Never in the annals of science and engineering has there been a phenomenon so ubiquitous, a paradigm so universal, or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg, for beneath it lies a much finer structure of immense complexity, a geometric labyrinth of endless convolutions, and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore, thereby forefeiting their only chance to grapple with reality.
 Leon O. Chua



INTRODUCTION
    

Most natural phenomena are nonlinear; yet even today, theoretical analysis of physical systems is usually based on linear mathematical models or ones having small deviations from linearity. Linear models are still routinely used because they are much easier to solve than the correct nonlinear ones. Within the last two decades, however, both theoretical and experimental investigations of nonlinear phenomena have shown that often behavior that appears to be random or chaotic is actually deterministic in its orgin. Nonlinear deterministic systems under these conditions are predictable only for short times. This paradoxical situation exists because the deterministic solutions depend very sensitively on initial conditions. It has also been found that several classes of systems show universal behavior at the onset of chaos. Thus, system as diverse as a dripping faucet and a heart in ventricular fibrillation show many common features in their dynamics.



RESEARCH  INTEREST

My area of research was in modeling and control of vibro-impact systems that have nonsmooth dynamics in phase space. In particular, I was interested in improving methods of recognizing and describing deterministic chaos and in studying methods for controlling chaos, in systems that show temporal chaos and spatially extended systems that show spatio-temporal chaos. Numerical studies of impact systems were mainly carried out on the following kinds of vibro-impact systems:

  • Driven Linear Oscillator impacting with symmetrically (about the equilibrium point) placed fixed walls.
  • Nonlinear Oscillator impact driven by a sinusoidally moving wall.
  • Impact coupled globally driven spaially extended linear oscillator systems.

    • Note in the first and thrid systems mentioned above are linear between impacts and the solution of the equations can be determined analytically. The response of the system is given in terms of an exponentially decaying term(transient solution) and a sinusoidal term with a constant amplitude term (dependent upon system parameters) and phase with a constant lag with respect to the drive phase. The source of the nonlinearity is in the impact process. The strong nonlinearity introduced due to the impact and its effect on the overall dynamics (novel bifurcation phenomenon like grazing bifurcation, intermittency near grazing collision, hysteresis, etc) of the system was topic of my thesis. Ofcourse, controlling and tracking too!



    		RELATED STUFF

    I have tried to organize this section on chaos, by introducing the concepts via the Logistic map. The choice of the system, besides historical, is because of the relative ease with which some analytical analysis can be done. Almost all figures are generated by the program provided within each subsection. The material is illustrated using the output of the programs provided.

    The section on GENERAL CHAOS is a general introduction to chaos via simple discrete time modeling (maps), ordinary differential equations (ODEs), delay differential equations(DDE), cellular automata (CA), coupled map lattice (CML) systems and coupled ordinary differential equations(CDE), and partial differential equations(pde).

    The section on BOUNCING BALL is a percursor to impact system - the area of my research. It is an illustration of chaos in impact systems. Given the simple nature of the system (freely falling ball impacting against a sinusoidally driven floor) it is surprising how the nonlinearity introduced by simple impact law leads to extremely complicated behavior of the ball. The second subsection is a simulation of an N-ball system with stationary 'floor' and the impact is assumed to be elastic. This is probably the only simulation of Hamiltonian system (energy is conserved) I have here.

    The following four sections, viz. IMPACT SYSTEMS, CHARACTERIZATION, CHAOS CONTROL, and SPATIAL CHAOS are edited, htmlized version of a few chapters from my thesis. Go there at your own risk! There are several subsections within these four sections mentioned. Check it out!

    The section on DEMONSTRATION is a java applet that demonstrates Occasional proportional feedback (OPF) linear control algorithm and nonlinear control algorithm on the chaotic Logistic map. Finally, LINKS & LISTINGS subsection, as of now, contains a list of programs that you can download from here.

    Note, the original tex files were converted to html using Ian Hutchinson's TEX to html converter - TTH. You may need to enable symbol fonts in Netscape running under X, in case your browser doesn't show the special symbols. The quick simple fix is to add the following line in your .Xdefaults (or .Xresources) file
    Netscape*documentFonts.charset*adobe-fontspecific: iso-8859-1
    Then from your command prompt run
    xrdb .Xdefaults
    Finally, restart Netscape!! This should fix it, if not click here for the details.

    The usual disclaimer holds for the programs that are provided here. There is no nothing that is guranteed by these programs, use it, modify it, or even trash it at your own risk! All that I know is that they have worked fine for me, if they help you in anyway good for you!