# PSU Math Department Announces PSU-PKU 2012 Summer Program in Dynamical Systems

**PSU/PKU Summer Program 2012**

We are pleased to announce the 2012 summer program in Dynamical Systems, organized by the Mathematics departments at Penn State and Peking University.

Time and place:

The program will take place from May 14 to June 1, 2012, at Peking University, China's oldest and most prestigious university and an internationally recognized educational institution. In addition, you will have the opportunity to visit the city of Beijing, the cultural and political center of China, and many of its historical sites.

**Course content:**

Lectures will be given by Prof. Mark Levi and Prof. Alexei Novikov from Penn State.

Ordinary differential equations, or dynamical systems, arise in an enormous variety of settings.

The entire field of classical mechanics, for instance, is a subfield of dynamical systems. Some

biological, chemical and electrical processes are also described by dynamical systems.

Virtually all that's understood about dynamical systems can be illustrated on very simple basic examples; the goal of this course is to provide such illustrations.

This course will describe some fascinating discoveries in dynamical systems of the 20th century,

yet will be accessible to students with prerequisites mentioned above. The approach will be

as intuitive and non-technical as possible. Some examples of the questions we will consider:

* What are some simple examples of chaos in mathematical models of some simple physical systems, e.g. of a pendulum?

*How do Cantor sets arise in some very simple differential equations of mechanics?

*If the pivot of the upside-down pendulum is vibrated rapidly, the pendulum becomes stable,

*What is the uncertainty principle in classical mechanics? How does it relate to divergence?

*How does the golden mean (or golden ratio) arise in simple differential equations?

*Can resonances self-destruct? Is the Solar system stable?

Some of these questions look special, but they capture in simple form

several of the most important mathematical developments of the 20th century.

Here is a more detailed syllabus; all the (possibly) mysterious terms will be explained during the course.

1. A REVIEW as needed of the background (1D flows, 2D autonomous systems - rest points (linearization, etc.), limit cycles,Poincare-Bendixson, bifurcations, tent/logistic maps and symbolic dynamics, fractal dimension.

2. LINEAR SYSTEMS: Floquet theory, stability of the inverted pendulum and 2 by 2 matrices with determinant =1. Explaining behavior of Schroedinger spectrum. Orthogonal and Hamiltonian flows. Lyapunov-Poincare theorem. Linear symplectic maps in R^4, with applications to optics, mechanics and differential geometry.

3. HAMILTONIAN SYSTEMS: From variational principles to Euler-Lagrange equations to Hamilton's equations - a brief introduction. Phase planes of Hamiltonian systems with one degree of freedom. Action--angle variables. Adiabatic invariants. Poincare's geometric theorem.

4. CHAOS. Smale's horseshoe, symbolic dynamics (topological Markov chains), chaotic motions of a pendulum, Lorenz attractor and an example of Cartwright and Littlewood. Strange attractors.

Participants:

Participants:

Participants will include students from Penn State and local students from Peking University.

Preparatory meetings:

In the Spring 2012 semester there will be several organizational meetings to review preliminary materials and to discuss pre-departure organizational matters.

**Prerequisites:**

Math 230 and 250 and consent of the instructor.

**Financial support:**

Financial support for qualified candidates will be available. In past years, most of the expenses, including travel, were covered.

**How to enroll**:

1. Apply using the form on the PSU-PKU web site (https://elements.science.psu.edu/psu-pku/classes-and-programs/PSU-PKU.application.2012-1.pdf)

2. If the application is approved, you will receive instructions on registering via eLion. In order to participate in this program and to be considered for financial support, you must complete and submit the application form by Jan. 20, 2012.

**Questions?**

Please contact:

Prof. Mark Levi, levi@math.psu.edu

Prof. Xiantao Li, xli@math.psu.edu

Prof. Jinchao Xu, xu@math.psu.edu