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Basin entropy: a new tool to analyze uncertainty in dynamical systems
Alvar Daza, Alexandre Wagemakers, Bertrand Georgeot, David Guéry-Odelin, Miguel A. F. Sanjuán
TL;DR
The paper addresses the lack of a quantitative basis for comparing uncertainty in basin predictions. It introduces basin entropy by discretizing phase space and applying Gibbs entropy to attractor outcomes, then uses it to quantify unpredictability across systems and parameters. It also gives a fixed-resolution sufficient test for fractal basin boundaries, subject to stated scope limitations.
Problem
Existing discussions of basin unpredictability lack a quantitative basis for comparing how difficult final-state prediction is.
Method
Basin entropy discretizes phase space into boxes, treats attractors as possible outcomes, and applies Gibbs entropy to quantify unpredictability.
Results
The basin entropy quantifies final-state unpredictability across parameter variations, including Hénon-Heiles energies and Newton-method root-finding problems.
Takeaways & Limitations
Basin entropy provides a tool for comparing unpredictability and assessing how parameter changes alter basin or iterative-process uncertainty.
Takeaways & Limitations
The log2 boundary-entropy criterion is sufficient but not necessary, works only for cases with three or more basins, and is most applicable to comparable basin sizes.
Abstract
from arXiv · showhide
In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than $\log 2 $, the basin is fractal.
1 Introduction
In multistable dynamical systems, initial conditions determine which attractor or escape exit trajectories reach, but basin geometry can make this final state difficult to predict. Existing measures capture aspects of basin structure or size, yet the paper identifies a need for a quantitative measure of basin uncertainty.
- Basin unpredictability: Multiple attractors make determining which initial conditions lead to which final states a fundamental question.A basin of attraction contains initial conditions leading to a specific attractor; open Hamiltonian systems admit analogous escape basins.
- Basin unpredictability: Fractal or intricately mixed basin boundaries can hinder prediction even when system evolution is deterministic.The paper relates chaotic dynamics to fractal basin structures and illustrates differing predictability for Hénon-Heiles escape basins at different energies.
- Need for measurement: There is no quantitative basis for the repeated claim that some basins are more uncertain than others.This gap is especially problematic when comparing basin figures associated with close parameter values, such as nearby energies.
- Proposed measure: The paper proposes basin entropy, obtained by discretizing phase space and applying Gibbs entropy to attractor outcomes within grid elements.The discretization reflects finite experimental precision and numerical resolution, both of which can affect final-state prediction.
- Existing approaches: The uncertainty exponent characterizes boundary topology, with α = 1 for smooth boundaries and α < 1 for fractal boundaries.Its value approaching zero indicates greater prediction difficulty, but mixed smooth and fractal boundaries make calculation cumbersome.
- Existing approaches: Basin stability compares relative basin sizes but does not account for how basins are mixed.Consequently, basins with similar sizes can differ in their geometric organization and associated unpredictability.
2 Concept and definition of basin entropy
Basin entropy quantifies uncertainty in basin assignments by applying Gibbs entropy to a discretized phase-space grid. It supports comparisons across basins and relates uncertainty to boundary fractality, while boundary entropy provides a sufficient fractality test.
- Basin entropy treats each phase-space box as a random variable whose possible outcomes are attractor labels, estimating uncertainty from trajectory colors.Probabilities are determined by the number of trajectories in each box leading to each attractor.
- The normalized entropy ranges from 0 for a sole attractor to logNA for completely randomized basins with NA equiprobable attractors.The upper value is seldom realized in practice, even for extremely chaotic systems.
- Boxes containing only one color do not contribute, so basin entropy is determined by boxes at boundaries between basins.The total entropy is normalized by the number of grid boxes to offset the growth in box count as ε decreases.
- For boundary k, Nk scales as nkε^-Dk, linking basin-entropy variation with box size to the uncertainty exponent αk = D−Dk and boundary fractality.Smooth boundaries have Dk = D−1 and αk = 1, whereas fractal boundaries have αk < 1.
- Fractal boundaries retain higher basin entropy as resolution increases: both smooth and fractal cases approach zero, but smooth cases converge faster.If αk = 0, basin entropy remains positive for every box size, as may occur in riddled basins.
- Boundary basin entropy isolates boxes containing multiple colors and exceeds log2 only as a sufficient, not necessary, indicator of fractal boundaries.The criterion can help identify fractal parameter regions without computing boundaries across multiple scales, but fractal boundaries may have Sbb < log2.
3 What does the basin entropy measure?
Basin entropy quantifies uncertainty by combining boundary size, uncertainty exponent, and number of attractors. Examples show how these ingredients affect entropy across damping, energy, attractor count, and forcing parameters.
- Basin entropy depends on boundary size, uncertainty exponent, and the number of attractors.These ingredients are examined across several dynamical-system examples.
- Boundary size: For the damped Duffing oscillator, all cases have α = 1 and NA = 2, so entropy differences arise entirely from boundary size n/˜n.The same uncertainty exponent does not imply the same basin entropy for a fixed ε.
- Uncertainty exponent: In the Hénon-Heiles system, changing energy modifies the boundary fractal dimension and uncertainty exponent while preserving the Wada property.Smaller energies produce smaller uncertainty exponents and more complex boundaries; basin-entropy scaling reflects boundary fractal dimension.
- Parameter dependence: For the periodically driven Duffing oscillator, the parameter map uses hot colors for larger entropy, while zero entropy marks a single attractor.The highest-entropy example combines eight attractors with highly mixed basins; cases with three or sixteen attractors illustrate the joint role of attractor count and boundary uncertainty.
- Number of attractors: Increasing the number of attractors increases basin entropy and raises the intercept of its log-log relation with box size.The effect cannot be fully separated from boundary-size changes because a new attractor also creates a new boundary.
4 Characterizing chaotic systems
The basin entropy parameter set quantitatively maps uncertainty across parameter choices, while Monte Carlo sampling makes this exploration faster. Boundary basin entropy supports a fixed-resolution test for fractality, but the log2 criterion is sufficient rather than necessary.
- Basin entropy parameter set: The basin entropy parameter set maps basin entropy across forcing amplitude F and frequency ω in the periodically driven Duffing oscillator.It uses a 200 × 200 grid with ε = 0.005 and 25 trajectories per box for each basin.
- Basin entropy parameter set: Higher basin entropy reflects both more attractors and greater mixing of their basins.Eight highly mixed attractors produce a higher value than three highly mixed attractors, while sixteen less intricate basins can produce a lower value.
- Monte Carlo sampling: 2000 sampled boxes per parameter point replace the million trajectories required for the usual procedure when estimating basin entropy.The random-sampling approach supports rapid parameter scans and later refinement of the most interesting basins.
- Monte Carlo sampling: 94% of evaluated parameter settings had relative basin-entropy error below 5% with random sampling.The error decreases as 1/N, so increasing the number of boxes can improve precision.
- Log2 criterion: Boundary basin entropy Sbb measures boundary uncertainty, with Sbb > log2 serving as a sufficient condition for fractal boundaries.For fixed ε, the method may confuse a smooth boundary separating more than two basins within one box with a fractal boundary.
- Log2 criterion: The log2 criterion is sufficient but not necessary for fractal boundaries and works only for cases with three or more basins.It is faster than direct fractal-dimension estimation because it requires only one resolution, making it useful when experimental resolution cannot be tuned.
5 Discussion
The paper presents basin entropy as a general measure of final-state uncertainty across dynamical systems and iterative algorithms. It also extends the framework to Wada uncertainty, fractal-boundary testing, and parameter exploration with Monte Carlo sampling.
- 5 Discussion: Basin entropy quantifies final-state unpredictability and is illustrated across systems including escape basins and iterative algorithms.The Newton method example quantifies uncertainty for different numbers of complex roots.
- 5 Discussion: Boundary basin entropy provides a fixed-resolution sufficient condition for fractal basin boundaries through the log2 criterion.Unlike box-counting dimension, the criterion does not require computation at different resolutions.
- 5 Discussion: The basin entropy parameter set supplements bifurcation diagrams and chaotic parameter sets by locating parameter regions with simple or complicated basins.Monte Carlo sampling makes it a quick guide for scanning parameter sets.
- 5 Discussion: The authors propose basin entropy as a tool for complex-systems studies, especially in settings involving multistability.The stated scope includes applications across multiple scientific fields.
Supplementary material
The paper derives the log2 criterion by bounding basin entropy for smooth boundaries, then discusses its practical accuracy and limitations. The criterion states that Sbb > log2 is sufficient, but not necessary, for fractal basin boundaries.
- Derivation of the log2 criterion: The boundary basin entropy Sbb differs from basin entropy Sb because Sb reflects basin sizes, whereas Sbb supports a sufficient test for fractal boundaries.Sbb cannot distinguish different basins with smooth boundaries, but it can establish fractality when it exceeds log2.
- Derivation of the log2 criterion: Smooth boundaries separating two basins occupy boxes whose entropy is at most log2.Boxes at intersections of more than two smooth boundaries can involve k colors, with entropy at most logk.
- Numerical accuracy: Using 25 trajectories per box keeps relative error below 5% in most cases while enabling fast basin-entropy computation.The trajectory count can be tuned according to the dynamical system and desired accuracy.
- Criterion and limitations: Sbb > log2 proves that basin boundaries are fractal, while fractal boundaries can also have Sbb ≤ log2.The log2 criterion is therefore sufficient but not necessary.
- Criterion and limitations: The criterion remains applicable at fixed experimental resolution when the number of boxes is sufficiently large.The strict inequality leaves room for deviations caused by the finite number of simulations or experiments.