Basin entropy: a new tool to analyze uncertainty in dynamical systems.

Miguel A F Sanjuán,Bertrand Georgeot,David Guéry-Odelin,Alvar Daza,Alexandre Wagemakers

doi:10.1038/srep31416

Abstract

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 log2, the basin is fractal.

Highlights

Dynamical systems describe magnitudes evolving in time according to deterministic rules
The Hénon-Heiles Hamiltonian is a well-known model for an axisymmetrical galaxy and it has been used as a paradigm in Hamiltonian nonlinear dynamics
We propose an analogy with the concept of chaotic parameter set[26], which is a plot that visually illustrates in a parameter plane when a dynamical system is chaotic or periodic by plotting the Lyapunov exponents for different pairs of parameters

Summary

Concept and definition of basin entropy

Suppose we have a dynamical system with NA attractors for a choice of parameters in a certain region Ω of the phase space. The basin entropy has three components: the term nk/n is a normalization constant that accounts for the boundary size which is independent of ε; the term of the uncertainty exponent αk, is related with the fractality of the boundaries and contains the variation of the basin entropy with the box size; there is a term that depends on the number of different colors mk All these terms depend on the dynamics of the system, while the scaling box size ε depends only on the geometry of the grid. If we take a sufficient number of boxes in the boundaries, the effect of those boxes containing more than two colors will be negligible for the computation of the basin entropy in the boundaries Sbb. the maximum possible value of Sbb that a smooth boundary can show is log[2], which would imply a pathological case where every box in the boundary contains equal proportions of two basins bpoi u=nd1a/2ry,. A detailed proof of the log[2] criterion can be found in the Supplementary Information

What does the basin entropy measure?

Characterizing chaotic systems

Discussion

Findings

Additional Information