Margazoglou Georgios, Magri Luca
Aeronautics Department, Imperial College London, South Kensington Campus, London, SW7 2AZ UK.
The Alan Turing Institute, 96 Euston Road, NW1 2DB London, UK.
Nonlinear Dyn. 2023;111(9):8799-8819. doi: 10.1007/s11071-023-08285-1. Epub 2023 Feb 10.
The prediction of the temporal dynamics of chaotic systems is challenging because infinitesimal perturbations grow exponentially. The analysis of the dynamics of infinitesimal perturbations is the subject of stability analysis. In stability analysis, we linearize the equations of the dynamical system around a reference point and compute the properties of the tangent space (i.e. the Jacobian). The main goal of this paper is to propose a method that infers the Jacobian, thus, the stability properties, from observables (data). First, we propose the echo state network (ESN) with the Recycle validation as a tool to accurately infer the chaotic dynamics from data. Second, we mathematically derive the Jacobian of the echo state network, which provides the evolution of infinitesimal perturbations. Third, we analyse the stability properties of the Jacobian inferred from the ESN and compare them with the benchmark results obtained by linearizing the equations. The ESN correctly infers the nonlinear solution and its tangent space with negligible numerical errors. In detail, we compute from data only (i) the long-term statistics of the chaotic state; (ii) the covariant Lyapunov vectors; (iii) the Lyapunov spectrum; (iv) the finite-time Lyapunov exponents; (v) and the angles between the stable, neutral, and unstable splittings of the tangent space (the degree of hyperbolicity of the attractor). This work opens up new opportunities for the computation of stability properties of nonlinear systems from data, instead of equations.
The online version contains supplementary material available at 10.1007/s11071-023-08285-1.
混沌系统时间动态的预测具有挑战性，因为无穷小扰动会呈指数增长。对无穷小扰动动态的分析是稳定性分析的主题。在稳定性分析中，我们围绕参考点将动态系统方程线性化，并计算切空间的性质（即雅可比矩阵）。本文的主要目标是提出一种从可观测量（数据）推断雅可比矩阵进而推断稳定性性质的方法。首先，我们提出将具有循环验证的回声状态网络（ESN）作为从数据中准确推断混沌动态的工具。其次，我们从数学上推导回声状态网络的雅可比矩阵，它提供了无穷小扰动的演化。第三，我们分析从ESN推断出的雅可比矩阵的稳定性性质，并将其与通过对方程进行线性化得到的基准结果进行比较。ESN能以可忽略的数值误差正确推断非线性解及其切空间。详细来说，我们仅从数据计算：（i）混沌状态的长期统计量；（ii）协变李雅普诺夫向量；（iii）李雅普诺夫谱；（iv）有限时间李雅普诺夫指数；（v）以及切空间稳定、中性和不稳定分裂之间的夹角（吸引子的双曲程度）。这项工作为从数据而非方程计算非线性系统的稳定性性质开辟了新机会。
在线版本包含可在10.1007/s11071-023-08285-1获取的补充材料。
