Abstract
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.
Highlights
Trying to imagine three regions separated by a unique boundary seems a difficult task
The Polish topologist Kazimierz Kuratowski showed that in the plane, continuous Wada boundaries must be indecomposable continua[3]. This discovery remained as a mathematical curiosity until James Yorke and his collaborators found that the basins of attraction of some dynamical systems presented the Wada property[4,5]
From the dynamical point of view, the most interesting feature of Wada basins is the fact that an arbitrarily small perturbation of a system with initial conditions lying in a Wada boundary can drive it to any of the possible attractors, which implies a special kind of unpredictability[6]
Summary
The set of all initial conditions leading to a particular attractor is called the basin of attraction of a dynamical system. We will say that the system is not Wada, and the method will determine which points are Wada and which ones are not This last step verifies if each pixel of the slim boundaries ∂Bi lies in the set ∂Bj. To connect with our previous definition of a basin with the Wada property, the algorithm checks if the points pi in the boundaries Bi are within a ball b(pj, r) of radius r (r is the fattening parameter) around the points pj of the boundary Bj. In the case of partially Wada basins[16], where Wada and non-Wada boundaries coexist, we can characterize them by the Wada parameter WNa defined in the grid method of Daza et al.[21]. The two paradigmatic dynamical systems under study are the Hénon-Heiles Hamiltonian[10] and the Newton method to find complex roots[8]
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