论文标题
高维的粗糙盆地边界：我们可以实验对它们进行分类吗？
Rough basin boundaries in high dimension: Can we classify them experimentally?
论文作者
论文摘要
我们表明，在Bistable 2D地图中具有粗糙盆地边界的已知条件可用于高维双态系统，该系统具有嵌入其盆地边界中的独特非吸引人的混沌集。粗糙度的条件是跨边界Lyapunov指数$λ_x$ {\ bfac在非吸引集合}上不是最大的。此外，我们为粗糙盆地边界的一般非全力共同量化提供了一个公式，该公式可以看作是坎茨 - 格拉斯伯格公式的概括。这种最多可以统一的共同度可以被认为是部分共同量化，因此，它可以与Lyapunov指数匹配。我们显示{\ bfac在2D非不可逆转 - 和3D可逆模型中，}，正式地，它不能与$λ_x$匹配。相反，部分尺寸$ d_0^{（x）} $，在粗糙边界的情况下，$λ_x$与之相关。进一步的结果暗示，后者也保持在更高的维度。这是粗糙分形的特征。但是，$ d_0^{（x）} $无法通过沿着边界的线的不确定性指数来测量。实际上，一个人无法通过测量分形尺寸来确定边界是粗糙还是丝状分形。取而代之的是，需要在数值或实验上同时测量最大和跨边界的Lyapunov指数。
We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent $λ_x$ {\bfac on the nonattracting set} is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show {\bfac in 2D noninvertible- and 3D invertible minimal models,} that, formally, it cannot be matched with $λ_x$. Rather, the partial dimension $D_0^{(x)}$ that $λ_x$ is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, $D_0^{(x)}$ cannot be measured via the uncertainty exponent along a line that traverses the boundary. Indeed, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.
