Discrete Dynamical System Calculator

Example Data Table

This sample uses the logistic map with x(0) = 0.25, r = 3.2, and K = 1.

Formula Used

A discrete system follows the recursive rule x(n+1) = f(x(n)).

Supported formulas

• Linear affine: x(n+1) = a × x(n) + b
• Logistic: x(n+1) = r × x(n) × [1 - x(n) / K]
• Exponential: x(n+1) = g × x(n)
• Beverton-Holt: x(n+1) = R × x(n) / [1 + ((R - 1) × x(n) / K)]
• Ricker: x(n+1) = x(n) × exp(r × [1 - x(n) / K])
• Quadratic: x(n+1) = a × x(n)^2 + b × x(n) + c

Stability test

The calculator estimates the derivative near the final value. If |f'(x)| < 1, the point is usually locally stable. If |f'(x)| > 1, it is usually locally unstable.

Fixed point residual

The residual is |f(x_final) - x_final|. A smaller residual means the final value is closer to a fixed point.

How to Use This Calculator

1. Select the system model that matches your problem.
2. Enter the starting value, written as x(0).
3. Set the needed model parameters.
4. Choose the number of iterations.
5. Set transient steps if early values should be ignored.
6. Use bounds when your model must stay inside limits.
7. Press the calculate button.
8. Review the summary, table, stability, and period result.
9. Download CSV or PDF when you need a saved report.

Discrete Dynamical System Guide

Step Based Modeling

A discrete dynamical system studies change step by step. It does not use continuous time. It moves from one state to the next by a rule. That rule is often written as x next equals f of x. The calculator above turns that idea into a clear table.

Where These Systems Appear

These systems appear in population models, finance, ecology, inventory planning, games, and scheduling. A small rule can create simple growth. It can also create oscillation, cycles, or chaotic looking behavior. For this reason, the first values matter. The chosen parameters also matter.

Model Choices

The tool supports several common maps. A linear map is useful for steady trend analysis. A logistic map is useful when growth slows near a capacity. The Ricker and Beverton-Holt maps are common in population work. The quadratic map helps explore nonlinear feedback. Each model creates a sequence from the starting value.

Advanced Options

Advanced options help with deeper checks. You can set the number of steps. You can ignore early transient values. You can select decimal precision. You can choose a tolerance for fixed point and cycle tests. Bounds are also included for safe modeling when values must stay inside a practical range.

Reading the Results

The result section shows the last value, total change, residual, local derivative, stability label, and detected period. The residual compares the final state with the next predicted state. A small residual means the sequence is close to a fixed point. The derivative estimate gives a local stability clue. When its absolute value is below one, nearby values usually move toward the point.

Cycle Detection

Cycle detection compares recent repeated blocks. It is a numerical test, not a formal proof. Use a smaller tolerance for strict checks. Use a larger tolerance when rounded data or noisy estimates are involved.

Exporting and Reviewing

The iteration table is useful for reports. It shows each current value, next value, change, and relative change. You can export this table as CSV for spreadsheets. You can also create a simple PDF for sharing. Always review units and assumptions before using results for decisions. For best results, compare several parameter sets, inspect early jumps, and validate the chosen map against real observations, not only a final number alone in practice.

FAQs