Basin Entropy ($S_b$) in Dynamical Systems

• Basin entropy is a quantitative measure that captures the unpredictability of final states in multistable dynamical systems.
• It computes local uncertainties by discretizing phase space and applying Gibbs entropy to distinct attractor basins, revealing fractal and complex boundaries.
• Applications span Boolean networks, celestial mechanics, and delayed systems, aiding in the detection of bifurcations and dynamic transitions.

Basin entropy ($S_b$) is a quantitative measure designed to characterize the unpredictability of multistable dynamical systems, focusing specifically on the global uncertainty in final states dictated by initial conditions. When a system exhibits multiple attractors—each with its own basin of attraction—the phase space may organize into regions of markedly different predictability, often with intricate, possibly fractal, boundaries separating the basins. Basin entropy formalizes these notions and serves as a robust indicator of the complexity and criticality in a wide array of dynamical systems, including Boolean networks, nonlinear oscillators, celestial mechanics, population dynamics, non-equilibrium statistical mechanics, and delayed systems.

1. Mathematical Formulation of Basin Entropy

The basin entropy $S_b$ is constructed by discretizing the phase space into $N$ boxes (or balls) of finite linear size $\epsilon$. For each box $i$, the probability $p_{ij}$ is determined—representing the fraction of initial conditions within box $i$ that converge to attractor $j$. The Gibbs (Shannon) entropy for each box is computed: $S_i = -\sum_{j=1}^{m_i} p_{ij} \ln p_{ij}$ where $m_i$ is the number of distinct attractors represented in box $i$. The total basin entropy is averaged over all boxes: $S_b = \frac{1}{N} \sum_{i=1}^{N} S_i$ A maximal value of $S_b = \ln N_A$ (for $N_A$ attractors) is achieved when all attractors are equally represented in each box (complete unpredictability), and $S_b = 0$ reflects perfect predictability (each box maps to a single attractor).

For quantifying the uncertainty specifically at basin boundaries, the boundary basin entropy $S_{bb}$ computes the average entropy over boxes intersecting more than one basin: $S_{bb} = \frac{1}{N_b} \sum_{i=1}^{N_b} S_i$ where $N_b$ is the number of boundary boxes.

In specific systems (e.g., relativistic chaotic scattering (Bernal et al., 2018)), $S_b$ is more generally expressed as: $$S_b = \sum_{k=1}^{k_{\text{max}}} \left(\frac{N^0_k}{N^0}\right) \cdot \delta^{\alpha_k} \cdot \ln m_k$$ with $N^0_k/N^0$ the fraction of phase space occupied by boundary $k$, $\alpha_k$ the uncertainty (fractal) dimension, $m_k$ the number of possible outcomes, and $\delta$ the discretization scale.

2. Uncertainty, Fractality, and Classification Criteria

Basin entropy quantifies not only the relative sizes ("basin stability") of attracted regions but also the geometric complexity of their boundaries—a key source of unpredictability. The uncertainty exponent $\alpha$, defined as

$f(\epsilon) \sim \epsilon^{\alpha}$

for the fraction $f(\epsilon)$ of uncertain initial conditions at scale $\epsilon$, appears explicitly in the scaling relations for $S_b$. Smooth boundaries correspond to $\alpha \simeq 1$, fractal boundaries to $0 < \alpha < 1$, and riddled boundaries to $\alpha = 0$.

A sufficient criterion for recognizing fractal boundaries is $S_{bb} > \ln 2$ (Daza et al., 2016), since boxes with more than two attractors are only possible for highly mixed (non-smooth) boundaries. This has been refined further (Puy et al., 2022); the theoretical value for $S_{bb}$ in the case of a flat (smooth) boundary is $S_{bb} \simeq 0.4395093$ for disk-box sampling in 2D, with deviations from this value (considering statistical and systematic errors) signaling fractality even in two-attractor systems.

3. Basin Entropy in Specific Systems: Boolean Networks, Scattering, and Delayed Dynamics

In asynchronous random Boolean networks (ARBNs) (Shreim et al., 2010), the occupation probability $\rho_j$ for state $j$ evolves to define long-time attractors. Normalized basin sizes for attractor $\alpha$ are $p_\alpha = \sum_{j \in \alpha} \rho_j$, and basin entropy for network $i$ is $h_i = -\sum_{\alpha=1}^{A_i} p_\alpha \ln p_\alpha$. Ensemble averages reveal that $S_b$ grows with system size $N$ only for critical connectivity ($K=2$)—where basin size and attractor length distributions follow power laws. For ordered ($K=1$) or chaotic ($K>2$) networks, $S_b$ remains essentially constant versus $N$. Analytical treatment is exact for $K=1$ via loop-counting combinatorics.

In relativistic chaotic scattering (Bernal et al., 2018), basin entropy tracks the transition from highly fractal exit basins (high $\alpha$, large $S_b$) at low relativistic parameter $\beta$ to smoother structure with diminished unpredictability at higher $\beta$. The crossover at $\beta \approx 0.625$ marks the disappearance of KAM islands and a shift from algebraic to exponential escape dynamics.

For infinite-dimensional time-delayed systems (Tarigo et al., 3 Sep 2024), standard grid discretization is intractable. Basin entropy is generalized using stochastic sampling: random boxes in initial condition function space (e.g., for the Mackey–Glass model), with local entropies averaged over many sampled boxes and trajectories. The basin fraction for attractor $j$, $f_j$, estimates the proportion of initial conditions converging to $j$—together, $S_b$ and $f_j$ elucidate the structure and dominance of attractors, especially in multistable and non-equilibrium regimes.

4. Sensitivity to Bifurcations and Dynamical Transitions

The evolution of basin entropy as system parameters vary allows detection and classification of bifurcations:

• Saddle-node bifurcations generate new basins, producing discontinuous increases in $S_b$ (Wagemakers et al., 2023).
• Pitchfork bifurcations may show a modest jump when attractor symmetry breaks, unless basin structure remains largely unchanged (as observed in certain delays (Tarigo et al., 5 Feb 2024)).
• Homoclinic and boundary crises cause sudden loss or transformation of attractors, sharply affecting $S_b$ and $S_{bb}$.
• Metamorphoses (transitions between smooth and fractal boundaries) are reflected by trends and drops in $S_{bb}$, often more clearly than classical bifurcation diagrams.

5. Applications, Computational Techniques, and Methodological Advances

Basin entropy finds broad application across domains:

• Nonlinear oscillators (Duffing): parameter space maps (“basin entropy parameter set”) guide characterization of multistability and unpredictability (Daza et al., 2016, Daza et al., 2022).
• Population and biodiversity models: structured lattice systems (e.g., cyclic rock-paper-scissors (Mugnaine et al., 2019)) show $S_b$ transitions from high (chaotic, diverse) to zero (ordered, extinct) as mobility or interaction parameters vary.
• Celestial mechanics and billiards: escape dynamics analyzed via $S_b$ in area-preserving maps and open billiards (Haerter et al., 21 Oct 2025, Haerter et al., 30 Sep 2024), with sensitivity to exit placement and emergence of KAM islands (reflected in $S_b$, mean escape time, and survival probability scaling).
• Non-equilibrium statistical mechanics: Computation of basin volumes replaces direct counting (Casiulis et al., 2022), with entropy relating to both Boltzmann and Shannon measures

$S_S = -\sum_{i=1}^{\Omega} p_i \ln p_i$

derived from estimated basin volumes $v_i$ via advanced biasing and sampling techniques (thermodynamic integration, MBAR, parallel tempering).

Recent methodological advances include single-scale fractality criteria based on $S_{bb}$ (Puy et al., 2022), which are computationally more efficient and experimentally viable versus traditional uncertainty exponent measurements that require multi-scale sampling.

6. Limitations, Open Challenges, and Future Directions

While basin entropy offers a unified and robust framework for quantifying global unpredictability and classifying basin types, certain limitations persist:

• Sensitivity depends on scale and sampling density; finite resolution can cause smooth basins to appear more unpredictable than genuinely fractal ones.
• For some bifurcations (e.g., pitchfork transitions in time-delayed systems), basin entropy may fail to signal qualitative changes if basin intermixing does not occur (Tarigo et al., 5 Feb 2024).
• High-dimensional and infinite-dimensional systems require stochastic or adaptive sampling; rigorous renormalization approaches for multi-scale integration remain an ongoing research focus (Tarigo et al., 3 Sep 2024).
• The relationship between basin entropy and other entropic measures such as the Kolmogorov–Sinai entropy is under current investigation; connections may enable bridging instantaneous (local) and asymptotic (global/final-state) unpredictability.

Future research directions involve extension of basin entropy analysis to systems with complex constraints (e.g., isobaric ensembles), deeper exploration in spin glasses and glassy liquids, and applications to energy landscapes in neural networks, combinatorial optimization, and high-dimensional control problems. Basin entropy continues to emerge as an indispensable quantitative tool in the paper of final-state unpredictability and the organization of basins in complex dynamical systems.