Study logistic map regimes with detailed numerical diagnostics. Compare transient decay, periodic windows, and sensitivity. See tiny parameter changes create dramatic long term behavior.
This model uses the logistic map as a classic route-to-chaos laboratory.
The calculator studies the logistic map, a standard nonlinear recurrence that demonstrates period doubling and chaos:
xn+1 = r xn (1 - xn)
Here, r controls growth intensity and xn is the state after the nth iteration. As r increases, the orbit can move from a stable fixed point to a period-2 cycle, then to period-4, period-8, and finally to chaotic behavior with embedded windows.
The Lyapunov exponent is estimated from the post-transient orbit:
λ = (1 / N) Σ ln | r (1 - 2xn) |
If λ is negative, nearby trajectories tend to converge. If λ is positive, nearby trajectories separate exponentially, which signals sensitive dependence and practical chaos.
Fixed point stability follows the derivative rule |f′(x*)| < 1. The period estimate is obtained by comparing recent tail values within a numeric tolerance.
Tip: Use r between 3.4 and 3.7 to see the classical transition from high-order periodic motion into chaotic bands.
This reference table shows commonly cited logistic-map milestones that many users explore with this calculator.
| Parameter r | Typical Behavior | Interpretation |
|---|---|---|
| 0.500000 | Stable zero state | The orbit contracts toward x = 0. |
| 2.800000 | Stable fixed point | The map settles to one attracting nonzero value. |
| 3.200000 | Period-2 orbit | The orbit alternates between two persistent states. |
| 3.500000 | Period-4 orbit | Another doubling appears, revealing the route to chaos. |
| 3.569950 | Near chaos onset | The cascade has nearly accumulated into broadband instability. |
| 3.830000 | Periodic window inside chaos | Chaotic intervals can still contain stable repeating pockets. |
It evaluates the logistic map, estimates the Lyapunov exponent, detects short repeating periods, summarizes tail behavior, and draws orbit and bifurcation plots across a parameter range.
It is simple, bounded, and famous for showing fixed points, period doubling, chaotic bands, and periodic windows using only one state variable and one control parameter.
A positive exponent means nearby initial conditions separate exponentially on average. That is a practical signature of sensitive dependence and chaotic behavior in the analyzed orbit.
Early iterations often reflect startup effects rather than long-run structure. Discarding them gives cleaner estimates for attractors, periods, and Lyapunov growth.
Zero means the recent tail did not repeat within the selected tolerance and maximum period cap. The orbit may be chaotic, weakly structured, or require different settings.
For the standard logistic map, users usually study 0 ≤ r ≤ 4. Values outside that interval are not the classic bounded route-to-chaos experiment.
Yes. Chaotic parameter intervals often include narrow periodic windows, such as period-3 and its descendants. The bifurcation plot helps reveal those islands.
Use CSV when you want the scan table for spreadsheets or further modeling. Use PDF when you want a printable report of the visible result section.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.