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

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Content of the material and its recommended use:

The material is basically two sets of video lectures, 40 leactures each, meant for full- semester courses. The titles of these courses are:

(a) Dynamics of Physical Systems
(b) Chaos, Fractals, and Dynamical Systems

These lectures were recorded as part of the National Programme of Technology Enhanced Learning (NPTEL) of the Government of India. These are available at the NPTEL website http://nptel.iitm.ac.in/ as well as on the YouTube. As a result of this wide and free availability, these lectures can be used by students anywhere in the world to learn the subject, as well as by teachers as supplementary material for their lectures. By January 2013, some of these lectures have recorded more than 25,000 views, and have attracted a large number of people to the subject of dynamics in general and nonlinear dynamics in particular (as evidenced from the comments posted in YouTube).

A brief description of the course “Dynamics of Physical Systems”

One important component of physics and engineering education is to impart in the student the abilities to formulate mathematical models of systems and to analyze the behavior of systems based on the mathematical description. In this respect, the current undergraduate textbooks on dynamics and control (studied in the engineering curricula) suffer from a serious deficiency, as they are heavily biased toward linear systems, with nonlinearity treated as oddity. Modeling approaches are generally aimed at deriving transfer functions, and the analysis of stability and other properties is then carried out with the standard tools in the Laplace domain. This restricts the exposure of the students to the behavior of linear systems only.

Over the past few decades there has been an increasing realization that most of the physical systems are nonlinear, and linearity is a very special case. This calls for adequate exposure to the techniques of time-domain formulations in addition to those in the Laplace-domain. The student has to learn the techniques of obtaining differential equations for any given physical system, and has to understand the dynamics in terms of the character of the vector field. A linear system description can then be understood as local linear approximation in the neighborhood of an equilibrium point.
Soumitro Banerjee

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 areas.
 Author Institutional Affiliation Soumitro Banerjee Department of Physical Science and Department of Mathematics & Statistics Indian Institute of Science Education & Research - Kolkata Tutorial Level Basic Tutorial Description Presentation Contest Entry Yes