Abstract

We consider a particle on the line moving in the presence of a force eld and colliding elastically with one or two innitely heavy walls performing a periodic motion. We will study topological, ergodic and statistical properties of this system paying a particular attention to the question of existence of accelerating orbits as well as the rate of acceleration (deceleration).

Outline

1. Introduction
   1. Basic properties of bouncing ball systems.
2. Adiabatic invariants.
3. Normal form for systems with singularities.
4. Applications of KAM theory.
   1. Absence of acceleration in strong potential.
   2. Accelerating tennis ball.
   3. Bounded orbits for piecewise smooth Ulam pingpong.
5. Recurrence.
   1. Applications of Poincare recurrence theorem.
   2. Ergodicity and recurrence.
6. Ergodicity of hyperbolic sawtooth maps.
7. Central Limit Theorem for Dynamical Systems.
8. Recurrence in very weak potentials.

References
All the author papers can be found at http://www2.math.umd.edu/~dolgop/papers.html.

Abstract

We begin with the basic concept of "information content" as the number of yes/no questions needed to identify an object. From this we will derive entropy of a partition and then discuss its properties. Next we will pass to the concept of dynamical entropy of a process understood as the average amount of information transmitted per unit of time. We will discuss the connection between entropy and horizontal compression rate. Next, we will prove one of the key entropy theorems - the Shannon-McMillan-Breiman theorem and provide its interpretation. Finally, we will discuss Kolmogorov-Sinai entropy of a dynamical system and relate it to the vertical data compression rate via generator theorems. 

Topics to be covered

1. Shannon information and entropy
2. Dynamical entropy of a process
3. Entropy theorems in processes (incl. Shannon-McMillan-Breiman Theorem)
4. Kolmogorov-Sinai entropy

Reference
Tomasz Downarowicz:  Entropy in Dynamical Systems, New Mathematical Monographs 18. Cambridge University Press 2011.  PART I

Abstract

The course will start by reviewing key ideas in neuroscience, including neurons and action potentials, synapses and neurotransmitters, so that those with little knowledge in the area can appreciate how mathematical models are constructed. After describing such phenomena and some of the mathematical tools (primarily dynamical systems and bifurcation theory), I will discuss cells as electrical circuits, the Hodgkin-Huxley equations that describe action potentials in single neurons, and generalizations to model bursting neurons (e.g. in locomotion, and other rhythmic patterns). I will then consider small networks of neurons coupled by chemical synapses and electrical gap junctions, and describe simplifying reductions to phase oscillators and integrate-and-fire models. I will also include topics such as Hopfield-Grossberg type neural networks, and the course will end with a survey of recent work at the interface of neuroscience and cognitive psychology, in particular, stochastic differential equations as models of human and primate decision making.