双同步混沌吸引子动力系统中的筛形域

RIDDLED BASINS IN DYNAMICAL SYSTEMS WITH TWO SYNCHRONIZED CHAOTIC ATTRACTORS

解析证明耦合映射混沌同步系统中的两个同步混沌吸引子的吸引域是筛形域.在特定耦合参数区间中,解析证明这两个同步混沌吸引子的吸引域不仅被无穷远吸引子的吸引域筛形,还通过数值证明它们的吸引域彼此互相筛形,展示出类似于wada性质的特征.但进一步的讨论表明这种复杂的被两个(或更多)吸引域共同筛形的结构并不是Wada域,而是由于筛形分岔和筛形域局部-全局分岔导致的.

Riddled basins have positive Lebegue measure with no open subsets. Since chaotic synchronized systems have many important applications, numerous investigators have studied them together with this new striking phenomenon, numerically or experimentally. In this paper, we have mainly studied a coupled skew tent maps with two synchronized chaotic attractors, A1 and A2. The basins of attraction of A1 and A2 are analytically proven to be riddled basins. Furthermore, in a special coupling parameter interval, it is also proven that these two basins are globally riddled not only by the basin of the infinity but also by each other. It is well known that intermingled basins will appear if two basins are riddled by each other when a system having two chaotic attractors. But if a system has three (or more than three) attractors, wada basin will appear if a basin has wada basin boundary with wada property that any point that is on the boundary of that basin is also simultaneously on the boundaries of at least two other basins. So it is obvious that the phenomenon discussed here is more complicated than intermingled basins, exhibiting similar wada property at the same time. That is, in any neighborhood of an arbitrary point in the basin of A1 (or A2) there is a point set with positive measure which are repelled to A2 (or A1) and the infinity attractor. But the further discussion tells us that these complicated multi-riddled basins are not wada basins. From the view angle of definition, it is seen that there are open sets in wada basins but there are no open sets in riddled basins. The calculations of uncertainty exponents also indicate that the uncertainty exponents of the basins of A1 and A2 are very close to zero. It is also an obvious character different from wada basins. So we think, it is not appropriate if we consider these complicated multi-riddled basins as wada basins sweepingly. The further study tells us that these complicated multi-riddled basins may be deduced by riddling bifurcation and local-global riddling bifurcation. Because of the ubiquity of multi-attractor systems the complicated multi-riddeld basins discussed here will be general in dynamics.