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arXiv · 2604.20541

The tentacles landscape: geometric properties of high-dimensional basins of attraction

Pablo Groisman
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Abstract

Basins of attraction in multistable high dimensional dynamical systems are expected to have universal features but very little has been proved rigorously. We consider phase oscillators coupled according to a cycle graph, $\dot\theta_i = f(\theta_{i+1}-\theta_i) + f(\theta_{i-1}-\theta_i)$, with coupling $f$ that is $C^1$, odd, $2\pi$-periodic, and strictly increasing on $(-\pi,\pi)$. We prove the full ``octopus'' picture of the basins of attraction observed numerically by Zhang and Strogatz [Phys.\ Rev.\ Lett.\ 127 (2021) 194101] and, beyond this model, across a wide family of high-dimensional multistable systems. In our case, we have a family of stable equilibria that can be indexed by their winding number $q \in \mathbb Z\cap(-n/2,n/2)$. Basin volumes obey a Gaussian law $\mu(\mathcal K_q)=\sqrt{6/(\pi n)}\,e^{-6q^2/n}(1+o(1))$ in the winding number. The distance from a uniform sample to its attractor, when divided by $\sqrt n$, concentrates at $\sqrt{\pi^2/3}\approx 1.814$. Along almost every straight line through any twisted state, the ray enters every other basin infinitely many times, with frequencies given by the basin volumes. The inscribed ball at a twisted state has radius $({\pi}/{\sqrt2})(1-{2|q|}/{n})$ for every $n$, while as $n\to\infty$, a typical ray travels distance $(\pi/2)\sqrt{n/\log n}$ before first leaving the basin: the head of the octopus is sharply anisotropic.

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Pablo Groisman. 2026-04-22. The tentacles landscape: geometric properties of high-dimensional basins of attraction. https://arxiv.org/abs/2604.20541

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