Rough basin boundaries in high dimension: Can we classify them experimentally?

Abstract

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 $\lambda_x$ {\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 $\lambda_x$. Rather, the partial dimension $D_0^{(x)}$ that $\lambda_x$ 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.