10. Nonlinear Dynamics in Simple Maps

Read the opening pages · printed page 139

The Bifurcation Diagram (bifur1d.c (and gen1d.c, phase1d.c))

The logistic map, x becomes 4rx(1-x), is a one-line model of a population with limited food. For each value of r the program runs the map a while and plots where it ends up. Small r gives a single steady value, then it splits in two, then four, then eight, at intervals shrinking by Feigenbaum's constant, until the values fill whole bands: chaos, with windows of order inside it. The view starts just before the first split and then zooms into the cascade to show that the whole diagram repeats inside itself. Full screen →

Sections

1. The Logistic Map
2. Stability and Instability
3. Bifurcations and Universality
4. Prediction, Layered Pastry, and Information Loss
5. The Shadowing Lemma
6. Characteristics of Chaos
7. Further Exploration
8. Further Reading

Figures

Figure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6Figure 10.7Figure 10.8Figure 10.9Figure 10.10