Basin Entropy in Dynamical Systems

• Basin entropy is a measure of unpredictability that averages the local Shannon entropy over phase space, quantifying the uncertainty in final attractor outcomes.
• It integrates aspects like boundary complexity, attractor multiplicity, and lacunarity to diagnose smooth, fractal, Wada, and riddled basin structures.
• Recent extensions use stochastic sampling to apply basin entropy in infinite-dimensional and delayed systems, tracking bifurcations and parameter-induced metamorphoses.

Basin entropy is a quantitative measure of final-state unpredictability in multistable dynamical systems. Introduced by Daza et al. as a tool to analyze uncertainty in basins of attraction, it assigns to each finite-resolution box in a region of phase space a Shannon or Gibbs entropy based on the probabilities of reaching the available attractors, and then averages this local entropy over the sampled boxes (Daza et al., 2016). The quantity is used to compare the organization of attraction basins in dissipative systems and escape basins in open Hamiltonian systems and area-preserving maps, to detect fractal and Wada boundaries, and to track parameter-induced transformations of phase-space structure, including basin boundary metamorphoses that are not reflected in standard bifurcation diagrams (Daza et al., 2022, Wagemakers et al., 2023). Recent work has extended the construction to delayed systems whose phase space is infinite dimensional, where initial conditions are functions on a time interval rather than points in a finite-dimensional state space (Tarigo et al., 2024, Tarigo et al., 2024).

1. Formal definition

For a system with multiple coexisting attractors, the standard construction partitions a region of phase space into small boxes, or more generally into $\varepsilon$-balls, and estimates for each box $i$ the probabilities $p_{i,j}$ that an initial condition in that box reaches attractor $j$ (Daza et al., 2022). If $m_i$ attractors are represented in box $i$, the local entropy is

$S_i=-\sum_{j=1}^{m_i} p_{i,j}\log p_{i,j},$

and the basin entropy is the average over all sampled boxes,

$$S_b=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{m_i} p_{i,j}\log p_{i,j}.$$

The maximum possible value is $\log N_A$, where $N_A$ is the total number of attractors, and $i$0 when every sampled box contains only one asymptotic outcome (Daza et al., 2022).

This definition gives basin entropy a finite-resolution interpretation. It measures the average uncertainty about the final attractor when the initial condition is only known up to the scale of the sampling box. High values indicate that several attractors are locally accessible within the same neighborhood; low values indicate that the neighborhood is effectively monochromatic in basin label and therefore predictable (Daza et al., 2016).

A related quantity is the boundary basin entropy,

$i$1

where the average is restricted to the $i$2 boxes that intersect basin boundaries (Daza et al., 2022). This isolates the uncertainty contributed by the boundary set rather than by the whole sampled domain.

2. Structural ingredients and interpretation

The 2022 review identifies three main aspects integrated by basin entropy: boundary size or lacunarity, uncertainty exponent or uncertainty dimension, and the number of attractors (Daza et al., 2022). In the classification framework of Wagemakers and collaborators, basin entropy synthesizes fractality, connectivity or lacunarity, and attractor multiplicity into a single local-statistical measure of unpredictability (Daza et al., 2022).

Under assumptions such as equal attractor probabilities per box and a single boundary, the review gives the expression

$i$3

where $i$4 is linked to lacunarity, $i$5 is the uncertainty exponent, and $i$6 is the box size (Daza et al., 2022). A more general expression given in the classification paper is

$i$7

with $i$8 indexing different boundary types, $i$9 the corresponding uncertainty exponents, and $p_{i,j}$0 the number of basins separated by boundary $p_{i,j}$1 (Daza et al., 2022). These formulas make explicit that basin entropy does not measure only one geometric property; it blends boundary abundance, boundary fractality, and the number of available outcomes.

This also explains why basin entropy is complementary to other indicators. The uncertainty exponent captures scaling of uncertain initial conditions with resolution, basin stability measures the volume of each basin, and the Wada index quantifies the extent to which boundaries are shared by multiple basins. Basin entropy incorporates information from all of these ingredients at a fixed observational scale (Daza et al., 2022, Daza et al., 2022).

3. Fractal, Wada, riddled, and intermingled boundaries

A central use of basin entropy is the diagnosis of basin-boundary complexity. Daza et al. introduced the $p_{i,j}$2 criterion: if the boundary basin entropy satisfies $p_{i,j}$3, then the boundary is fractal (Daza et al., 2016). The review emphasizes that this is a sufficient but not necessary condition, so fractal boundaries can exist even when the criterion is not triggered (Daza et al., 2022).

Puy, Daza, Wagemakers, and Sanjuán later extended this idea into the $p_{i,j}$4 fractality test, designed to improve sensitivity while retaining the single-scale character of the original approach (Puy et al., 2022). For two basins with smooth boundaries in two dimensions and disk-shaped boxes, the expected smooth-boundary value is stated as $p_{i,j}$5, and the paper proposes declaring the boundary fractal when the observed $p_{i,j}$6 differs significantly from that smooth reference after accounting for systematic and statistical errors (Puy et al., 2022). In the driven Duffing oscillator, only about $p_{i,j}$7 of the fractal boundaries detected by the new $p_{i,j}$8 test were detected by the older $p_{i,j}$9 criterion, which illustrates the increased sensitivity of the refined test (Puy et al., 2022).

The classification paper further uses basin entropy to distinguish smooth, fractal, Wada, riddled, and intermingled basins (Daza et al., 2022). In that scheme, riddled and intermingled basins correspond to the extreme case in which the boundary effectively fills the sampled region, so that boundary uncertainty and global basin uncertainty coincide. The review also states that a special relation exists between boundary basin entropy and the Wada index for Wada basins (Daza et al., 2022). This suggests that basin entropy is not only a scalar unpredictability score but also part of a broader diagnostic framework for classifying basin topology.

4. Basin entropy and bifurcations

Basin entropy has been developed into a numerical tool for exploring bifurcations by plotting $j$0 or $j$1 against a control parameter (Wagemakers et al., 2023). In this usage it acts as an “entropy diagram” that tracks transformations of phase-space structures as parameters evolve. The 2023 study reports quantitative effects of saddle-node, pitchfork, Hopf, and Neimark-Sacker bifurcations, as well as of boundary crises, homoclinic bifurcations, basin bifurcations, and smooth-fractal or fractal-fractal metamorphoses (Wagemakers et al., 2023). A key point is that some basin boundary metamorphoses can be identified with basin entropy even though they are not reflected in the bifurcation diagram.

The time-delayed study of a bistable system with linear delayed feedback shows both the promise and the limits of this perspective (Tarigo et al., 2024). There, basin entropy increases as the system approaches a Hopf bifurcation because the basins of attraction become more intricate and intermingled, and then drops when most initial conditions lead to the new oscillatory attractor. In contrast, it fails to capture the proximity of a pitchfork bifurcation because the basin structure does not significantly change near that transition (Tarigo et al., 2024).

A related pattern appears in the Mackey-Glass delay equation, where peaks in basin entropy are reported near parameter regions associated with bifurcations and attractor extinction (Tarigo et al., 2024). The six-body study likewise finds sharp peaks in $j$2 and $j$3 at parameter values where the number of libration points changes from $j$4 to $j$5 (Kumar et al., 2020). These examples support a restricted but important claim: basin entropy is sensitive to bifurcations insofar as they reorganize basin structure, but it is not a universal detector of every local change in attractor stability.

5. High-dimensional and infinite-dimensional extensions

In delayed systems, the phase space is infinite dimensional because the initial condition is a function on a finite time interval rather than a finite-dimensional state vector (Tarigo et al., 2024, Tarigo et al., 2024). This makes exhaustive gridding infeasible and motivates stochastic versions of the basin-entropy algorithm. The 2024 Mackey-Glass paper extends basin entropy by randomly sampling arbitrarily high-dimensional spaces through random boxes in the space of initial functions (Tarigo et al., 2024).

The method proceeds by selecting random centers of hypercubic boxes in the functional space, generating random initial-condition functions within a physically relevant range, and launching a specified number $j$6 of trajectories from each box (Tarigo et al., 2024). For each attractor $j$7 found from box $j$8, the empirical probability $j$9 is estimated as the fraction of sampled initial functions in that box that evolve toward attractor $m_i$0, and the basin entropy is computed as

$m_i$1

The paper reports rapid convergence of this stochastic estimate and states that about $m_i$2 boxes and about $m_i$3 trajectories per box suffice for convergence in the Mackey-Glass system (Tarigo et al., 2024).

That work also combines basin entropy with basin fraction, meaning the fraction of the sampled initial-condition space occupied by each attractor (Tarigo et al., 2024). Basin entropy then measures how mixed the basins are locally, while basin fraction measures how dominant each attractor is globally. The paper argues that two-dimensional visualizations based on special families of initial functions, such as sinusoidal slices, are useful but can miss global features that stochastic sampling captures (Tarigo et al., 2024). A plausible implication is that basin entropy becomes more informative in infinite-dimensional settings when it is paired with a statistically representative sampling strategy rather than with a low-dimensional visualization alone.

6. Representative applications

The review surveys applications across astrophysics, cold atom experiments, relativistic chaotic scattering, plasma physics, nano/micro-electromechanics, biophysics, complex networks, and ecology (Daza et al., 2022). Within open Hamiltonian and area-preserving systems, a recent tokamak study shows that the positioning of exits significantly affects the complexity and behavior of escape basins, with abrupt changes in basin entropy linked to the choice of exits (Haerter et al., 2024). In that setting basin entropy quantifies how the exit geometry intersects sticky or transport-structured regions of phase space.

In ecological network simulations of the rock-paper-scissors type, basin entropy is computed by dividing the lattice into non-overlapping square boxes and evaluating local occupation probabilities of the three species plus empty sites (Mugnaine et al., 2019). The reported behavior is regime dependent: in coexistence regimes the final basin entropy is positive, largely independent of lattice size for fixed box size, and increases with box size, whereas for sufficiently high mobility, where biodiversity is lost and one species dominates, $m_i$4 drops to zero regardless of initial conditions or box size (Mugnaine et al., 2019). The paper positions basin entropy as complementary to the Hamming distance density: the former quantifies local unpredictability and mixing within a single realization, while the latter measures sensitivity to initial conditions by comparing paired realizations (Mugnaine et al., 2019).

In celestial mechanics and chaotic scattering, basin entropy has been used to quantify how exit-basin topology changes with system parameters. In the relativistic Hénon-Heiles system, the entropy reaches a maximum near $m_i$5 and then decreases as the boundary area and fractality are reduced; the same study reports Wada basins for $m_i$6 (Bernal et al., 2018). In the restricted six-body problem with square configuration, the boundary basin entropy is reported to be greater than $m_i$7 for all tested values of the mass parameter $m_i$8, while the global basin entropy exceeds $m_i$9 specifically for $i$0 and $i$1, indicating that the basin of attraction is unpredictable throughout at those values (Kumar et al., 2020).

7. Limits, challenges, and variant usages

The review identifies several interpretive limitations (Daza et al., 2022). Basin entropy depends on sampling scale, and at finite resolution the question of which basin organization is “most unpredictable” may not have a unique answer. The measure also blends several ingredients—number of attractors, topology, lacunarity, and fractality—so identical entropy values do not imply identical basin geometries. These limitations motivate multiscale or renormalization approaches, stronger links to quantities such as the uncertainty exponent and Wada index, extensions to higher-dimensional systems, and adaptations to stochastic dynamics (Daza et al., 2022).

A common misconception is to treat basin entropy as a direct proxy for all notions of dynamical complexity. The rock-paper-scissors study explicitly distinguishes it from Hamming distance density, which probes divergence between nearby realizations rather than local mixing within one realization (Mugnaine et al., 2019). The delayed bifurcation study shows that basin entropy is not a generic early-warning signal for every bifurcation; it responds strongly to Hopf transitions that restructure the basins, but not to pitchfork bifurcations when basin organization remains simple (Tarigo et al., 2024).

The term also has distinct usages in neighboring literatures. In asynchronous random Boolean networks, basin entropy denotes the Shannon entropy of the normalized basin-size distribution,

$i$2

where $i$3 is the probability of reaching attractor $i$4 from a randomly chosen initial state (Shreim et al., 2010). In that setting the average basin entropy grows with system size only for critical networks, namely $i$5 networks under the study’s conventions (Shreim et al., 2010). In Bayesian deep learning, the EMCMC paper uses local entropy as a flatness measure around a parameter value and states that higher values correspond to flatter, broader minima, i.e. higher “basin entropy” in the energy-landscape sense (Li et al., 2023). This suggests that “basin entropy” is a stable concept within nonlinear dynamics, but not a fully uniform term across all fields that discuss basins and landscapes.