How to detect Wada Basins

Annals of the New York Academy of Sciences, 1987

International Journal of Bifurcation and Chaos, 2003

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical tools. The proposed example connects the two-dimensional maps of the real plane to the well-known complex map.

International Journal of Bifurcation and Chaos, 2003

Basins generated by a noninvertible mapping formed by two symmetrically coupled logistic maps are studied when the only parameter λ of the system is modified. Complex patterns on the plane are visualized as a consequence of the basins' bifurcations. According to the already established nomenclature in the literature, we present the relevant phenomenology organized in different scenarios: fractal islands disaggregation, finite disaggregation, infinitely disconnected basin, infinitely many converging sequences of lakes, countable self-similar disaggregation and sharp fractal boundary. By use of critical curves, we determine the influence of zones with different number of first rank preimages in the mechanisms of basin fractalization.

Science, 1996

Many remarkable properties related to chaos have been found in the dynamics of nonlinear physical systems. These properties are often seen in detailed computer studies, but it is almost always impossible to establish these properties rigorously for specific physical systems. This article presents some strange properties about basins of attraction. In particular, a basin of attraction is a "Wada basin" if every point on the common boundary of that basin and another basin is also on the boundary of a third basin. The occurrence of this strange property can be established precisely because of the concept of a basin cell.

Communications in Nonlinear Science and Numerical Simulation, 2020

First conceived as a topological construction, Wada basins abound in dynamical systems. Basins of attraction showing the Wada property possess the particular feature that any small perturbation of an initial condition lying on the boundary can lead the system to any of its possible outcomes. The saddle-straddle method, described here, is a new method to identify the Wada property in a dynamical system based on the computation of its chaotic saddle in the fractalized phase space. It consists of finding the chaotic saddle embedded in the boundary between the basin of one attractor and the remaining basins of attraction by using the saddle-straddle algorithm. The simple observation that the chaotic saddle is the same for all the combinations of basins is sufficient to prove that the boundary has the Wada property.

Ergodic Theory and Dynamical Systems, 2003

In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary ∂B as its limit set. The latter property reflects a complete entangled basin and its boundary. § Permanent address:

International Journal of Bifurcation and Chaos, 2006

Two-dimensional (Z 1 -Z 3 -Z 1 ) maps are such that the plane is divided into three unbounded open regions: a region Z 3 , whose points generate three real rank-one preimages, bordered by two regions Z 1 , whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

Fractals

We define fractal continuations and the fast basin of the IFS and investigate which properties they inherit from the attractor. Some illustrated examples are provided.

Fractal and Fractional

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

2004

The paper deals with the solutions generated by a two-dimensional noninvertible map defined by a cubic polynomial. The map is of the (Z 1 − Z 3 − Z 1) type, i.e. the plane is divided into three unbounded open areas: one Z 3 generating three real rank-one preimages, bordered by two regions Z 1 generating only one real rank-one preimage. This short study concerns the bifurcations leading a basin in a non fractal situation into a fractal one (multiply connected basin, or non connected basin). In this route to fractalization the first step is the existence of a chaotic attractor which after destabilization gives rise to a strange repeller, limit set for the the basin elements, when it is no longer simply connected. More specially one of such routes is considered, when the strange repeller is said to have a structure with "high density".