Bifurcation Diagram Simulator — Logistic Map and Chaos
Set growth rate 0–4 and initial value 0–1. Compare the same orbit across bifurcation, cobweb and time-series views, with reproducible exponent checks and explicit finite-computation limits.
Parameters
3.500
The bifurcation diagram, cobweb, time series and exponent use the same initial value.
Finite-sample Lyapunov estimate. A positive value indicates local expansion, not proof that the displayed orbit is nonperiodic. Also check its period candidate and initial value.
The Feigenbaum constant is the limiting ratio of period-doubling intervals. Annotations approximate theoretical values; this tool does not measure the constant. Check fixed-point and period-two exponents in the examples below.
Explore fixed points, cycles and irregular orbits from the same recurrence
This is an educational numerical experiment with the logistic map \(x_{n+1}=r x_n(1-x_n)\). Set r from 0 to 4 and the initial value from 0 to 1. Each bifurcation column starts from the same selected initial value. The diagram does not combine many seeds, nor directly model a real ecosystem or vehicle vibration.
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Student: At r=4 I expected chaos, but starting at 0.5 the time series reaches zero. Is that a bug?
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Professor: This is an exceptional orbit: 0.5→1→0→0 exactly. Once at zero the derivative is 4, so the post-transient exponent is ln(4)≈1.386. Despite that positive exponent, the displayed orbit is a fixed point. Distinguish an unstable orbit from a typical chaotic attractor.
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Student: What changes if I start at 0.123456789?
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Professor: At r=4 this test produces an irregular sequence and a finite-sample exponent close to ln(2)≈0.693. Compare the cobweb, time series and bifurcation diagram under identical conditions. Finite-precision computation does not prove the behavior of every seed over infinite time.
Reproducible input examples
Fixed-point check: r=2.5 and x₀=0.5 converge to x*=1−1/r=0.6. The derivative is 2−r=−0.5, giving exponent ln(0.5)≈−0.693.
Superstable fixed point: At r=2 and x₀=0.5 the orbit starts at its fixed point, with zero derivative. The exponent −∞ from ln(0) is the defined result, not a NaN calculation failure.
Period-two check: At r=3.2 and x₀=0.5 two values repeat. The derivative product is −r²+2r+4=0.16, so the exponent is ln(0.16)/2≈−0.916. Also compare period four at r=3.5 and period three at r=3.83.
Same r=4, different seeds: Starting at 0.5 gives the exceptional orbit reaching zero; 0.123456789 gives an irregular orbit under these computation settings. An exponent alone cannot establish that the displayed orbit is nonperiodic.
Method and limits of classification
The bifurcation diagram, time series and exponent discard 4,096 transient iterations. The exponent averages \(\ln|r(1-2x_n)|\) over the next 4,096 iterations. An exactly zero derivative gives −∞; small derivatives are not silently omitted. The cobweb starts at the selected initial value.
Period detection compares 512 post-transient values with absolute tolerance 10⁻⁹, seeking the smallest candidate from 1 to 64. Otherwise the result is no period detected or unresolved. Long transients, high periods, bifurcation boundaries and floating-point rounding are not automatically classified as chaos. A positive finite-sample exponent is neither an infinite-time mathematical proof nor a real-system safety assessment.
The initial horizontal view is 2.4–4 to focus on bifurcations. Entering a lower r expands it to 0–4; resetting zoom also restores that full range. The shaded region includes periodic windows: not every r there is chaotic. Bifurcation annotations are approximate theoretical values, not numerical measurements of the Feigenbaum constant.
For the recurrence and the exact r=4 transformation and solution, see Wolfram MathWorld: Logistic Map and the r=4 analysis. Check the fixed-point and period-two derivatives independently; the implementation can be inspected in this page's source.
Frequently asked questions
Does a positive Lyapunov exponent make the displayed orbit nonperiodic?
Not necessarily. At r=4, x₀=0.5 follows 0.5→1→0. Its post-transient exponent is ln(4), but its orbit is a fixed point. Examine the exponent, period candidate and initial value together.
Is −∞ a calculation error?
It represents ln(0) when the derivative is zero. Reproduce it with the superstable fixed point r=2, x₀=0.5. This differs from discarding small derivatives and incorrectly reporting zero.
Is the bifurcation diagram independent of the initial value?
Each r starts at the same selected seed. Ordinary stable orbits may converge to the same branches, but exceptional seeds such as 0, 1 or 0.5 at r=4 can change the result. Independence from the initial value is not guaranteed.
Does no period detected mean chaos?
No. This finite-precision calculation searches candidates only from 1 to 64. Long transients, higher periods and boundary cases may escape detection. A positive exponent estimate also differs from a mathematical proof of chaos.
Set growth rate r from 0 to 4 and initial value x₀ from 0 to 1. Selecting a low r expands the diagram to 0–4; all plots and the exponent use the same seed.
Each bifurcation column discards 4,096 transient iterations. Move the r cursor and compare the cobweb iteration playback and time series at the same settings.
Use the worked-example buttons above for a fixed point, a period-2 orbit, a zero derivative and the exceptional r=4 orbit. Read the finite-period candidate together with the exponent; a positive estimate alone does not prove aperiodicity.
Worked Example
With r=2.5 and x₀=0.5, the fixed point is x*=0.6 and the exponent is ln(0.5)≈−0.693. At r=3.2, the period-2 values are about 0.5130445 and 0.7994555, with exponent ln(0.16)/2≈−0.916. At r=4, x₀=0.5 follows the exceptional orbit 0.5→1→0; its post-transient exponent is ln(4)≈1.386. Use the example buttons above to compare it with x₀=0.123456789.
Practical Notes
The exponent averages 4,096 logarithmic derivatives after 4,096 transient iterations. The diagram plots 180 points per column; period candidates 1–64 are checked against 512 tail values with absolute tolerance 10⁻⁹. There is no iteration-count input.
An exactly zero derivative gives the defined value −∞. A positive finite-time exponent does not prove that the displayed orbit is aperiodic. Periodic windows, long transients and exceptional seeds need separate consideration.
Bifurcation annotations and δ≈4.669 are approximate theoretical values. This educational map does not measure the constant, predict arbitrary nonlinear systems or certify engineering stability or safety.
🎬 Watch it in motion
Bead on a Spinning Hoop | A Bifurcation from Rotation #Shorts
Feigenbaum Cascade | Order Branching into Chaos #Shorts
Logistic Map | order splits in two, again and again, into chaos #Shorts
