TFPIE2012: Recursion Schemes of Scientific Models
The Recursion Schemes of Scientific Models
By: Baltasar Trancón y Widemann
- Draft proceedings version File:TFPIE2012 Draft RecursionSchemesOfScientificModels Widemann.pdf
- Literate Haskell Archive File:TFPIE2012 Supplement RecursionSchemesOfScientificModels Widemann.zip
The following diagrams can be produced from the Literate Haskell code (provided some not quite hassle-free Haskell packages are installed, sorry).
- Stable limit points that attract the initial state, either one-sided (<math>r<2</math>) or with overshooting (<math>r>2</math>). These correspond to the initial branch of the logistic map's bifurcation diagram.
- Stable periodic solution of period length 2 that attracts the initial state; corresponds to the first bifurcation at <math>r=3</math>.
- Stable periodic solution of period length 2 that attracts the initial state; more difficult to find in the bifurcation diagram in a small window of orderly behaviour among chaos.
- Stable periodic solution of period length 4 that attracts the initial state; corresponds to the second bifurcation at <math>r \approx 3.45</math>.
- Chaotic behaviour at <math>r = 4</math>, ranging over the whole domain. Illustrates one of the defining properties of chaos: orbits (points connected by a trajectory) are dense.
- Chaotic behaviour at <math>r = 4</math>, starting from a tiny neighbourhood of an unstable limit point. Illustrates another defining property of chaos: near trajectories diverge exponentially.
- The initial state is 0.35 in all cases except the last, where it is 0.7500000000001.
- The inference fails after one step, because there is no such state.
- The inference stagnates, because the observation is not unique.
- The inference converges slowly.
- The inference converges quickly.