Quick guide
How to use this calculator
- Select the stated protocol, strategy, approximation, or conversion mode when one is available.
- Enter the labelled finite bound, probabilities, payoffs, sequence state, or special-function argument.
- Read the theoretical or deterministic result separately from any seeded empirical result.
Calculation method
Apply the stated mathematical model
xₖ₊₁=r xₖ(1−xₖ), with 0≤r≤4 and 0≤x₀≤1.
The calculator validates model hypotheses, finite work limits, probability domains, singular cases, and numerical approximation ranges before returning a result.
Worked example
Worked example
At r=4,x₀=0.2, the orbit exhibits sensitive behavior.
xₖ₊₁=r xₖ(1−xₖ), with 0≤r≤4 and 0≤x₀≤1.
Supported inputs
Precision and limits
Model scope
Paradoxes depend on their information or randomization protocol. Queueing, games, fractals, and cellular automata use exactly the visible assumptions; changing those assumptions changes the answer.
Deterministic versus simulated
Exact combinatorial and theoretical probabilities are labelled separately from seeded pseudorandom simulations. Reusing a seed reproduces the same finite experiment.
Numerical methods
Finite double-precision arithmetic is used for probability models and special functions. Gamma uses a bounded Lanczos approximation, Bessel J uses a bounded convergent series, and erf/erfc disclose their approximation error.
Limits
Iteration, grid, candidate, trial, and series caps are visible in field labels or calculator-specific notes. A bounded trace never claims to prove an unresolved conjecture.