Testing for Basins of Wada

Physical review, 1999

Motivated by recent numerical observations on a four-dimensional continuous-time dynamical system, we consider different types of basin boundary structures for chaotic systems. These general structures are essentially mixtures of the previously known types of basin boundaries where the character of the boundary assumes features of the previously known boundary types at different points arbitrarily finely interspersed in the boundary. For example, we discuss situations where an everywhere continuous boundary that is otherwise smooth and differentiable at almost every point has an embedded uncountable, zero Lebesgue measure set of points at which the boundary curve is nondifferentiable. Although the nondifferentiable set is only of zero Lebesgue measure, the curve's fractal dimension may ͑depending on parameters͒ still be greater than one. In addition, we discuss bifurcations from such a mixed boundary to a ''pure'' boundary that is a fractal nowhere differentiable curve or surface and to a pure nonfractal boundary that is everywhere smooth.

Physica D: Nonlinear Phenomena, 1994

Recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby (the basin is "riddled" with holes). Here we consider the dynamics near the transition to this situation as a parameter is varied. Using a simple analyzable model, we obtain the characteristic behaviors near this transition. Numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered. 1 This situation is to be distinguished from the case where the basin is a solid volume with a fractal boundary (e.g., see Ref. [31).

Physical Review E, 1998

Physics Letters A, 1985

We discuss the structure of fractal basin boundaries in typical nonanalytic maps of the plane and describe a new type of crisis phenomenon. Basin boundaries for dynamical systems with coexisting attractors can be smooth or fractal. An understanding of fractal boundaries may be important for a number of physically significant reasons [1,2] (e.g., they affect the degree to which final states can be predicted in situations where error is present in the specification of initial conditions). Until recently [1-5], however, studies focusing on the properties of fractal basin boundaries have been primarily restricted to mappings of a single complex variable, z ~ F(z), where z-x + iy and F is an analytic function (henceforth referred to as analytic maps). In this case the formalism of analytic function theory can be applied, and many beautiful results have been obtained [6], dating back to the classic work of Julia and Fatou at the beginning of this century. We note, however, that analytic maps represent a very restricted class of twodimensional mappings (e.g., due to the Cauchy-Riemann conditions they cannot possess chaotic attractors). The work reported in this letter was initially motivated by the desire to see whether the properties of fractal basin boundaries of analytic maps extend to basin boundaries of more general mappings of the plane (such as those that might arise from a surface of section for systems of ordinary differential equations). In addition, we shall also describe a new type of boundary crisis [7] (in a boundary crisis, a

Communications in Nonlinear Science and Numerical Simulation, 2021

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 ln 2 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 S bb fractality test. Using the same single scale information, the S bb fractality test allows for the detection of fractal boundaries in many more cases than the ln 2 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.

Journal of Physics: Conference Series, 2010

Chaotic dynamical systems with certain phase space symmetries may exhibit riddled basins of attraction, which can be viewed as extreme fractal structures in the sense that, regardless of how small is the uncertainty in the determination of an initial condition, we cannot decrease the fraction of such points that are certain to converge to a given attractor. We investigate a mechanical system exhibiting riddled basins of attraction: a particle under a two-dimensional potential with friction and time-periodic forcing. The verification of riddling is made by checking its mathematical requirements through computation of finite-time Lyapunov exponents as well as by scaling laws describing the fine structure of basin filaments densely intertwined in phase space.

Chaos, Solitons & Fractals, 2007

The nonlinear interaction of two wave triplets, with a non-zero frequency mismatch, has been found to present many indications of complex behavior. Considering the presence of dissipation, there are coexistent periodic and chaotic attractors in the high-dimensional phase space of the system. Their basin structure is also complex, parts of it being densely mixed in arbitrarily fine scales. We claim that the fractal part of the basin boundary presents the so-called Wada property, for which points on the boundary are arbitrarily close to points of all coexisting basins. The practical consequence of this fact is that, given a small yet non-zero uncertainty on determining the initial condition, the asymptotic state to which the system goes is almost completely uncertain.

SPIE Proceedings, 2003

We study fluctuational transitions between two co-existing chaotic attractors separated by a fractal basin boundary in a discrete dynamical system. It is shown that the mechanism of fluctuational transition through a fractal boundary is generic, and determined by a hierarchy of homoclinic original saddles. The most probable escape path from a chaotic attractor to the fractal boundary is found using both statistical analyses of fluctuational trajectories and the Hamiltonian theory of fluctuations.

Physical Review Letters, 1986

A basin boundary can undergo sudden changes in its character as a system parameter passes through certain critical values. In particular, basin boundaries can suddenly jump in position and can change from being smooth to being fractal. e describe these changes ("metamorphoses") and f&nd that they involve certain special unstable orbits on the basin boundary which are aceessibIe from inside one of the basins. The forced damped pendulum (Josephson junction) is used to illustrate these phenomena.

Physica D: Nonlinear Phenomena, 2005

Basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology, chemistry, economics, engineering, and physics. One of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins. A basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins, is called a Wada basin, and its boundary is called a Wada basin boundary. Here we consider maps on the interval. We present a sufficient and necessary condition guaranteeing that three Wada basins are emerging from a tangent bifurcation for certain one-dimensional maps having negative Schwarzian derivative, two fixed point attractors on one side of the tangent bifurcation, and three fixed point attractors on the other side of the tangent bifurcation. All the conditions involved are numerically verifiable.