Testing for Basins of Wada

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

doi:10.1038/srep16579

Cited by 48 publications

(33 citation statements)

References 19 publications

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“…, N a ; then we will say that the basin is Wada. Otherwise, we will say that the system is not Wada, and the method will determine In the case of partially Wada basins [16], where Wada and non-Wada boundaries coexist, we can characterize them by the Wada parameter W Na defined in the grid method of Daza et al [21]. This parameter W Na provides the ratio of Wada points to boundary points (Wada and non-Wada), in such a way that W Na = 1 means that the system has the full Wada property, whereas W Na < 1 indicates only partially Wada basins.…”

Section: A Description Of the Merging Methodsmentioning

confidence: 99%

“…This contrasts with previous methods to test Wada basins. The grid method [21] needs to compute new trajectories at finer resolutions, which can take several hours or even days of parallel computation in a cluster with one hundred cores. The Nusse-Yorke method [18,19] requires detailed knowledge of the dynamics of the system and, in general, it cannot be automated.…”

Section: A Description Of the Merging Methodsmentioning

confidence: 99%

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Ascertaining when a basin is Wada: the merging method

Daza

Wagemakers

Sanjuán

2018

Sci Rep

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Trying to imagine three regions separated by a unique boundary seems a difficult task. However, this is exactly what happens in many dynamical systems showing Wada basins. Here, we present a new perspective on the Wada property: A Wada boundary is the only one that remains unaltered under the action of merging the basins. This observation allows to develop a new method to test the Wada property, which is much faster than the previous ones. Furthermore, another major advantage of the merging method is that a detailed knowledge of the dynamical system is not required.

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Section: A Description Of the Merging Methodsmentioning

confidence: 99%

Section: A Description Of the Merging Methodsmentioning

confidence: 99%

Ascertaining when a basin is Wada: the merging method

Daza

Wagemakers

Sanjuán

2018

Sci Rep

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“…Each exit of the system has its own associated exit basin. Previous research described some algorithms to obtain the numerical verification of the Wada property in dynamical systems [24][25][26][27]. In this paper, we resort to the appearance of the exit basin boundaries to give visual indications about the persistence of the Wada property as the parameter β is varied.…”

Section: Exit Basins Descriptionmentioning

confidence: 99%

Uncertainty dimension and basin entropy in relativistic chaotic scattering

2018

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Chaotic scattering is an important topic in nonlinear dynamics and chaos with applications in several fields in physics and engineering. The study of this phenomenon in relativistic systems has received little attention as compared to the Newtonian case. Here we focus our work on the study of some relevant characteristics of the exit basin topology in the relativistic Hénon-Heiles system: the uncertainty dimension, the Wada property, and the basin entropy. Our main findings for the uncertainty dimension show two different behaviors insofar as we change the relativistic parameter β, in which a crossover behavior is uncovered. This crossover point is related with the disappearance of KAM islands in phase space, which happens for velocity values above the ultrarelativistic limit, v>0.1c. This result is supported by numerical simulations and by qualitative analysis, which are in good agreement. On the other hand, the computation of the exit basins in the phase space suggests the existence of Wada basins for a range of β<0.625. We also studied the evolution of the exit basins in a quantitative manner by computing the basin entropy, which shows a maximum value for β≈0.2. This last quantity is related to the uncertainty in the prediction of the final fate of the system. Finally, our work is relevant in galactic dynamics, and it also has important implications in other topics in physics such as as in the Störmer problem, among others.

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“…The associated basin appearing in Fig. 9(b) is a basin of Wada, as can be demonstrated using recent numerical techniques [Daza et al, 2015[Daza et al, , 2018.…”

Section: Basins Of Wadamentioning

confidence: 56%

Computing Complex Horseshoes by Means of Piecewise Maps

López

Daza

Seoane

et al. 2018

Int. J. Bifurcation Chaos

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A systematic procedure to numerically compute a horseshoe map is presented. This new method uses piecewise functions and expresses the required operations by means of elementary transformations, such as translations, scalings, projections and rotations. By repeatedly combining such transformations, arbitrarily complex folding structures can be created. We show the potential of these horseshoe piecewise maps to illustrate several central concepts of nonlinear dynamical systems, as for example the Wada property.

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Testing for Basins of Wada

Cited by 48 publications

References 19 publications

Ascertaining when a basin is Wada: the merging method

Ascertaining when a basin is Wada: the merging method

Uncertainty dimension and basin entropy in relativistic chaotic scattering

Computing Complex Horseshoes by Means of Piecewise Maps

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