Many important high-dimensional dynamical systems exhibit complex chaotic behaviour. Their complexity means that their dynamics are necessarily comprehended under strong reducing assumptions. It is therefore important to have a clear picture of these reducing assumptions' range of validity. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively hyperbolic, which implies many felicitous statistical properties. We demonstrate, contrary to the chaotic hypothesis, the existence of non-hyperbolic large-scale dynamics in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev Galerkin transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be assumed to hold in all systems, and a better understanding of the domain of its validity is required.

中文翻译：

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混沌系统大规模动力学中的非双曲性

许多重要的高维动力系统表现出复杂的混沌行为。它们的复杂性意味着它们的动态必须在强还原假设下得到理解。因此，重要的是要清楚地了解这些减少假设的有效性范围。Gallavotti 和 Cohen 的极具影响力的混沌假设指出，高维系统的大规模动力学实际上是双曲线的，这意味着许多恰当的统计特性。我们证明，与混沌假设相反，平均场耦合系统中存在非双曲大尺度动力学。为此，我们将系统降低到其热力学极限，我们用 Chebyshev Galerkin 传递算子离散化在数值上近似。这使我们能够获得同宿相切的高精度估计，这意味着双曲线的失败。在扰动下预计会有稳健的非双曲线行为。因此，不应假设混沌假设在所有系统中都成立，并且需要更好地理解其有效性的域。