Topic: Mathematics and Physics
Forrest L. Maddox III
School of Mathematics
Bifurcation diagram for random perturbations of
f : [0, 1] , f(x) = (33/10)x(1-x)
The watercolor splashes on this paper canvas were made by the splatter of paint-laden sponges using a custom-made sponge launcher also fabricated by the artist. It is a depiction of qualitative changes in the behavior of the dynamical system f in the presence of random noise applied at each timestep. No two splashes are quite the same, and their irregular, seemingly random patterns evoke the random noise applied to the system.
The closed interval [0, 1] describes the state of the system, while the function f : [0, 1] → [0, 1] describes how the system at an initial state x in [0, 1] evolves to a state f(x) after one timestep. Each vertical “slice” of the canvas (e.g., the three vertical black lines) is an individual portrait of behavior of f at a fixed noise amplitude, with the presence or absence of color indicating where trajectories are expected. At left, the noise amplitude is small and the state of the system is confined to the two separated color bands, while further to the right the noise is larger and the system can move more freely. Made in consultation with Alex Blumenthal.
For slow-motion video of the artist using the sponge cannon:
For a more in-depth explanation of the math behind this piece: