A test for fractal boundaries based on the basin entropy

Abstract

In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the $\ln 2$ criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the \textit{$S_{bb}$ fractality test}. Using the same single scale information, the $S_{bb}$ fractality test allows for the detection of fractal boundaries in many more cases than the $\ln 2$ criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.

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Cited by in corpus (7)

• Effortless estimation of basins of attraction
• Transient chaos enforces uncertainty in the British power grid
• Fractal Patterns in the Parameter Space of Bi-stable Duffing Oscillator
• Unpredictability and basin entropy
• Using the basin entropy to explore bifurcations
• Deep Learning-based Analysis of Basins of Attraction
• Classifying basins of attraction using the basin entropy