This project explores bifurcation in nonlinear dynamical systems, focusing on the famous logistic map and its transition from stability to chaos. The bifurcation diagram visualized here demonstrates the period-doubling route to chaos, a phenomenon extensively studied by Mitchell Feigenbaum in the 1970s.
Bifurcation refers to a system's sudden qualitative change in behavior when a parameter is varied. The logistic map, given by:
models population growth with resource constraints. As the control parameter (r) increases, the system exhibits:
- Stable fixed points for small (r)
- Periodic doubling (bifurcations)
- Chaotic behavior beyond a critical threshold (~3.57)
Feigenbaum discovered that the ratios of successive bifurcation intervals approach a universal constant (~4.669). This Feigenbaum constant applies to a wide range of systems, establishing a deep connection between nonlinear dynamics and universality in chaos theory.
The provided Python script generates the bifurcation diagram by iterating the logistic map for different values of (r), discarding initial transients, and plotting the stabilized values. This reveals:
- Single fixed points for small (r)
- Period doubling (2, 4, 8, ...) as (r) increases
- Chaotic regions with seemingly random fluctuations
- Windows of periodicity where order re-emerges within chaos