basic definitions for a metric space. (These may be encountered in Math 112, Math 101, Math 55, or others.)
See for example section 1.2 in reference [D]. See also A.1 in reference [HK].

Assessment:
80% Psets, 20% final project.
Please take a look at the Syllabus for more details. Specifically: take a look at rules regarding Psets and the grading philosophy.
Information about the final project.