SYLLABUS
“panta rei“
HrakleitosoEfesios
AUBG, Fall 2017
MAT 497: Dynamical Systems (meeting time: TF 14:15-15:35)
A course where deep theory meets nontrivial applications.
prerequisites: Calculus 3 (MAT 212) and Linear Algebra (MAT 105)
Alexander GANCHEV
, analysis/ ,
office 304 BAC, phone: 480,
office hours: T 17:30-18:30, W 13:00-14:00, and by appointment
Course description: Everything flows hence everything is a dynamical system. The examples of real life dynamical systems include phenomena from economics, computer science, biology, ecology, engineering, finance, physics, etc. The local dynamics (how things change for small time increments) is given. It is described either by differential (continuous time) or difference (discrete time) equations. The goal is from local information to deduce information about the behavior of the system at large scales. The emphasis is not on finding an explicit solution but on qualitative analysis of the behavior of the system – hence the need for tools from topology, geometry, measure theory, etc. Since most interesting dynamical systems are nonlinear the aid of machines and software, in particular the use of computer algebra systems such as MatLab, Octave, Maxima, etc., are indispensable in the dynamical systems toolbox The course will be an introduction to discrete and continuous dynamical systems. Possible topics will include some of the following: linear and nonlinear phase portraits, limit sets (fixed points, orbits, etc.), stability, bifurcations, chaos, fractals, etc.
Concepts and methods from analysis, topology and geometry will be introduced along
the way. We will not shy away from rigorous proofs but an equal emphasis will be on numerical methods and the use software.
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Expected Outcomes:
By the end of the course the student should:
- Be familiar with basic concepts from the theory of dynamical systems (such as: phase portraits, limit sets, stability, chaos, etc.)
- Be familiar with a concrete CAS and be able to use it as a tool to find solutions to concrete dynamical systems and graphically display the information.
- Have advanced her/his skill to reason in a rigorous fashion and communicate it in oral or written form.
- Work in a team to translate a real-life phenomena into a mathematical model.
- Work in a team using numerical and/or rigorous technics to analyze a mathematical model.
- Acquire a deep appreciation for the power and beauty of mathematics in general and dynamical systems in particular.
- Learn to read a mathematical text and to obtain information from numerous sources. Read research articles. Develop basic skills in writing mathematical proofs
The above is a list of basic notions and technical mathematical skills we want to develop but what is more important we want to learn to think creatively, be able to attack a problem you have not seen before, develop tools for that and produce a rigorous proof of the basic statements – in other words develop further your mathematical maturity.
Prerequisites: A good knowledge of the standard course of Calculus (1 & 2) and Linear Algebra is a necessary prerequisite. The courses in Numerical Methods or Differential Equations will be helpful but I do not expect that you have taken them.
Textbook: Steven H. Strogatz. Nonlinear Dynamical Systems and Chaos (2nd edition, Westview Press, 2015 )
We will use several other texts as sources of information (see the reading list below).
Assessment: Assessment will be based on several components: homework, mini-projects, short quizzes, bigger project, midterm, and final exam. Students will be assigned mini-projects and report on them in 5 to 10 min either to me or to the class. Homework will be assigned due in one week. Homework will not be collected. The homework grade will be based on discussions of the homework by the students. Obviously under these circumstances late homework is meaningless. There will be a midterm, its date will be announce at least one week in advance. Students will have to make a project with a wtitten part and a presentation. The homework, mini-projects, participation, and short quizzes will count for 25%, the midterm for 25%, the project for 10%, and the final for 40% of the final grade. The Points/Grade correspondence is: D > 40, D+ > 50, C- > 55, C > 60, C+ > 65, B- > 70, B > 75, B+ > 80, A- > 85, A > 90.
Exam policies: the standard AUBG honor code applies for exams.
Attendance: Students are expected to attend classes regularly and come prepared (having read the assigned text if there is such) and to show active participation during the lecture. You are responsible for all material covered in class, whether or not you attended this class
Office hours and help: If the “official” office hours above are not convenient for you please contact me to arrange some other time. Don’t be afraid to come and ask. There are no stupid questions. You can also try the – a place where you can ask questions (but don’t ask to have your homework done ).
Reading & Writing: I encourage you to explore the topics we cover on your one. At the AUBG library we have a very good collection of textbooks on Dynamical Systems and Differential Equations. Look into several of these books, find the one(s) that best fits your background and your style. Having several different viewpoints on one topic will only help. And take notes – take notes in class, take notes when you read a book. Ideally you will turn your own notes into a coherent text – your personal lecture notes. This is a process that we start here and could continue into graduate school. In order to do this I encourage you to learn TeX (its dialect LaTeX and if you are scared by programming languages you could use the user-friendly front-end LyX). This is the informal side.
Online Components of the Course: The Course Schedule will be updated as we go along on and placed on my web space. You will be able to find additional material there that I will add as we proceed.
Disclaimer: This syllabus is subject to modification. The instructor will communicate with students on any changes.
Additional/optional reading:
discrete:
R. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, 1989) QA614.8 .D48 1989
R.A. Holmgren. A First Course in Discrete Dynamical Systems (Springer 1994) QA614.8 .H65 2000
discrete and continuous:
K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos (An Introduction to Dynamical Systems), (Springer, 1996).
R.C. Robinson. An introduction to dynamical systems: continuous and discrete (AMS 2012) QA614.8 .R65 2012
Luis Barrera, Claudia Valls, Dynamical Systems: an introduction (Springer 2013) QA614.8.B37713 2013
Stephen Wiggins, Introduction to Applied Dynamical Systems and Chaos (Springer 2003) QA614.8.W544 2003
continuous:
S. Strogatz. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Addison-Wesley Pub., 1994) Q172.5.C45 S767 1994
M. W. Hirsch, S. Smale, R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. (Academic Press, 2004) QA372 .H67 2004
S. Lynch, Dynamical Systems with Applications Using Maple, (Birkhauser, 2010). QA614.8 .L96 2010
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,
(Springer-Verlag, 1983) QA1 .A647 vol.42
(O)DEs:
D. Betounes. Differential Equationss. Theory and Application (2 edition, Springer 2010) QA371.5.D37 B47 2010
free online:
G. Teschl, Ordinary Differential Equations and Dynamical Systems (free draft 2012)
J. Leydold, M. Petry, Introduction to Maxima for Economics (Creative Commons license 2011)
J.E. Villate, Introduction to Dynamical Systems. A Hands-on Approach with Maxima (CC license 2006)
J.E. Villate, Introducao aos sistemas dinamicos: uma abordagem pratica com Maxima (CC license 2007)
A. Morante, J. A. Vallejo, Chaotic dynamics with Maxima, arXiv 1301.3240
CAS (computer algebra systems) – free and/or open source:
MAXIMA: SAGE:
OCTAVE:
When it comes to formatting (any text but especially mathematics) there is nothing to compare to TeX and LaTeX. Find yourself stuff on the web or use Formatting information - A beginner's introduction to typesetting with LaTeX by Peter Flynn.
Weekly Schedule (tentative subject to changes)
week 01 / Introduction (continuous versus discrete time dynamical systems, reversible versus irreversible systems, examples of real life dynamical systems, the notions of time, state and evolution, flows)week 02 / Fixed points of discrete-time one-dimensional systems (graphical/cobweb/staircase analysis).
week 03 / Continuous-time systems (flows, integral curves, orbits, equilibrium points). One-dimensional systems. Bifurcations in one dimensional systems..
week 04 / Two-dimensional continuous-time systems. Use of CAS such as Maxima or dedicated tools such as Pplane.
week 05 / Two-dimensional continuous-time linear systems. Analysis and classification of equilibrium points. (Trace-determinant plane)
week 06 / Concrete examples: Duffing equation, Competing species, Exact pendulum.
week 07 / Phase portrait of two-dimensional continuous-time systems. (Nulclines, equilibrium points, Hartman-Grobman Theorem)
week 08 / Phase portrait of two-dimensional continuous-time systems. (limit cycles, van der Pol systems, Poincare-Bendixson Theorem)
week 09 / One-dimensional discrete-time systems. Bifurcations. The logistic equation and the quadratic map.
week 10 / One-dimensional discrete-time systems. Period doubling. Bifurcation diagram.
week 11 / One-dimensional discrete-time systems. (Sarkovskii Theorem)
week 12 / One-dimensional discrete-time systems. Chaotic dynamics. Cantor set as non-wandering set.
week 13 / Symbolic dynamics.
week 14 / Review. Student presentations.
Finals / Final exam
New Year and vacation