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Graduate (MAGIC) course Dynamical Systems II: Maps

Graduate (MAGIC) course Dynamical Systems II: Maps

Access course material through the website. The material was used to deliver the module MAGIC060 online via "Access Grid" technology. The course was developed originally by Toby Hall and taught in 2013 by Lasse Rempe-Gillen, on behalf of the EPSRC-funded MAGIC (Mathematics Access Grid Instruction and Collaboration) network. The support of EPSRC for the Magic network is gratefully acknowledged. 

The course materials include slides that are projected during lectures, and handouts of these slides that can be read off-line. There are some exercises and an official examination available that can test your understanding of these resources. The material is arranged into 10 lectures. An online bibliography provides additional references and some web-links for programs that allow computer experiments.

The course aims to give an overview and introduction to a number of topics in discrete one-dimensional topological dynamics (symbolic dynamics; unimodal maps; the horseshoe map; complex dynamics) rather than to treat any one concept in great detail. Proofs are usually omitted in favour of broad ideas and motivation, unless they are considered to be particularly enlightening.
Graduate Introductory Survey Course on Nonlinear Dynamics

Graduate Introductory Survey Course on Nonlinear Dynamics

A web site for a graduate introductory survey course on nonlinear dynamics, originally intended for physical science students, but also appropriate for mathematics students with no prior background.

The web site includes detailed lecture notes, xppaut input files to illustrate certain concepts (with detailed calculation procedures often described in the notes), and Maple worksheets to illustrate certain computations, as well as past assignments.

How to use: Start with the lecture notes. The computer files are intended to illustrate calculation procedures and to be used side-by-side with the lecture notes. The assignments and lectures are numbered alike, i.e. assignment n is to be completed after lecture n.
Dynamics of Physical Systems; Chaos, Fractals, and Dynamical Systems

Dynamics of Physical Systems; Chaos, Fractals, and Dynamical Systems

The set of lectures is aimed at addressing this pedagogical issue, and is divided into two parts. In the first part, the readers are introduced to the methods and techniques for translating a physical problem into mathematical language by formulating differential equations. In general, the methods of obtaining differential equations follow from the Lagrangian and Hamiltonian techniques of classical mechanics. It is generally believed that these techniques are not applicable to engineering systems, and consequently are not taught in engineering courses. In these lectures we show that, in contrary to common belief, they are quite applicable to engineering systems: mechanical as well as electrical. For complex electrical circuits, the graph theoretic method also offers a systematic procedure. Engineers have developed a very convenient and systematic procedure of automating the process of deriving differential equations using a method called bond graphs. However, this technique is not yet known to the physics community even though it can be very useful in modeling most systems a physicist has to deal with. This course gives a beginner's exposure to these techniques. In the second part, the method of local linearization is introduced, and the dynamics of linear systems are then analyzed using eigenvalues and eigenvectors. The objective is to develop a geometric understanding of dynamics in the state space. This book also introduces those aspects of nonlinear and discrete dynamics which any 21st century scientist and engineer, irrespective of discipline, should know.

A brief description of the course “Chaos, Fractals & Dynamical Systems”

This is an undergraduate-level introductory course on nonlinear dynamics and chaos theory. It starts with the different models of dynamical systems: differential equations and difference equations. It then introduces different types of stable dynamical behaviors in the context of flows: equilibrium points, limit cycles, quasiperiodic orbits, and chaos. The method of Poincare section is introduced, and the rest of the bifurcation theory is treated in the context of maps. This includes the local bifurcations like saddle-node bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation, and global bifurcations like interior crisis and boundary crisis.

A proper treatment of nonlinear dynamics cannot be done without introducing fractals, which is taken up in the next stage. The concept of fractional dimension is introduced. Mandelbrot set and Julia sets are placed in the context of parameter space and phase space respectively. The iterated function system is introduced.

Then we take up the characterization of different dynamical behaviors, especially in the context of analyzing experimental data. The Lyapunov exponent, correlation dimension, invariant density (including the idea of Frobenius-Perron operator), and spectral analysis is presented. Finally issues like control of chaos, targeting in the phase space, etc., are taken up. All the concepts in this course are illustrated using examples from electrical, mechanical and fluid-dynamic systems. An unique feature of this course is its treatment of the dynamics of non-smooth systems—a topic that is becoming important and finding use in many application
An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

An Introduction to Coupled Oscillators: Exploring the Kuramoto Model

Prize winner, DSWeb Student Competition, 2007

This tutorial provides an introduction to the application and non-linear dynamics of globally coupled oscillator systems by considering the popular and well researched Kuramoto model. 
Dynamical Systems and Fractals

Dynamical Systems and Fractals

Lecture notes from an Oklahoma State University course on symbolic and analytic dynamics, with an overview of fractal geometry.
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