Fractals and Strange Attractors
Stub. The geometry of chaos: countable vs. uncountable sets and the Cantor set as the prototypical fractal, dimension for self-similar fractals, box-counting and pointwise/correlation dimensions for fractals that aren't exactly self-similar — then strange attractors in maps and flows: the Hénon map, the Rössler system, attractor reconstruction from experimental chemical-chaos data, and the forced double-well oscillator.
Corresponds to Strogatz, Chapters 11–12. A strange attractor (like the Lorenz attractor of the previous note) is typically a fractal: a set with structure at every scale, often of non-integer dimension — which is why this note starts with the set-theoretic and geometric machinery needed to make sense of that (countable vs. uncountable sets, the Cantor set as the cleanest example, and several notions of fractal dimension, since “dimension” stops being unambiguous once self-similarity isn’t exact). Applies this to concrete strange attractors beyond Lorenz: the Hénon map (a 2D discrete map with a strange attractor, simpler to iterate than a 3D flow), the Rössler system (a simpler-looking 3D flow than Lorenz’s, with its own strange attractor), attractor reconstruction (recovering attractor geometry from a single measured time series of a real chemical-chaos experiment), and the forced double-well (Duffing-type) oscillator as a further physical route to a strange attractor.
Outline (TODO — flesh out each)
- Countable and uncountable sets
- The Cantor set
- Dimension of self-similar fractals
- Box dimension
- Pointwise and correlation dimensions
- The simplest strange-attractor examples
- The Hénon map
- The Rössler system
- Chemical chaos and attractor reconstruction
- The forced double-well oscillator