Calculator Inputs
Example Data Table
This sample setup shows one practical way to explore coupled behavior before entering your own values.
| Parameter | Example Value | Purpose |
|---|---|---|
| Map type | Logistic | Uses the classic nonlinear map for local site updates. |
| Sites | 24 | Creates a moderate ring size for visible spatial patterns. |
| Iterations | 80 | Tracks transient behavior and later-stage lattice structure. |
| Control parameter r | 3.8 | Places the logistic map in a strongly nonlinear regime. |
| Coupling ε | 0.22 | Blends self-dynamics with nearest-neighbor influence. |
| Initialization | Random from 0.20 to 0.80 | Introduces heterogeneous starting states across the lattice. |
Formula Used
For a one-dimensional lattice with nearest-neighbor coupling, the update rule is:
xᵢ(t+1) = (1 − ε) f(xᵢ(t)) + (ε / 2) [ f(xᵢ₋₁(t)) + f(xᵢ₊₁(t)) ]
Here, ε is the coupling strength, and f(x) is the selected local map.
- Logistic map:
f(x) = r x (1 − x) - Tent map:
f(x) = r · min(x, 1 − x) - Sine map:
f(x) = r · sin(πx)
Key summary statistics are computed as follows:
- Mean field:
M(t) = (1 / N) Σ xᵢ(t) - Variance:
V(t) = (1 / N) Σ [xᵢ(t) − M(t)]² - Synchronization error:
E(t) = √V(t) - Spatial roughness: average absolute difference between neighboring sites.
How to Use This Calculator
- Choose a local map type and set the number of sites.
- Enter the iteration count, control parameter
r, and couplingε. - Select a boundary condition and an initialization method.
- Provide random, uniform, sine-profile, or manual starting values.
- Pick a site index to monitor in the trajectory plot.
- Optionally discard early iterations when averaging summary metrics.
- Click Simulate Lattice to generate the results.
- Review the charts, table, and summary cards. Then export with CSV or PDF if needed.
Frequently Asked Questions
1) What does this calculator simulate?
It simulates a one-dimensional coupled map lattice, where every site follows a nonlinear map and exchanges influence with neighboring sites during each discrete time step.
2) Why does coupling strength matter?
Coupling strength controls how strongly each site responds to its neighbors. Low coupling preserves local independence, while higher coupling can promote synchronization, pattern formation, or smoother spatial behavior.
3) What is synchronization error?
Synchronization error is the square root of spatial variance. Smaller values indicate sites are behaving more similarly, while larger values imply stronger spatial differences across the lattice.
4) When should I discard early iterations?
Discard early steps when you want averages that reflect long-run behavior instead of startup transients. This is useful when initial conditions strongly affect the first part of the simulation.
5) What do periodic boundaries mean?
Periodic boundaries wrap the lattice into a ring. The first and last sites become neighbors, preventing open-edge effects and supporting circular spatial coupling.
6) Why would I clamp states to [0, 1]?
Clamping prevents values from leaving the unit interval when you test aggressive parameters. It keeps charts readable and avoids runaway values during exploratory analysis.
7) What does the heatmap show?
The heatmap displays site values over time. Rows represent iterations, columns represent sites, and colors reveal emerging spatial structures, coherence, and irregular dynamics.
8) Can I reproduce a previous run exactly?
Yes. Keep the same parameters and random seed, or use a manual initial state list. That will reproduce the same trajectory and summary results.