Rössler Network

System Description

Network of 100 coupled Rössler oscillators studying synchronization dynamics:

\[ \begin{aligned} \dot{x}_i &= -y_i - z_i + K \sum_{j \in \mathcal{N}_i} (x_j - x_i) \\ \dot{y}_i &= x_i + ay_i \\ \dot{z}_i &= b + z_i(x_i - c) \end{aligned} \]

where \(i = 1, \ldots, 100\) and \(\mathcal{N}_i\) denotes the neighbors of node \(i\) in the network.

System Parameters

Parameter	Symbol	Value
Rössler parameter	\(a\)	0.2
Rössler parameter	\(b\)	0.2
Rössler parameter	\(c\)	7.0
Coupling strength	\(K\)	0.218 (baseline)
Network topology	—	Scale-free, 100 nodes

Sampling

Dimension: \(D = 300\) (3 states \(\times\) 100 nodes)
Sample size: \(N = 500\)
Distribution: \(\rho\) = Uniform
Region of interest: \(\mathcal{Q}(x_i, y_i, z_i) : [-15, 15] \times [-15, 15] \times [-5, 35]\) per node

Solver

Setting	Value
Method	Dopri5 (Diffrax)
Time span	\([0, 1000]\)
Steps	1000 (\(f_s\) = 1 Hz)
Relative tolerance	1e-03
Absolute tolerance	1e-06
Event function	Divergence at \(\lvert y \rvert > 400\)

Feature Extraction

Maximum pairwise deviation at final time step:

States: all \(x_i, y_i, z_i\)
Formula: \(\Delta_x = \max_i(x_i) - \min_i(x_i)\), similarly for \(y\), \(z\); plus \(\Delta_{\text{all}} = \max(\Delta_x, \Delta_y, \Delta_z)\)
Transient cutoff: \(t^* = 950.0\)

Clustering

Method: Threshold classifier (SynchronizationClassifier)
Threshold: \(\epsilon = 1.5\)
Rule: Synchronized if \(\Delta_{\text{all}} < \epsilon\), desynchronized otherwise

Attractors

The system exhibits three types of behavior:

Synchronized: All oscillators converge to a common trajectory
Desynchronized: Oscillators remain coupled but do not synchronize
Unbounded: Some trajectories diverge to infinity (detected by event function)

Basin stability is computed for non-unbounded states (synchronized + desynchronized).

Reproduction Code

Setup

def setup_rossler_network_system() -> SetupProperties:
    """Setup the Rössler network system for basin stability estimation.

    Uses coupling strength K=0.218 (expected S_B ≈ 0.496 from paper).

    :return: Configuration dictionary for BasinStabilityEstimator.
    """
    k = 0.218
    n = 500

    device = "cuda" if torch.cuda.is_available() else "cpu"
    print(f"Setting up Rössler network system on device: {device}")
    print(f"  N = {N_NODES} nodes, k = {k}")

    params: RosslerNetworkParams = {
        "a": 0.2,
        "b": 0.2,
        "c": 7.0,
        "K": k,
    }

    ode_system = RosslerNetworkJaxODE(params, n=N_NODES, edges_i=EDGES_I, edges_j=EDGES_J)

    min_limits = (
        [-15.0] * N_NODES  # x_i in [-15, 15]
        + [-15.0] * N_NODES  # y_i in [-15, 15]
        + [-5.0] * N_NODES  # z_i in [-5, 35]
    )
    max_limits = (
        [15.0] * N_NODES  # x_i
        + [15.0] * N_NODES  # y_i
        + [35.0] * N_NODES  # z_i
    )

    sampler = UniformRandomSampler(
        min_limits=min_limits,
        max_limits=max_limits,
        device=device,
    )

    solver = JaxSolver(
        t_span=(0, 1000),
        t_steps=200,
        t_eval=(980.0, 1000.0),
        device=device,
        rtol=1e-3,
        atol=1e-6,
        cache_dir=".pybasin_cache/rossler_network",
        event_fn=rossler_stop_event,
    )

    feature_extractor = SynchronizationFeatureExtractor(
        n_nodes=N_NODES,
        time_steady=950,
        device=device,
    )

    sync_classifier = SynchronizationClassifier(
        epsilon=1.5,
    )

    return {
        "n": n,
        "ode_system": ode_system,
        "sampler": sampler,
        "solver": solver,
        "feature_extractor": feature_extractor,
        "estimator": sync_classifier,
    }

Single K Value

def main() -> tuple[BasinStabilityEstimator, StudyResult]:
    """Run basin stability estimation for Rössler network.

    Uses coupling strength K=0.218 (expected S_B ≈ 0.496 from paper).

    :return: Basin stability estimator with results.
    """
    props = setup_rossler_network_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",
        feature_selector=None,
        detect_unbounded=False,
    )

    result = bse.run()

    return bse, result

K Parameter Sweep

def main() -> BasinStabilityStudy:
    """Run parameter study for Rössler network coupling strength.

    Sweeps through K values from paper to analyze basin stability variation.

    :return: Basin stability study with results.
    """
    props = setup_rossler_network_system()

    study_params = SweepStudyParams(
        **{'ode_system.params["K"]': K_VALUES_FROM_PAPER.tolist()},
    )

    solver = props.get("solver")
    feature_extractor = props.get("feature_extractor")
    estimator = props.get("estimator")
    template_integrator = props.get("template_integrator")
    assert solver is not None
    assert feature_extractor is not None
    assert estimator is not None

    bse = BasinStabilityStudy(
        n=props["n"],
        ode_system=props["ode_system"],
        sampler=props["sampler"],
        solver=solver,
        feature_extractor=feature_extractor,
        estimator=estimator,
        study_params=study_params,
        template_integrator=template_integrator,
        output_dir="results_k_study",
    )

    bse.run()

    return bse

Baseline Results (K=0.218)

Comparison with Paper Results

Attractor	pybasin BS ± SE	Paper BS ± SE	Difference	95% CI	Status
synchronized	0.48400 ± 0.02235	0.49600 ± 0.02236	-0.0120	±0.0620	✓
unbounded	0.51600 ± 0.02235	0.50400 ± 0.02236	+0.0120	±0.0620	✓

Visualizations

Basin Stability

State Space

Feature Space

K Parameter Sweep

Comparison with Paper Results

Parameter	Attractor	pybasin BS ± SE	Paper BS ± SE	Difference	95% CI	Status
0.119	synchronized	0.09200 ± 0.01293	0.22600 ± 0.01870	-0.13400	±0.04456	✗
	unbounded	0.79800 ± 0.01796	0.77400 ± 0.01870	+0.02400	±0.05082
0.139	synchronized	0.21600 ± 0.01840	0.27400 ± 0.01995	-0.05800	±0.05319	✗
	unbounded	0.75200 ± 0.01931	0.72600 ± 0.01995	+0.02600	±0.05442
0.159	synchronized	0.30400 ± 0.02057	0.33000 ± 0.02103	-0.02600	±0.05766	✓
	unbounded	0.69200 ± 0.02065	0.67000 ± 0.02103	+0.02200	±0.05776
0.179	synchronized	0.36000 ± 0.02147	0.34600 ± 0.02127	+0.01400	±0.05924	✓
	unbounded	0.63800 ± 0.02149	0.65400 ± 0.02127	-0.01600	±0.05927
0.198	synchronized	0.42000 ± 0.02207	0.47200 ± 0.02233	-0.05200	±0.06153	✓
	unbounded	0.58000 ± 0.02207	0.52800 ± 0.02233	+0.05200	±0.06153
0.218	synchronized	0.48400 ± 0.02235	0.49600 ± 0.02236	-0.01200	±0.06196	✓
	unbounded	0.51600 ± 0.02235	0.50400 ± 0.02236	+0.01200	±0.06196
0.238	synchronized	0.55000 ± 0.02225	0.59400 ± 0.02196	-0.04400	±0.06127	✓
	unbounded	0.45000 ± 0.02225	0.40600 ± 0.02196	+0.04400	±0.06127
0.258	synchronized	0.60800 ± 0.02183	0.62800 ± 0.02162	-0.02000	±0.06022	✓
	unbounded	0.39200 ± 0.02183	0.37200 ± 0.02162	+0.02000	±0.06022
0.278	synchronized	0.65600 ± 0.02124	0.65600 ± 0.02124	+0.00000	±0.05889	✓
	unbounded	0.34400 ± 0.02124	0.34400 ± 0.02124	+0.00000	±0.05889
0.297	synchronized	0.71200 ± 0.02025	0.69400 ± 0.02061	+0.01800	±0.05663	✓
	unbounded	0.28800 ± 0.02025	0.30600 ± 0.02061	-0.01800	±0.05663
0.317	synchronized	0.68000 ± 0.02086	0.69000 ± 0.02068	-0.01000	±0.05758	✓
	unbounded	0.28400 ± 0.02017	0.31000 ± 0.02068	-0.02600	±0.05662

Visualizations

Basin Stability Variation

References

Menck, P. J., Heitzig, J., Marwan, N., & Kurths, J. (2013). How basin stability complements the linear-stability paradigm. Nature Physics, 9(2), 89-92.