### Chaotic Substrate

Program Notes
Chaotic Substrate is an improvised work for live electronics (audio and video) constructed around the logistic map - a commonly used example of chaos arising from a simple, deterministic formula.
It is defined as:

x2 = r∙x1∙(1-x1)
x3 = r∙x2∙(1-x2)
etc.

where all xs are between 0 and 1, and r is between 0 and 4.

As the number of times x is updated approaches infinity, the possible solutions to x converges to a finite number of values. That number is determined by r. When r is below 3, x converges to a single value. After that,
x converges to 2 values, and then 4 values, and then 8 values, etc. As r increases, the number of values that x can achieve as it approaches infinity continues to double until about r = 3.6 at which point this number explodes into an uncountably large amount, shown in the diagram.

For the remaining values of r, the number of values x can achieve becomes noisy and chaotic. However, in the chaos there is still structure: the outer extremes smoothly expand and clusters of points create patterns and lines that move in interesting ways. One striking pattern is the number of possible values suddenly collapses to a much smaller amount, at points as low as 3. Much like the beginning, the number starts to double until chaos is again reached.

In Chaotic Substrate, the logistic map becomes an instrument that the performer improvises with. As they move through different values of r, up to 256 separate sine oscillators sound in proportion to the corresponding x values, which are also displayed visually. Since the convergence of x is only guaranteed as x approaches infinity, computing these values (through a necessarily finite number of iterations) yields interesting artifacts where convergence does not quite occur. For brief periods, extra oscillators suddenly sound, all approaching the convergent values, but not quite reaching them.

Biographical Sketch
Nicholas Shaheed is a composer of electronic and instrumental music, often combining the two. His electronic music emphasizes live performance - typically with acoustic instruments - and explores the procedural generation of musical textures and systems, as well as ways to better integrate electronics with the natural musicality of the live performer. For his music, he has received a number awards and grants such as the Edward Mattila Award for Excellence in Electronic Music, the Brosseau Creativity Award, the James K. Hitt Award for Outstanding Undergraduate Research, and the University of Kansas Undergraduate Research Award. His music has been performed internationally. His works have been played by Quartetto Indaco, Helianthus Contemporary Music Ensemble, and the University of Kansas Tuba-Euphonium Consort. He is currently pursuing a B.M. in music composition and a B.S. in computer science from the University of Kansas, and has studied under Kip Haaheim, Ingrid Stolzel, and James Barnes.