I recently finished my doctoral thesis on "Non-smooth dynamics of Vibro-impact Systems" under Prof. Roger W. Rollins, from the Department of Physics and Astronomy, at Ohio University (you can have a look at my resume, or as word doc format).

More specifically, I was concentrating on modeling, analysis and control of systems (referred to as impact systems) that have discontinuous dynamics, in phase space, due to recurrent impacts. The study was done from the point of view of modern dynamical systems theory. Methods like, phase space analysis, bifurcational analysis, and local stability analysis, were used to understand the dynamics. Estimation of invariants of the attractor, like Lyapunov exponents and Lyapunov dimension, detection and tracking of embedded unstable periodic orbits, were carried numerically. Implementation of Recursive Proportional Feedback (RPF) control algorthm, in conjunction with an adaptive learning algorithm were done to stabilize and track the unstable period orbit targetted for stabilization. An extension of the above methods in the study of spatially extended impact coupled systems were also done to gain a qualitative and quantitative understanding of spatio-temporal chaotic dynamics of the system.

Be warned this site is annoyingly disorganized. I took a stab at organizing the material after my thesis and got around to getting some of it hosted here. May be one of these days, without a care in the world (so to say), I can sit and work on stuff here. But as of now here is a brief description of this site.

I have divided this webpage into seven different sections. Ofcourse, there are considerable amount of overlaps. Almost every section is accompanied by several computer codes written by me. Instead of providing one monolithic code that does several things, I have broken down the codes as separate independent programs. The implementation of various algorithms are either in C, C++, Java, or in scripting languages like perl, awk or shell-script. Most of the graphics with interactive capabilities are written in XWindow (Unix/Linux), there are a few programs with Motif and Tk interface, There are a couple of full blown control simulations written in Borland(5.0). As a matter of fact almost all the plots are generated by programs that you can download from this website (check individual sections or go to links & people section).

  ORDER & CHAOS introduces concepts and mathematical tools which are part and parcel of chaoticians. Simple notions are described using the logistic map (the simplest one dimensional nonlinear map that shows chaos), Henon map, Lorenz and Rossler models (using ODEs), Mackey-Glass system (modeled using delay differential equations), coupled map lattices, coupled ODEs and partial differential equations (pdes) and the rest of the stuff in there is about impact systems related to my Ph.D. thesis.

  FRACTALS starts with a discussion on fractal dimension and introduces some classical strictly self similar fractals and the subsections deal with Julia set and some mathematical remarks, Mandelbrot set and some simple mathematical properties, a rather elaborate description of iterated function systems (IFS), and a rather sorry description of (I need to complete it!) Lindemayer system (L-systems).

The following two sections are under construction and it will take a while before the material is completed. Some of the sections are partially complete.

  TIME SERIES ANALYSIS, deals with the method of reconstructing the dynamics based on delay time embedding and issues related to the choice of the delay time and embedding dimension. Methods like noise reduction, false nearest neighbor detection(FNN), corelation function, mutual information, Corelation Dimension and Lyapunov exponent estimation and nonlinear time series prediction techniques are discussed.

  COMPLEXITY starts with a discussion of Cellular automata (CA) as modeling tool for dynamical systems. Both 1-D and 2-D CAs are discussed and shows an example of one particular choice of CA rule that show spiral waves like the ones seen in B-Z reaction. Autonomous agents and self-organization, is discussed in the context of how simple autonomous interacting components self-organize to form potentially evolving structures exhibiting a hierarchy of emergent system properties. From the point of strategy evolution, in competitive situations a brief discussion of competition and cooperation systems in the context of iterated prisoners dilemma problem, is done. One of the search techniques, motivated by biological evolution, genetic algorithm, is described and applied to problems of combinatorial optimization (job-scheduling), continuous function optimization, and evolution of strategies in zero some games like iterated prisoner's dilemma. Artificial neural networks and learning technique used in pattern recall, combinatorial optimization, and prediction of chaotic time series (Ikeda map).

  REFERENCE contains links to tutorials, documentation or plain informal description on various things that interest me. As of now the link are mainly to texi documentation on various unix utilities, programming languages and the likes.

  ODDS 'N ENDS is a lumping together of topics that I don't know where to fit. They are mostly my indulgence with stuff that I think are cool. It is hard to describe the things that are in there. Check it out for yourself. There are some very cool rendering, done using POV-ray and Geomview, of parametric and non-parametric surface equations. An amazingly simple IFS algorithm, using two affine transformations, that generates spiral like attractors, an algorithm to generate symmetric chaotic attractors, a few tips on algorithm for coloring the attractor, etc. etc.

  LINKS & PEOPLE, as of now, is a comprehensive list of most of the programs, demonstrations and animations, that are scattered among various sections. Almost all graphics on this site are algorithmically generated and you will find complete or partial code in the relevant section.

Having said that a word or two about the navigation bar on the left (just in case u didn't notice ;-) ). The idea and code for it is stolen from GIMP site, all that I have done is figured how the code assembles the pieces of the graphics, changed the color and placed it here.