Lorenz System
System Description
Lorenz system:
\[
\begin{aligned}
\dot{x} &= \sigma(y - x) \\
\dot{y} &= rx - xz - y \\
\dot{z} &= xy - bz
\end{aligned}
\]
System Parameters
| Parameter | Symbol | Value |
|---|---|---|
| Prandtl number | \(\sigma\) | 0.12 |
| Rayleigh number | \(r\) | 0.0 |
| Geometric factor | \(b\) | -0.6 |
Sampling
- Dimension: \(D = 3\)
- Sample size: \(N = 20000\)
- Distribution: \(\rho\) = Uniform
- Region of interest: \(\mathcal{Q}(x, y, z) : [-10, 10] \times [-20, 20] \times [0]\)
Solver
| Setting | Value |
|---|---|
| Method | Dopri5 (Diffrax) |
| Time span | \([0, 1000]\) |
| Steps | 4000 (\(f_s\) = 4 Hz) |
| Relative tolerance | 1e-08 |
| Absolute tolerance | 1e-06 |
| Event function | Divergence at \(\vert y \vert > 200\) |
Feature Extraction
Mean of \(x\) coordinate after transient:
- States: \(x\) (state 0)
- Formula: \(\bar{x} = \text{mean}(x_{t > t^*})\)
- Transient cutoff: \(t^* = 900.0\)
Clustering
- Method: k-NN (k=1)
- Template ICs:
- chaotic attractor 1: \([0.8, -3.0, 0.0]\) — Positive wing chaotic attractor
- chaotic attractor 2: \([-0.8, 3.0, 0.0]\) — Negative wing chaotic attractor
- unbounded: \([10.0, 50.0, 0.0]\) — Diverging trajectories
Key Feature
Demonstrates unboundedness detection with event_fn.
Reproduction Code
Setup
def setup_lorenz_system() -> SetupProperties:
n = 20_000
device = "cuda" if torch.cuda.is_available() else "cpu"
print(f"Setting up Lorenz system on device: {device}")
params: LorenzParams = {"sigma": 0.12, "r": 0.0, "b": -0.6}
ode_system = LorenzJaxODE(params)
sampler = UniformRandomSampler(
min_limits=[-10.0, -20.0, 0.0], max_limits=[10.0, 20.0, 0.0], device=device
)
solver = JaxSolver(
t_span=(0, 1000),
t_steps=4000,
t_eval=(900.0, 1000.0),
device=device,
rtol=1e-8,
atol=1e-6,
cache_dir=".pybasin_cache/lorenz",
event_fn=lorenz_stop_event,
)
feature_extractor = JaxFeatureExtractor(
time_steady=900.0,
normalize=False,
features_per_state={
0: {"mean": None},
1: None,
2: None,
},
)
classifier_initial_conditions = [
[0.8, -3.0, 0.0],
[-0.8, 3.0, 0.0],
[10.0, 50.0, 0.0],
]
classifier_labels = ["chaotic attractor 1", "chaotic attractor 2", "unbounded"]
knn = KNeighborsClassifier(n_neighbors=1)
template_integrator = TemplateIntegrator(
template_y0=classifier_initial_conditions,
labels=classifier_labels,
ode_params=params,
)
return {
"n": n,
"ode_system": ode_system,
"sampler": sampler,
"solver": solver,
"feature_extractor": feature_extractor,
"estimator": knn,
"template_integrator": template_integrator,
}
Main Estimation
def main() -> tuple[BasinStabilityEstimator, StudyResult]:
props = setup_lorenz_system()
bse = BasinStabilityEstimator(
n=props["n"],
ode_system=props["ode_system"],
sampler=props["sampler"],
solver=props.get("solver"),
feature_extractor=props.get("feature_extractor"),
predictor=props.get("estimator"),
template_integrator=props.get("template_integrator"),
output_dir="results_case1",
# feature_selector=None,
)
result = bse.run()
print("Basin Stability:", result["basin_stability"])
# bse.save()
return bse, result
Case 1: Baseline Results
Comparison with MATLAB bSTAB
Overall Classification Quality:
- Matthews Correlation Coefficient: 0.9985
| Attractor | pybasin BS ± SE | bSTAB BS ± SE |
|---|---|---|
| chaotic attractor 1 | 0.08940 ± 0.00202 | 0.08940 ± 0.00202 |
| chaotic attractor 2 | 0.08750 ± 0.00200 | 0.08745 ± 0.00200 |
| unbounded | 0.82310 ± 0.00270 | 0.82315 ± 0.00270 |
Visualizations
Basin Stability
State Space
Feature Space
Template Phase Space
Case 2: Sigma Parameter Sweep
Comparison with MATLAB bSTAB
Average MCC = 0.9999
| Parameter | Attractor | pybasin BS ± SE | bSTAB BS ± SE | MCC |
|---|---|---|---|---|
| 0.12 | chaotic attractor 1 | 0.08440 ± 0.00197 | 0.08430 ± 0.00196 | 0.9982 |
| chaotic attractor 2 | 0.08625 ± 0.00199 | 0.08590 ± 0.00198 | ||
| unbounded | 0.82935 ± 0.00266 | 0.82980 ± 0.00266 | ||
| 0.1225 | chaotic attractor 1 | 0.10195 ± 0.00214 | 0.10195 ± 0.00214 | 1.0000 |
| chaotic attractor 2 | 0.10470 ± 0.00216 | 0.10470 ± 0.00216 | ||
| unbounded | 0.79335 ± 0.00286 | 0.79335 ± 0.00286 | ||
| 0.125 | chaotic attractor 1 | 0.11220 ± 0.00223 | 0.11220 ± 0.00223 | 1.0000 |
| chaotic attractor 2 | 0.11385 ± 0.00225 | 0.11385 ± 0.00225 | ||
| unbounded | 0.77395 ± 0.00296 | 0.77395 ± 0.00296 | ||
| 0.1275 | chaotic attractor 1 | 0.11405 ± 0.00225 | 0.11405 ± 0.00225 | 1.0000 |
| chaotic attractor 2 | 0.11625 ± 0.00227 | 0.11625 ± 0.00227 | ||
| unbounded | 0.76970 ± 0.00298 | 0.76970 ± 0.00298 | ||
| 0.13 | chaotic attractor 1 | 0.11480 ± 0.00225 | 0.11480 ± 0.00225 | 1.0000 |
| chaotic attractor 2 | 0.11375 ± 0.00225 | 0.11375 ± 0.00225 | ||
| unbounded | 0.77145 ± 0.00297 | 0.77145 ± 0.00297 | ||
| 0.1325 | chaotic attractor 1 | 0.10795 ± 0.00219 | 0.10795 ± 0.00219 | 1.0000 |
| chaotic attractor 2 | 0.11265 ± 0.00224 | 0.11265 ± 0.00224 | ||
| unbounded | 0.77940 ± 0.00293 | 0.77940 ± 0.00293 | ||
| 0.135 | chaotic attractor 1 | 0.11050 ± 0.00222 | 0.11050 ± 0.00222 | 1.0000 |
| chaotic attractor 2 | 0.11060 ± 0.00222 | 0.11060 ± 0.00222 | ||
| unbounded | 0.77890 ± 0.00293 | 0.77890 ± 0.00293 | ||
| 0.1375 | chaotic attractor 1 | 0.11175 ± 0.00223 | 0.11175 ± 0.00223 | 1.0000 |
| chaotic attractor 2 | 0.11260 ± 0.00224 | 0.11260 ± 0.00224 | ||
| unbounded | 0.77565 ± 0.00295 | 0.77565 ± 0.00295 | ||
| 0.14 | chaotic attractor 1 | 0.11225 ± 0.00223 | 0.11225 ± 0.00223 | 1.0000 |
| chaotic attractor 2 | 0.11025 ± 0.00221 | 0.11025 ± 0.00221 | ||
| unbounded | 0.77750 ± 0.00294 | 0.77750 ± 0.00294 | ||
| 0.1425 | chaotic attractor 1 | 0.11195 ± 0.00223 | 0.11190 ± 0.00223 | 0.9999 |
| chaotic attractor 2 | 0.10920 ± 0.00221 | 0.10920 ± 0.00221 | ||
| unbounded | 0.77885 ± 0.00293 | 0.77890 ± 0.00293 | ||
| 0.145 | chaotic attractor 1 | 0.11430 ± 0.00225 | 0.11430 ± 0.00225 | 1.0000 |
| chaotic attractor 2 | 0.10930 ± 0.00221 | 0.10930 ± 0.00221 | ||
| unbounded | 0.77640 ± 0.00295 | 0.77640 ± 0.00295 | ||
| 0.1475 | chaotic attractor 1 | 0.11345 ± 0.00224 | 0.11345 ± 0.00224 | 1.0000 |
| chaotic attractor 2 | 0.11220 ± 0.00223 | 0.11220 ± 0.00223 | ||
| unbounded | 0.77435 ± 0.00296 | 0.77435 ± 0.00296 | ||
| 0.15 | chaotic attractor 1 | 0.11385 ± 0.00225 | 0.11385 ± 0.00225 | 1.0000 |
| chaotic attractor 2 | 0.10815 ± 0.00220 | 0.10815 ± 0.00220 | ||
| unbounded | 0.77800 ± 0.00294 | 0.77800 ± 0.00294 | ||
| 0.1525 | chaotic attractor 1 | 0.11550 ± 0.00226 | 0.11550 ± 0.00226 | 1.0000 |
| chaotic attractor 2 | 0.11165 ± 0.00223 | 0.11165 ± 0.00223 | ||
| unbounded | 0.77285 ± 0.00296 | 0.77285 ± 0.00296 | ||
| 0.155 | chaotic attractor 1 | 0.11120 ± 0.00222 | 0.11120 ± 0.00222 | 1.0000 |
| chaotic attractor 2 | 0.11635 ± 0.00227 | 0.11635 ± 0.00227 | ||
| unbounded | 0.77245 ± 0.00296 | 0.77245 ± 0.00296 | ||
| 0.1575 | chaotic attractor 1 | 0.11160 ± 0.00223 | 0.11160 ± 0.00223 | 1.0000 |
| chaotic attractor 2 | 0.10920 ± 0.00221 | 0.10920 ± 0.00221 | ||
| unbounded | 0.77920 ± 0.00293 | 0.77920 ± 0.00293 | ||
| 0.16 | chaotic attractor 1 | 0.11415 ± 0.00225 | 0.11415 ± 0.00225 | 1.0000 |
| chaotic attractor 2 | 0.11140 ± 0.00222 | 0.11140 ± 0.00222 | ||
| unbounded | 0.77445 ± 0.00296 | 0.77445 ± 0.00296 | ||
| 0.1625 | chaotic attractor 1 | 0.11310 ± 0.00224 | 0.11310 ± 0.00224 | 1.0000 |
| chaotic attractor 2 | 0.11565 ± 0.00226 | 0.11565 ± 0.00226 | ||
| unbounded | 0.77125 ± 0.00297 | 0.77125 ± 0.00297 | ||
| 0.165 | chaotic attractor 1 | 0.11135 ± 0.00222 | 0.11135 ± 0.00222 | 1.0000 |
| chaotic attractor 2 | 0.11210 ± 0.00223 | 0.11210 ± 0.00223 | ||
| unbounded | 0.77655 ± 0.00295 | 0.77655 ± 0.00295 | ||
| 0.1675 | chaotic attractor 1 | 0.11230 ± 0.00223 | 0.11230 ± 0.00223 | 0.9999 |
| chaotic attractor 2 | 0.11935 ± 0.00229 | 0.11940 ± 0.00229 | ||
| unbounded | 0.76835 ± 0.00298 | 0.76830 ± 0.00298 | ||
| 0.17 | chaotic attractor 1 | 0.11360 ± 0.00224 | 0.11360 ± 0.00224 | 1.0000 |
| chaotic attractor 2 | 0.11815 ± 0.00228 | 0.11815 ± 0.00228 | ||
| unbounded | 0.76825 ± 0.00298 | 0.76825 ± 0.00298 | ||
| 0.1725 | chaotic attractor 1 | 0.12005 ± 0.00230 | 0.12005 ± 0.00230 | 1.0000 |
| chaotic attractor 2 | 0.11710 ± 0.00227 | 0.11710 ± 0.00227 | ||
| unbounded | 0.76285 ± 0.00301 | 0.76285 ± 0.00301 | ||
| 0.175 | chaotic attractor 1 | 0.11795 ± 0.00228 | 0.11795 ± 0.00228 | 1.0000 |
| chaotic attractor 2 | 0.11180 ± 0.00223 | 0.11180 ± 0.00223 | ||
| unbounded | 0.77025 ± 0.00297 | 0.77025 ± 0.00297 | ||
| 0.1775 | chaotic attractor 1 | 0.11620 ± 0.00227 | 0.11620 ± 0.00227 | 1.0000 |
| chaotic attractor 2 | 0.11810 ± 0.00228 | 0.11810 ± 0.00228 | ||
| unbounded | 0.76570 ± 0.00300 | 0.76570 ± 0.00300 | ||
| 0.18 | chaotic attractor 1 | 0.11720 ± 0.00227 | 0.11720 ± 0.00227 | 1.0000 |
| chaotic attractor 2 | 0.12195 ± 0.00231 | 0.12195 ± 0.00231 | ||
| unbounded | 0.76085 ± 0.00302 | 0.76085 ± 0.00302 |
Visualizations
Basin Stability Variation
Bifurcation Diagram
Case 3: Solver rtol Convergence Study
This hyperparameter study demonstrates the effect of ODE solver relative tolerance on basin stability estimation. Coarse tolerances (rtol=1e-3) produce inaccurate results, while finer tolerances converge to consistent values.
Comparison with MATLAB bSTAB
Average MCC = 0.9024
| Parameter | Attractor | pybasin BS ± SE | bSTAB BS ± SE | MCC |
|---|---|---|---|---|
| 1.0e-03 | chaotic attractor 1 | 0.02355 ± 0.00107 | 0.08950 ± 0.00202 | 0.4478 |
| chaotic attractor 2 | 0.02115 ± 0.00102 | 0.08585 ± 0.00198 | ||
| unbounded | 0.95530 ± 0.00146 | 0.82465 ± 0.00269 | ||
| 1.0e-04 | chaotic attractor 1 | 0.08705 ± 0.00199 | 0.08745 ± 0.00200 | 0.9771 |
| chaotic attractor 2 | 0.08595 ± 0.00198 | 0.08615 ± 0.00198 | ||
| unbounded | 0.82700 ± 0.00267 | 0.82640 ± 0.00268 | ||
| 1.0e-05 | chaotic attractor 1 | 0.08710 ± 0.00199 | 0.08705 ± 0.00199 | 0.9952 |
| chaotic attractor 2 | 0.08550 ± 0.00198 | 0.08500 ± 0.00197 | ||
| unbounded | 0.82740 ± 0.00267 | 0.82795 ± 0.00267 | ||
| 1.0e-06 | chaotic attractor 1 | 0.08710 ± 0.00199 | 0.08715 ± 0.00199 | 0.9984 |
| chaotic attractor 2 | 0.08875 ± 0.00201 | 0.08870 ± 0.00201 | ||
| unbounded | 0.82415 ± 0.00269 | 0.82415 ± 0.00269 | ||
| 1.0e-07 | chaotic attractor 1 | 0.08600 ± 0.00198 | 0.08620 ± 0.00198 | 0.9979 |
| chaotic attractor 2 | 0.08820 ± 0.00201 | 0.08825 ± 0.00201 | ||
| unbounded | 0.82580 ± 0.00268 | 0.82555 ± 0.00268 | ||
| 1.0e-08 | chaotic attractor 1 | 0.08760 ± 0.00200 | 0.08740 ± 0.00200 | 0.9984 |
| chaotic attractor 2 | 0.08720 ± 0.00199 | 0.08710 ± 0.00199 | ||
| unbounded | 0.82520 ± 0.00269 | 0.82550 ± 0.00268 |
Visualizations
Basin Stability Variation
Bifurcation Diagram
Case 4: Sample Size Convergence Study
This hyperparameter study varies the number of initial conditions \(N\) from 200 to 20,000 (using \(2 \times \text{logspace}(2, 4, 50)\)) to assess how basin stability estimates converge as sample size increases. The relative standard error decreases as \(\text{SE}/\mathcal{S}_{\mathcal{B}} \sim 1/\sqrt{N}\).
Comparison with MATLAB bSTAB
Average MCC = 0.9981
| Parameter | Attractor | pybasin BS ± SE | bSTAB BS ± SE | MCC |
|---|---|---|---|---|
| 200 | chaotic attractor 1 | 0.10000 ± 0.02121 | 0.10000 ± 0.02121 | 1.0000 |
| chaotic attractor 2 | 0.07500 ± 0.01862 | 0.07500 ± 0.01862 | ||
| unbounded | 0.82500 ± 0.02687 | 0.82500 ± 0.02687 | ||
| 219.7082 | chaotic attractor 1 | 0.11364 ± 0.02140 | 0.10909 ± 0.02102 | 0.9844 |
| chaotic attractor 2 | 0.05455 ± 0.01531 | 0.05455 ± 0.01531 | ||
| unbounded | 0.83182 ± 0.02522 | 0.83636 ± 0.02494 | ||
| 241.3585 | chaotic attractor 1 | 0.05372 ± 0.01449 | 0.05372 ± 0.01449 | 1.0000 |
| chaotic attractor 2 | 0.07851 ± 0.01729 | 0.07851 ± 0.01729 | ||
| unbounded | 0.86777 ± 0.02178 | 0.86777 ± 0.02178 | ||
| 265.1423 | chaotic attractor 1 | 0.06391 ± 0.01500 | 0.06391 ± 0.01500 | 1.0000 |
| chaotic attractor 2 | 0.10526 ± 0.01882 | 0.10526 ± 0.01882 | ||
| unbounded | 0.83083 ± 0.02299 | 0.83083 ± 0.02299 | ||
| 291.2697 | chaotic attractor 1 | 0.07877 ± 0.01576 | 0.07877 ± 0.01576 | 1.0000 |
| chaotic attractor 2 | 0.08219 ± 0.01607 | 0.08219 ± 0.01607 | ||
| unbounded | 0.83904 ± 0.02151 | 0.83904 ± 0.02151 | ||
| 319.9717 | chaotic attractor 1 | 0.10000 ± 0.01677 | 0.10000 ± 0.01677 | 1.0000 |
| chaotic attractor 2 | 0.09688 ± 0.01654 | 0.09688 ± 0.01654 | ||
| unbounded | 0.80312 ± 0.02223 | 0.80312 ± 0.02223 | ||
| 351.5021 | chaotic attractor 1 | 0.09375 ± 0.01554 | 0.09375 ± 0.01554 | 1.0000 |
| chaotic attractor 2 | 0.09091 ± 0.01532 | 0.09091 ± 0.01532 | ||
| unbounded | 0.81534 ± 0.02068 | 0.81534 ± 0.02068 | ||
| 386.1395 | chaotic attractor 1 | 0.08786 ± 0.01439 | 0.08786 ± 0.01439 | 1.0000 |
| chaotic attractor 2 | 0.08010 ± 0.01380 | 0.08010 ± 0.01380 | ||
| unbounded | 0.83204 ± 0.01900 | 0.83204 ± 0.01900 | ||
| 424.1902 | chaotic attractor 1 | 0.09176 ± 0.01400 | 0.09176 ± 0.01400 | 1.0000 |
| chaotic attractor 2 | 0.08706 ± 0.01368 | 0.08706 ± 0.01368 | ||
| unbounded | 0.82118 ± 0.01859 | 0.82118 ± 0.01859 | ||
| 465.9904 | chaotic attractor 1 | 0.09013 ± 0.01327 | 0.09013 ± 0.01327 | 1.0000 |
| chaotic attractor 2 | 0.09657 ± 0.01368 | 0.09657 ± 0.01368 | ||
| unbounded | 0.81330 ± 0.01805 | 0.81330 ± 0.01805 | ||
| 511.9096 | chaotic attractor 1 | 0.08594 ± 0.01239 | 0.08594 ± 0.01239 | 1.0000 |
| chaotic attractor 2 | 0.07812 ± 0.01186 | 0.07812 ± 0.01186 | ||
| unbounded | 0.83594 ± 0.01637 | 0.83594 ± 0.01637 | ||
| 562.3537 | chaotic attractor 1 | 0.07638 ± 0.01119 | 0.07638 ± 0.01119 | 1.0000 |
| chaotic attractor 2 | 0.07638 ± 0.01119 | 0.07638 ± 0.01119 | ||
| unbounded | 0.84725 ± 0.01516 | 0.84725 ± 0.01516 | ||
| 617.7687 | chaotic attractor 1 | 0.09709 ± 0.01191 | 0.09709 ± 0.01191 | 1.0000 |
| chaotic attractor 2 | 0.07605 ± 0.01066 | 0.07605 ± 0.01066 | ||
| unbounded | 0.82686 ± 0.01522 | 0.82686 ± 0.01522 | ||
| 678.6444 | chaotic attractor 1 | 0.07806 ± 0.01029 | 0.07806 ± 0.01029 | 1.0000 |
| chaotic attractor 2 | 0.08100 ± 0.01047 | 0.08100 ± 0.01047 | ||
| unbounded | 0.84094 ± 0.01404 | 0.84094 ± 0.01404 | ||
| 745.5187 | chaotic attractor 1 | 0.09786 ± 0.01088 | 0.09786 ± 0.01088 | 1.0000 |
| chaotic attractor 2 | 0.10456 ± 0.01120 | 0.10456 ± 0.01120 | ||
| unbounded | 0.79759 ± 0.01471 | 0.79759 ± 0.01471 | ||
| 818.983 | chaotic attractor 1 | 0.06960 ± 0.00889 | 0.06960 ± 0.00889 | 1.0000 |
| chaotic attractor 2 | 0.08913 ± 0.00996 | 0.08913 ± 0.00996 | ||
| unbounded | 0.84127 ± 0.01277 | 0.84127 ± 0.01277 | ||
| 899.6865 | chaotic attractor 1 | 0.08778 ± 0.00943 | 0.08778 ± 0.00943 | 1.0000 |
| chaotic attractor 2 | 0.08444 ± 0.00927 | 0.08444 ± 0.00927 | ||
| unbounded | 0.82778 ± 0.01259 | 0.82778 ± 0.01259 | ||
| 988.3427 | chaotic attractor 1 | 0.08898 ± 0.00905 | 0.08898 ± 0.00905 | 1.0000 |
| chaotic attractor 2 | 0.08392 ± 0.00882 | 0.08392 ± 0.00882 | ||
| unbounded | 0.82710 ± 0.01202 | 0.82710 ± 0.01202 | ||
| 1085.7351 | chaotic attractor 1 | 0.08287 ± 0.00837 | 0.08379 ± 0.00841 | 0.9912 |
| chaotic attractor 2 | 0.09945 ± 0.00908 | 0.09761 ± 0.00901 | ||
| unbounded | 0.81768 ± 0.01172 | 0.81860 ± 0.01169 | ||
| 1192.7247 | chaotic attractor 1 | 0.09304 ± 0.00841 | 0.09220 ± 0.00838 | 0.9972 |
| chaotic attractor 2 | 0.08047 ± 0.00788 | 0.08047 ± 0.00788 | ||
| unbounded | 0.82649 ± 0.01096 | 0.82733 ± 0.01094 | ||
| 1310.2571 | chaotic attractor 1 | 0.08467 ± 0.00769 | 0.08467 ± 0.00769 | 0.9975 |
| chaotic attractor 2 | 0.09230 ± 0.00799 | 0.09306 ± 0.00802 | ||
| unbounded | 0.82304 ± 0.01054 | 0.82227 ± 0.01056 | ||
| 1439.3713 | chaotic attractor 1 | 0.07639 ± 0.00700 | 0.07639 ± 0.00700 | 1.0000 |
| chaotic attractor 2 | 0.08750 ± 0.00745 | 0.08750 ± 0.00745 | ||
| unbounded | 0.83611 ± 0.00975 | 0.83611 ± 0.00975 | ||
| 1581.2086 | chaotic attractor 1 | 0.08534 ± 0.00702 | 0.08534 ± 0.00702 | 0.9979 |
| chaotic attractor 2 | 0.09039 ± 0.00721 | 0.08976 ± 0.00719 | ||
| unbounded | 0.82427 ± 0.00957 | 0.82491 ± 0.00956 | ||
| 1737.0227 | chaotic attractor 1 | 0.08285 ± 0.00661 | 0.08285 ± 0.00661 | 1.0000 |
| chaotic attractor 2 | 0.07480 ± 0.00631 | 0.07480 ± 0.00631 | ||
| unbounded | 0.84235 ± 0.00874 | 0.84235 ± 0.00874 | ||
| 1908.191 | chaotic attractor 1 | 0.08486 ± 0.00638 | 0.08381 ± 0.00634 | 0.9963 |
| chaotic attractor 2 | 0.07648 ± 0.00608 | 0.07648 ± 0.00608 | ||
| unbounded | 0.83866 ± 0.00842 | 0.83971 ± 0.00840 | ||
| 2096.2263 | chaotic attractor 1 | 0.09299 ± 0.00634 | 0.09251 ± 0.00633 | 0.9985 |
| chaotic attractor 2 | 0.08965 ± 0.00624 | 0.08965 ± 0.00624 | ||
| unbounded | 0.81736 ± 0.00844 | 0.81784 ± 0.00843 | ||
| 2302.7908 | chaotic attractor 1 | 0.08207 ± 0.00572 | 0.08207 ± 0.00572 | 0.9986 |
| chaotic attractor 2 | 0.09249 ± 0.00604 | 0.09205 ± 0.00602 | ||
| unbounded | 0.82545 ± 0.00791 | 0.82588 ± 0.00790 | ||
| 2529.7104 | chaotic attractor 1 | 0.08696 ± 0.00560 | 0.08696 ± 0.00560 | 0.9986 |
| chaotic attractor 2 | 0.07984 ± 0.00539 | 0.08024 ± 0.00540 | ||
| unbounded | 0.83320 ± 0.00741 | 0.83281 ± 0.00742 | ||
| 2778.991 | chaotic attractor 1 | 0.08312 ± 0.00524 | 0.08312 ± 0.00524 | 0.9976 |
| chaotic attractor 2 | 0.08924 ± 0.00541 | 0.08924 ± 0.00541 | ||
| unbounded | 0.82764 ± 0.00716 | 0.82764 ± 0.00716 | ||
| 3052.8359 | chaotic attractor 1 | 0.08680 ± 0.00510 | 0.08680 ± 0.00510 | 0.9967 |
| chaotic attractor 2 | 0.08090 ± 0.00494 | 0.08123 ± 0.00494 | ||
| unbounded | 0.83230 ± 0.00676 | 0.83197 ± 0.00677 | ||
| 3353.6659 | chaotic attractor 1 | 0.08289 ± 0.00476 | 0.08318 ± 0.00477 | 0.9980 |
| chaotic attractor 2 | 0.08468 ± 0.00481 | 0.08438 ± 0.00480 | ||
| unbounded | 0.83244 ± 0.00645 | 0.83244 ± 0.00645 | ||
| 3684.1399 | chaotic attractor 1 | 0.09389 ± 0.00480 | 0.09362 ± 0.00480 | 0.9937 |
| chaotic attractor 2 | 0.08033 ± 0.00448 | 0.08033 ± 0.00448 | ||
| unbounded | 0.82578 ± 0.00625 | 0.82605 ± 0.00624 | ||
| 4047.1793 | chaotic attractor 1 | 0.08622 ± 0.00441 | 0.08547 ± 0.00439 | 0.9968 |
| chaotic attractor 2 | 0.09313 ± 0.00457 | 0.09289 ± 0.00456 | ||
| unbounded | 0.82065 ± 0.00603 | 0.82164 ± 0.00602 | ||
| 4445.993 | chaotic attractor 1 | 0.08232 ± 0.00412 | 0.08232 ± 0.00412 | 0.9977 |
| chaotic attractor 2 | 0.08884 ± 0.00427 | 0.08862 ± 0.00426 | ||
| unbounded | 0.82883 ± 0.00565 | 0.82906 ± 0.00565 | ||
| 4884.1062 | chaotic attractor 1 | 0.09110 ± 0.00412 | 0.09110 ± 0.00412 | 0.9980 |
| chaotic attractor 2 | 0.08393 ± 0.00397 | 0.08373 ± 0.00396 | ||
| unbounded | 0.82497 ± 0.00544 | 0.82518 ± 0.00543 | ||
| 5365.3916 | chaotic attractor 1 | 0.08405 ± 0.00379 | 0.08386 ± 0.00378 | 0.9982 |
| chaotic attractor 2 | 0.09057 ± 0.00392 | 0.09020 ± 0.00391 | ||
| unbounded | 0.82538 ± 0.00518 | 0.82594 ± 0.00518 | ||
| 5894.1034 | chaotic attractor 1 | 0.08753 ± 0.00368 | 0.08753 ± 0.00368 | 0.9994 |
| chaotic attractor 2 | 0.08957 ± 0.00372 | 0.08974 ± 0.00372 | ||
| unbounded | 0.82290 ± 0.00497 | 0.82273 ± 0.00497 | ||
| 6474.9151 | chaotic attractor 1 | 0.08525 ± 0.00347 | 0.08541 ± 0.00347 | 0.9979 |
| chaotic attractor 2 | 0.08541 ± 0.00347 | 0.08494 ± 0.00346 | ||
| unbounded | 0.82934 ± 0.00468 | 0.82965 ± 0.00467 | ||
| 7112.9606 | chaotic attractor 1 | 0.08421 ± 0.00329 | 0.08449 ± 0.00330 | 0.9962 |
| chaotic attractor 2 | 0.08787 ± 0.00336 | 0.08759 ± 0.00335 | ||
| unbounded | 0.82792 ± 0.00448 | 0.82792 ± 0.00448 | ||
| 7813.8799 | chaotic attractor 1 | 0.08984 ± 0.00323 | 0.08984 ± 0.00323 | 0.9980 |
| chaotic attractor 2 | 0.09073 ± 0.00325 | 0.09061 ± 0.00325 | ||
| unbounded | 0.81943 ± 0.00435 | 0.81955 ± 0.00435 | ||
| 8583.8685 | chaotic attractor 1 | 0.09052 ± 0.00310 | 0.09063 ± 0.00310 | 0.9988 |
| chaotic attractor 2 | 0.08329 ± 0.00298 | 0.08329 ± 0.00298 | ||
| unbounded | 0.82619 ± 0.00409 | 0.82607 ± 0.00409 | ||
| 9429.7327 | chaotic attractor 1 | 0.08706 ± 0.00290 | 0.08706 ± 0.00290 | 0.9982 |
| chaotic attractor 2 | 0.08515 ± 0.00287 | 0.08484 ± 0.00287 | ||
| unbounded | 0.82778 ± 0.00389 | 0.82810 ± 0.00389 | ||
| 10358.9494 | chaotic attractor 1 | 0.08485 ± 0.00274 | 0.08514 ± 0.00274 | 0.9974 |
| chaotic attractor 2 | 0.08698 ± 0.00277 | 0.08727 ± 0.00277 | ||
| unbounded | 0.82817 ± 0.00371 | 0.82759 ± 0.00371 | ||
| 11379.7321 | chaotic attractor 1 | 0.08524 ± 0.00262 | 0.08533 ± 0.00262 | 0.9967 |
| chaotic attractor 2 | 0.08234 ± 0.00258 | 0.08269 ± 0.00258 | ||
| unbounded | 0.83243 ± 0.00350 | 0.83199 ± 0.00350 | ||
| 12501.1039 | chaotic attractor 1 | 0.08935 ± 0.00255 | 0.08943 ± 0.00255 | 0.9966 |
| chaotic attractor 2 | 0.08911 ± 0.00255 | 0.08927 ± 0.00255 | ||
| unbounded | 0.82155 ± 0.00342 | 0.82131 ± 0.00343 | ||
| 13732.9769 | chaotic attractor 1 | 0.08906 ± 0.00243 | 0.08913 ± 0.00243 | 0.9988 |
| chaotic attractor 2 | 0.08687 ± 0.00240 | 0.08687 ± 0.00240 | ||
| unbounded | 0.82407 ± 0.00325 | 0.82400 ± 0.00325 | ||
| 15086.2401 | chaotic attractor 1 | 0.08517 ± 0.00227 | 0.08511 ± 0.00227 | 0.9975 |
| chaotic attractor 2 | 0.08484 ± 0.00227 | 0.08484 ± 0.00227 | ||
| unbounded | 0.82999 ± 0.00306 | 0.83005 ± 0.00306 | ||
| 16572.8555 | chaotic attractor 1 | 0.08755 ± 0.00220 | 0.08749 ± 0.00219 | 0.9976 |
| chaotic attractor 2 | 0.08628 ± 0.00218 | 0.08635 ± 0.00218 | ||
| unbounded | 0.82616 ± 0.00294 | 0.82616 ± 0.00294 | ||
| 18205.9636 | chaotic attractor 1 | 0.08569 ± 0.00207 | 0.08552 ± 0.00207 | 0.9978 |
| chaotic attractor 2 | 0.08893 ± 0.00211 | 0.08876 ± 0.00211 | ||
| unbounded | 0.82539 ± 0.00281 | 0.82572 ± 0.00281 | ||
| 20000 | chaotic attractor 1 | 0.08895 ± 0.00201 | 0.08895 ± 0.00201 | 0.9974 |
| chaotic attractor 2 | 0.08670 ± 0.00199 | 0.08660 ± 0.00199 | ||
| unbounded | 0.82435 ± 0.00269 | 0.82445 ± 0.00269 |
Visualizations
Basin Stability Variation
Bifurcation Diagram
