WADA FRACTAL BASINS OF ATTRACTION IN A ZERO-STIFFNESS VIBRATION ISOLATION SYSTEM

A fractal basin boundary is a Wada fractal basin boundary if it contains at least three different basins. The corresponding basin is called a Wada fractal basin. Previous results show that some oscillators possess Wada fractal basins with common basin boundaries. Here we find that a nonlinear vibration isolation system can possess abundant coexisting basins and every basin is a Wada fractal basin. These Wada fractal basin boundaries separate different basins in the different regions. A proper classification of these Wada fractal basins is provided according to the order of saddles and Wada numbers. Basin organization is systematic and all basins spiral outward toward the infinity. The entangled basin boundaries are described by the manifolds of saddles and basins (tongues) accumulation.

CHARACTERIZING FRACTAL BASIN BOUNDARIES FOR PLANAR SWITCHED SYSTEMS

This paper is to introduce some analytical tools to characterize the properties of fractal basin boundaries for planar switched systems (with time-dependent switching). The characterizing methods are based on the view point of limit sets and prime ends. By constructing the auxiliary dynamical system, the fractal basin boundaries of planar switched systems can be proved if every diverging path in the basin of associated auxiliary system has the entire basin boundary as its limit set. Fractal property is also verified if every prime end that is defined in the basin of associated auxiliary system is a prime end of type 3 and all other prime ends are of type 1. Bifurcations of fractal basin boundary are investigated by analyzing what types of prime ends in the basin are involved. The fractal basin boundary of switched system is also described by the indecomposable continuum.

ALFVÉN COMPLEXITY

The complex dynamics of Alfvén waves described by the derivative nonlinear Schrödinger equation is investigated. In a region of the parameters space where multistability is observed, this complex system is driven towards an intermittent regime by the addition of noise. The effects of Gaussian and non-Gaussian noise are compared. In the intermittent regime, the Alfvén wave exhibits random qualitative changes in its dynamics as the result of a competition between three attractors and a chaotic saddle embedded in the fractal basin boundary.

On the Fractal Structure of Basin Boundaries in Two-Dimensional Noninvertible Maps

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.

Iterated Differentiable Maps with Nowhere Differentiable Basin Boundaries

The fractal basin boundary of a two-dimensional discrete dynamical system modelling a chaotic forcing applied to bistability is shown to be identical to the graph of an infinite series F(x,t)= of weighted iterates of an ergodic unimodal interval function f. In the special case, when f is the logistic map in "full chaos", i.e. ƒ: x ↦ 4x(1 - x), F is a nowhere differentiable function of x for each t > exp(-λf) (even equal to the Weierstrass function), where λf >0 is denoting the Lyapunov exponent of f. For further chaotic functions f, nowhere-differentiability is shown to be obvious from computer simulations.