A test for fractal boundaries based on the basin entropy

• Published in: Communications in nonlinear science & numerical simulation 2021-04, Vol.95, p.105588, Article 105588
• Main Authors: Puy, Andreu, Daza, Alvar, Wagemakers, Alexandre, Sanjuán, Miguel A.F.
• Format: Article
• Language: English
• Subjects: Attraction; Basin entropy; Basins; Basins of attraction; Boundaries; Criteria; Duffing oscillators; Dynamical systems; Entropy; Fractals; Initial conditions; Perturbation; Uncertainty; Uncertainty exponent
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2020.105588

•The basin entropy allows to measure the final state unpredictability of dynamical systems.•A new test for fractality based on the basin entropy is presented. This test only requires one scale to detect fractal boundaries.•The fractality test can be used with dynamical systems of any number of final states and dimensions.•The periodically forced Duffing oscillator is used to illustrate the methodology. 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 ln2 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 Sbbfractality test. Using the same single scale information, the Sbb fractality test allows for the detection of fractal boundaries in many more cases than the ln2 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.