Chaos: initial perturbations and integration error

Question

How do initial-state perturbations and step refinement affect computed Lorenz trajectories?

Mathematical background

The Lorenz system here has parameters (10,28,8/3) and reference initial state (1,1,1). A second trajectory starts at (1+ε,1,1).

Initial-condition sensitivity and integration error are investigated by different controlled comparisons. A finite trajectory comparison is not a proof of every property of chaos.

\dot x=\sigma(y-x),\qquad \dot y=x(\rho-z)-y,\qquad \dot z=xy-\beta z

Sources: [M7]

(x,y,z)=\left(\pm\sqrt{\beta(\rho-1)},\ \pm\sqrt{\beta(\rho-1)},\ \rho-1\right)

Sources: [M7]

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

When ε = 0, identical solver inputs are expected to give identical trajectories.

On short windows, reducing the RK4 step is expected to reduce discretization differences. Long-time trajectory separation is not required to shrink monotonically.

Parameters and assumptions

Perturbations are 0,10⁻⁸,10⁻⁵,0.01; steps are 0.01,0.005,0.0025.
Each pair is run to T = 40 for sensitivity and T = 1 for short-window comparison. Half-step comparison is enabled for the short runs.
The sampled distance threshold is 1 in model units.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Use the shared fixed-step RK4 kernel and compare trajectories only at common computed times.

Retain complete trajectories, distance curves, first sampled threshold crossings or not-observed states, and the short-window step comparison. Playback frequency is not an integration input.

Methods reported by the executed kernel

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Software identifier: 0.1.0+0e6017e6a747.dirty.02665ba9098c
Git revision: 0e6017e6a747d3c610b956612b4125e232283001
Uncommitted worktree state: Present when the batch was generated
Source SHA-256: 02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b
Runtime: v26.5.1

Results from an actual run

This saved batch contains 24 completed calculations. The table and downloads below refer to those actual runs.

The largest recorded absolute diagnostic residual was 0. 24 of 24 residual-bearing cases were within their own reported numerical tolerance.

Across 24 recorded readings, Initial separation ranged from 0 to 0.01.

Across 24 recorded readings, Final separation ranged from 0 to 20.25781.

Across 24 recorded readings, Maximum sampled separation ranged from 0 to 53.87257.

Across 12 recorded readings, Short-window maximum step difference ranged from 2.88163e-6 to 0.0007490856.

6 of 6 saved zero-perturbation cases had exactly zero maximum sampled separation. Identical binary64 inputs follow identical solver operations in these runs.

At h = 0.005 and T = 40, ε = 0: final separation 0; first sampled d(t) > 1 at Not observed. These are observations on the saved time grid.

At h = 0.005 and T = 40, ε = 1e-8: final separation 9.035401; first sampled d(t) > 1 at 27.725. These are observations on the saved time grid.

At h = 0.005 and T = 40, ε = 0.00001: final separation 5.825322; first sampled d(t) > 1 at 21.56. These are observations on the saved time grid.

At h = 0.005 and T = 40, ε = 0.01: final separation 12.19672; first sampled d(t) > 1 at 14.02. These are observations on the saved time grid.

Short-window comparison with reference initial state (1,1,1), h = 0.01: maximum difference from its own h/2 run 0.0007490856; endpoint difference 9.27629e-5.

Short-window comparison with reference initial state (1,1,1), h = 0.005: maximum difference from its own h/2 run 4.64173e-5; endpoint difference 2.42895e-6.

Short-window comparison with reference initial state (1,1,1), h = 0.0025: maximum difference from its own h/2 run 2.88163e-6; endpoint difference 1.58815e-7.

Download summary CSV Download complete summary JSON

Recorded short-window step comparisons

Vertical: Maximum ‖u_h − u_h/2‖₂ (logarithmic scale)

Same initial state, through T = 1

Lines connect the recorded cases to aid comparison. Intermediate settings are not additional experiments.

Actual case outputs. JSON retains every observable and the complete computed data.
Case	Selected measurements	Residual and tolerance	Data
1. ε=0 · h=0.01 · T=40	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed	0 Tolerance: 5.68434e-13	JSON · CSV
2. ε=0 · h=0.01 · T=1	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed Short-window maximum step difference: 0.0007490856 Short-window endpoint difference: 9.27629e-5	0 Tolerance: 1.42109e-14	JSON · CSV
3. ε=0 · h=0.005 · T=40	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed	0 Tolerance: 5.68434e-13	JSON · CSV
4. ε=0 · h=0.005 · T=1	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed Short-window maximum step difference: 4.64173e-5 Short-window endpoint difference: 2.42895e-6	0 Tolerance: 1.42109e-14	JSON · CSV
5. ε=0 · h=0.0025 · T=40	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed	0 Tolerance: 5.68434e-13	JSON · CSV
6. ε=0 · h=0.0025 · T=1	Initial separation: 0 Final separation: 0 Maximum sampled separation: 0 First sampled threshold crossing: Not observed Short-window maximum step difference: 2.88163e-6 Short-window endpoint difference: 1.58815e-7	0 Tolerance: 1.42109e-14	JSON · CSV
7. ε=1e-8 · h=0.01 · T=40	Initial separation: 1.00000e-8 Final separation: 20.25781 Maximum sampled separation: 42.7736 First sampled threshold crossing: 28.48 model time	0 Tolerance: 5.68434e-13	JSON · CSV
8. ε=1e-8 · h=0.01 · T=1	Initial separation: 1.00000e-8 Final separation: 1.11603e-8 Maximum sampled separation: 2.01074e-7 First sampled threshold crossing: Not observed Short-window maximum step difference: 0.0007490856 Short-window endpoint difference: 9.27629e-5	0 Tolerance: 1.42109e-14	JSON · CSV
9. ε=1e-8 · h=0.005 · T=40	Initial separation: 1.00000e-8 Final separation: 9.035401 Maximum sampled separation: 45.85778 First sampled threshold crossing: 27.725 model time	0 Tolerance: 5.68434e-13	JSON · CSV
10. ε=1e-8 · h=0.005 · T=1	Initial separation: 1.00000e-8 Final separation: 1.11600e-8 Maximum sampled separation: 2.01078e-7 First sampled threshold crossing: Not observed Short-window maximum step difference: 4.64173e-5 Short-window endpoint difference: 2.42895e-6	0 Tolerance: 1.42109e-14	JSON · CSV
11. ε=1e-8 · h=0.0025 · T=40	Initial separation: 1.00000e-8 Final separation: 9.726384 Maximum sampled separation: 48.93659 First sampled threshold crossing: 30.44 model time	0 Tolerance: 5.68434e-13	JSON · CSV
12. ε=1e-8 · h=0.0025 · T=1	Initial separation: 1.00000e-8 Final separation: 1.11599e-8 Maximum sampled separation: 2.01078e-7 First sampled threshold crossing: Not observed Short-window maximum step difference: 2.88163e-6 Short-window endpoint difference: 1.58815e-7	0 Tolerance: 1.42109e-14	JSON · CSV
13. ε=0.00001 · h=0.01 · T=40	Initial separation: 1.00000e-5 Final separation: 12.94598 Maximum sampled separation: 48.40676 First sampled threshold crossing: 22.94 model time	0 Tolerance: 5.68434e-13	JSON · CSV
14. ε=0.00001 · h=0.01 · T=1	Initial separation: 1.00000e-5 Final separation: 1.11603e-5 Maximum sampled separation: 0.0002010732 First sampled threshold crossing: Not observed Short-window maximum step difference: 0.0007490856 Short-window endpoint difference: 9.27629e-5	0 Tolerance: 1.42109e-14	JSON · CSV
15. ε=0.00001 · h=0.005 · T=40	Initial separation: 1.00000e-5 Final separation: 5.825322 Maximum sampled separation: 48.58068 First sampled threshold crossing: 21.56 model time	0 Tolerance: 5.68434e-13	JSON · CSV
16. ε=0.00001 · h=0.005 · T=1	Initial separation: 1.00000e-5 Final separation: 1.11599e-5 Maximum sampled separation: 0.000201077 First sampled threshold crossing: Not observed Short-window maximum step difference: 4.64173e-5 Short-window endpoint difference: 2.42895e-6	0 Tolerance: 1.42109e-14	JSON · CSV
17. ε=0.00001 · h=0.0025 · T=40	Initial separation: 1.00000e-5 Final separation: 5.395123 Maximum sampled separation: 49.62968 First sampled threshold crossing: 21.5575 model time	0 Tolerance: 5.68434e-13	JSON · CSV
18. ε=0.00001 · h=0.0025 · T=1	Initial separation: 1.00000e-5 Final separation: 1.11599e-5 Maximum sampled separation: 0.0002010773 First sampled threshold crossing: Not observed Short-window maximum step difference: 2.88163e-6 Short-window endpoint difference: 1.58815e-7	0 Tolerance: 1.42109e-14	JSON · CSV
19. ε=0.01 · h=0.01 · T=40	Initial separation: 0.01 Final separation: 15.65999 Maximum sampled separation: 53.87257 First sampled threshold crossing: 14.02 model time	0 Tolerance: 5.68434e-13	JSON · CSV
20. ε=0.01 · h=0.01 · T=1	Initial separation: 0.01 Final separation: 0.01113269 Maximum sampled separation: 0.2005162 First sampled threshold crossing: Not observed Short-window maximum step difference: 0.0007490856 Short-window endpoint difference: 9.27629e-5	0 Tolerance: 1.42109e-14	JSON · CSV
21. ε=0.01 · h=0.005 · T=40	Initial separation: 0.01 Final separation: 12.19672 Maximum sampled separation: 51.95915 First sampled threshold crossing: 14.02 model time	0 Tolerance: 5.68434e-13	JSON · CSV
22. ε=0.01 · h=0.005 · T=1	Initial separation: 0.01 Final separation: 0.01113231 Maximum sampled separation: 0.20052 First sampled threshold crossing: Not observed Short-window maximum step difference: 4.64173e-5 Short-window endpoint difference: 2.42895e-6	0 Tolerance: 1.42109e-14	JSON · CSV
23. ε=0.01 · h=0.0025 · T=40	Initial separation: 0.01 Final separation: 3.774176 Maximum sampled separation: 51.97781 First sampled threshold crossing: 14.0175 model time	0 Tolerance: 5.68434e-13	JSON · CSV
24. ε=0.01 · h=0.0025 · T=1	Initial separation: 0.01 Final separation: 0.01113231 Maximum sampled separation: 0.2005202 First sampled threshold crossing: Not observed Short-window maximum step difference: 2.88163e-6 Short-window endpoint difference: 1.58815e-7	0 Tolerance: 1.42109e-14	JSON · CSV

Interpretation

The reported diagnostic residual checks only the requested and integrated time-grid endpoints. Its small value is not an integration-error certificate. Use the short-window step differences to investigate discretization; use the fixed-step perturbation cases to investigate sensitivity.

Compare perturbation cases at fixed h, then compare step sizes at fixed initial conditions. A reported crossing is conditional on this threshold and observation window.

A missing crossing remains not observed. It must not be replaced by the last simulated time or interpreted as a universal prediction limit.

Limitations and possible errors

Floating-point arithmetic and a finite observation window limit the result. This study does not estimate a maximum Lyapunov exponent.
The model does not predict actual weather, and its state-space coordinates are not particle positions.

Warnings recorded during these runs

Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.
The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.
4001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.
The separation threshold was not exceeded in this observation window.
101 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.
Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately.
8001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.
201 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.
16001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.
401 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.

Reproduce this study

Restore the saved Lorenz parameters, perturbation, step, duration and comparison setting in Chaos.
Import the full JSON or inspect the CSV at common sample times. Use the recorded method and build metadata when reproducing numerical values.

Open chaos

1. ε=0 · h=0.01 · T=40

Recorded 17 September 2026 at 00:39 UTC. 4001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.01,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 40 model timeRequested T=40; 4000 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 4001

Download complete case JSON · Download case CSV

2. ε=0 · h=0.01 · T=1

Recorded 17 September 2026 at 00:39 UTC. 201 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.01,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 1 model timeRequested T=1; 100 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 201
Short-window maximum step difference: 0.00074908558Same initial state; h versus h/2 at 101 common points through t=1.
Short-window endpoint difference: 9.276e-5

Download complete case JSON · Download case CSV

3. ε=0 · h=0.005 · T=40

Recorded 17 September 2026 at 00:39 UTC. 8001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.005,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 40 model timeRequested T=40; 8000 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 8001

Download complete case JSON · Download case CSV

4. ε=0 · h=0.005 · T=1

Recorded 17 September 2026 at 00:39 UTC. 401 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.005,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 1 model timeRequested T=1; 200 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 401
Short-window maximum step difference: 4.642e-5Same initial state; h versus h/2 at 201 common points through t=1.
Short-window endpoint difference: 2.429e-6

Download complete case JSON · Download case CSV

5. ε=0 · h=0.0025 · T=40

Recorded 17 September 2026 at 00:39 UTC. 16001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.0025,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 40 model timeRequested T=40; 16000 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 16001

Download complete case JSON · Download case CSV

6. ε=0 · h=0.0025 · T=1

Recorded 17 September 2026 at 00:39 UTC. 801 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0,
  "step": 0.0025,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 1 model timeRequested T=1; 400 complete fixed steps.
Initial separation: 0
Final separation: 0
Maximum sampled separation: 0
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 801
Short-window maximum step difference: 2.882e-6Same initial state; h versus h/2 at 401 common points through t=1.
Short-window endpoint difference: 1.588e-7

Download complete case JSON · Download case CSV

7. ε=1e-8 · h=0.01 · T=40

Recorded 17 September 2026 at 00:39 UTC. 4001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.01,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 40 model timeRequested T=40; 4000 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 20.257811
Maximum sampled separation: 42.773601
First sampled threshold crossing: 28.48 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 4001

Download complete case JSON · Download case CSV

8. ε=1e-8 · h=0.01 · T=1

Recorded 17 September 2026 at 00:39 UTC. 201 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.01,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 1 model timeRequested T=1; 100 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 1.116e-8
Maximum sampled separation: 2.011e-7
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 201
Short-window maximum step difference: 0.00074908558Same initial state; h versus h/2 at 101 common points through t=1.
Short-window endpoint difference: 9.276e-5

Download complete case JSON · Download case CSV

9. ε=1e-8 · h=0.005 · T=40

Recorded 17 September 2026 at 00:39 UTC. 8001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.005,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 40 model timeRequested T=40; 8000 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 9.0354014
Maximum sampled separation: 45.857778
First sampled threshold crossing: 27.725 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 8001

Download complete case JSON · Download case CSV

10. ε=1e-8 · h=0.005 · T=1

Recorded 17 September 2026 at 00:39 UTC. 401 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.005,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 1 model timeRequested T=1; 200 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 1.116e-8
Maximum sampled separation: 2.011e-7
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 401
Short-window maximum step difference: 4.642e-5Same initial state; h versus h/2 at 201 common points through t=1.
Short-window endpoint difference: 2.429e-6

Download complete case JSON · Download case CSV

11. ε=1e-8 · h=0.0025 · T=40

Recorded 17 September 2026 at 00:39 UTC. 16001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.0025,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 40 model timeRequested T=40; 16000 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 9.726384
Maximum sampled separation: 48.936586
First sampled threshold crossing: 30.44 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 16001

Download complete case JSON · Download case CSV

12. ε=1e-8 · h=0.0025 · T=1

Recorded 17 September 2026 at 00:39 UTC. 801 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 1e-8,
  "step": 0.0025,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 1 model timeRequested T=1; 400 complete fixed steps.
Initial separation: 1.000e-8
Final separation: 1.116e-8
Maximum sampled separation: 2.011e-7
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 801
Short-window maximum step difference: 2.882e-6Same initial state; h versus h/2 at 401 common points through t=1.
Short-window endpoint difference: 1.588e-7

Download complete case JSON · Download case CSV

13. ε=0.00001 · h=0.01 · T=40

Recorded 17 September 2026 at 00:39 UTC. 4001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.01,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 40 model timeRequested T=40; 4000 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 12.945976
Maximum sampled separation: 48.406761
First sampled threshold crossing: 22.94 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 4001

Download complete case JSON · Download case CSV

14. ε=0.00001 · h=0.01 · T=1

Recorded 17 September 2026 at 00:39 UTC. 201 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.01,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 1 model timeRequested T=1; 100 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 1.116e-5
Maximum sampled separation: 0.00020107318
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 201
Short-window maximum step difference: 0.00074908558Same initial state; h versus h/2 at 101 common points through t=1.
Short-window endpoint difference: 9.276e-5

Download complete case JSON · Download case CSV

15. ε=0.00001 · h=0.005 · T=40

Recorded 17 September 2026 at 00:39 UTC. 8001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.005,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 40 model timeRequested T=40; 8000 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 5.825322
Maximum sampled separation: 48.580679
First sampled threshold crossing: 21.56 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 8001

Download complete case JSON · Download case CSV

16. ε=0.00001 · h=0.005 · T=1

Recorded 17 September 2026 at 00:39 UTC. 401 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.005,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 1 model timeRequested T=1; 200 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 1.116e-5
Maximum sampled separation: 0.00020107705
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 401
Short-window maximum step difference: 4.642e-5Same initial state; h versus h/2 at 201 common points through t=1.
Short-window endpoint difference: 2.429e-6

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17. ε=0.00001 · h=0.0025 · T=40

Recorded 17 September 2026 at 00:39 UTC. 16001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.0025,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 40 model timeRequested T=40; 16000 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 5.3951233
Maximum sampled separation: 49.629678
First sampled threshold crossing: 21.5575 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 16001

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18. ε=0.00001 · h=0.0025 · T=1

Recorded 17 September 2026 at 00:39 UTC. 801 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.00001,
  "step": 0.0025,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 1 model timeRequested T=1; 400 complete fixed steps.
Initial separation: 1.000e-5
Final separation: 1.116e-5
Maximum sampled separation: 0.00020107727
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 801
Short-window maximum step difference: 2.882e-6Same initial state; h versus h/2 at 401 common points through t=1.
Short-window endpoint difference: 1.588e-7

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19. ε=0.01 · h=0.01 · T=40

Recorded 17 September 2026 at 00:39 UTC. 4001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.01,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 40 model timeRequested T=40; 4000 complete fixed steps.
Initial separation: 0.01
Final separation: 15.659985
Maximum sampled separation: 53.872567
First sampled threshold crossing: 14.02 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 4001

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20. ε=0.01 · h=0.01 · T=1

Recorded 17 September 2026 at 00:39 UTC. 201 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.01,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.01 model time
Integrated duration: 1 model timeRequested T=1; 100 complete fixed steps.
Initial separation: 0.01
Final separation: 0.01113269
Maximum sampled separation: 0.20051616
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 201
Short-window maximum step difference: 0.00074908558Same initial state; h versus h/2 at 101 common points through t=1.
Short-window endpoint difference: 9.276e-5

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21. ε=0.01 · h=0.005 · T=40

Recorded 17 September 2026 at 00:39 UTC. 8001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.005,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 40 model timeRequested T=40; 8000 complete fixed steps.
Initial separation: 0.01
Final separation: 12.19672
Maximum sampled separation: 51.959145
First sampled threshold crossing: 14.02 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 8001

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22. ε=0.01 · h=0.005 · T=1

Recorded 17 September 2026 at 00:39 UTC. 401 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.005,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.005 model time
Integrated duration: 1 model timeRequested T=1; 200 complete fixed steps.
Initial separation: 0.01
Final separation: 0.011132313
Maximum sampled separation: 0.20052002
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 401
Short-window maximum step difference: 4.642e-5Same initial state; h versus h/2 at 201 common points through t=1.
Short-window endpoint difference: 2.429e-6

Download complete case JSON · Download case CSV

23. ε=0.01 · h=0.0025 · T=40

Recorded 17 September 2026 at 00:39 UTC. 16001 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.0025,
  "duration": 40,
  "threshold": 1,
  "compareStep": false
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 40 model timeRequested T=40; 16000 complete fixed steps.
Initial separation: 0.01
Final separation: 3.7741762
Maximum sampled separation: 51.977813
First sampled threshold crossing: 14.0175 model timeFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 16001

Download complete case JSON · Download case CSV

24. ε=0.01 · h=0.0025 · T=1

Recorded 17 September 2026 at 00:39 UTC. 801 exported data rows.

{
  "sigma": 10,
  "rho": 28,
  "beta": 2.6666666666666665,
  "epsilon": 0.01,
  "step": 0.0025,
  "duration": 1,
  "threshold": 1,
  "compareStep": true
}

Model version: chaos-rk4-1.0.0

Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.

Base integration step h: 0.0025 model time
Integrated duration: 1 model timeRequested T=1; 400 complete fixed steps.
Initial separation: 0.01
Final separation: 0.01113231
Maximum sampled separation: 0.20052025
First sampled threshold crossing: Not observedFirst d(t) > 1; conditional on this run and sample grid.
Full exported sample rows: 801
Short-window maximum step difference: 2.882e-6Same initial state; h versus h/2 at 401 common points through t=1.
Short-window endpoint difference: 1.588e-7

Download complete case JSON · Download case CSV

Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

[M7] Deterministic Nonperiodic Flow — Edward N. Lorenz
Journal of the Atmospheric Sciences 20 (1963), 130–141; equations and equilibrium analysis on p. 135, default parameter choices and original integration method on p. 136. The project’s fixed-step RK4 solver is its own documented choice and differs from the original double-approximation procedure. Source checked 16 September 2026.
[M12] Runge–Kutta methods — Richard Fitzpatrick
University of Texas at Austin computational physics course, 29 March 2006. Equations (25)–(29) give the four classical RK4 stages and local remainder. The project applies the vector form to its Lorenz system; the model paper is a separate source. Source checked 16 September 2026.