Unpredictability and basin entropy

Daza, Álvar; Wagemakers, Alexandre; Sanjuán, Miguel A. F.

doi:10.1209/0295-5075/acb5be

EPL

2023

DOI: 10.1209/0295-5075/acb5be

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Unpredictability and basin entropy

Álvar Daza¹,

Alexandre Wagemakers²,

Miguel A. F. Sanjuán³

Abstract: The basin entropy is a simple idea that aims to measure the the final state unpredictability of multistable systems. Since 2016, the basin entropy has been widely used in different contexts of physics, from cold atoms to galactic dynamics. Furthermore, it has provided a natural framework to study basins of attraction in nonlinear dynamics and new criteria for the detection of fractal boundaries. In this article, we describe the concept as well as fundamental applications. In addition, we provide our perspectiv… Show more

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Cited by 3 publications

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Using the basin entropy to explore bifurcations

Wagemakers,

Daza,

Sanjuán

2023

Chaos, Solitons & Fractals

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Using the basin entropy to explore bifurcations

Wagemakers,

Daza,

Sanjuán

2023

Chaos, Solitons & Fractals

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Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows

Viana,

Mathias,

Souza

et al. 2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The advection of passive scalars in time-independent two-dimensional incompressible fluid flows is an integrable Hamiltonian system. It becomes non-integrable if the corresponding stream function depends explicitly on time, allowing the possibility of chaotic advection of particles. We consider for a specific model (double gyre flow), a given number of exits through which advected particles can leak, without disturbing the flow itself. We investigate fractal escape basins in this problem and characterize fractality by computing the uncertainty exponent and basin entropy. Furthermore, we observe the presence of basin boundaries with points exhibiting the Wada property, i.e., boundary points that separate three or more escape basins.

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Basin entropy as an indicator of a bifurcation in a time-delayed system

Tarigo,

Stari,

Masoller

et al. 2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The basin entropy is a measure that quantifies, in a system that has two or more attractors, the predictability of a final state, as a function of the initial conditions. While the basin entropy has been demonstrated on a variety of multistable dynamical systems, to the best of our knowledge, it has not yet been tested in systems with a time delay, whose phase space is infinite dimensional because the initial conditions are functions defined in a time interval [−τ,0], where τ is the delay time. Here, we consider a simple time-delayed system consisting of a bistable system with a linear delayed feedback term. We show that the basin entropy captures relevant properties of the basins of attraction of the two coexisting attractors. Moreover, we show that the basin entropy can give an indication of the proximity of a Hopf bifurcation, but fails to capture the proximity of a pitchfork bifurcation. The Hopf bifurcation is detected because before the fixed points become unstable, a oscillatory, limit-cycle behavior appears that coexists with the fixed points. The new limit cycle modifies the structure of the basins of attraction, and this change is captured by basin entropy that reaches a maximum before the Hopf bifurcation. In contrast, the pitchfork bifurcation is not detected because the basins of attraction do not change as the bifurcation is approached. Our results suggest that the basin entropy can yield useful insights into the long-term predictability of time-delayed systems, which often have coexisting attractors.

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Unpredictability and basin entropy

Cited by 3 publications

References 41 publications

Using the basin entropy to explore bifurcations

Using the basin entropy to explore bifurcations

Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows

Basin entropy as an indicator of a bifurcation in a time-delayed system

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