基于短轨迹集合的无方程周期感知预测误差收缩负最大李雅普诺夫指数估计
Equation-Free Period-Aware Forecast-Error Contraction for Estimating Negative Largest Lyapunov Exponents from Short Trajectory Ensembles
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中文总结 AI 辅助
该研究提出无方程周期感知预测误差收缩法，从短轨迹集合中估计负最大李雅普诺夫指数，在两个映射上取得低误差、高$R^2$的良好结果，为未知控制方程的动力学研究提供了新方法。
中文摘要 AI 辅助
从数据中估计正最大李雅普诺夫指数相对自然，因为相邻轨迹会分离，而稳定动力学需要在测量噪声或有限精度抹去信号前解析收缩。我们提出一种周期感知预测误差收缩方法，用于从短标量轨迹集合中估计主导负李雅普诺夫指数，无需使用控制方程或解析雅可比矩阵。该方法在轨迹历史上训练k近邻预测器，在相位一致的预测 horizon 处评估几何平均绝对预测误差，指数由对数误差曲线的斜率得到。与重构局部演化矩阵或对学习的代理求导的数据驱动方法不同，所提方法直接从样本外预测误差中提取收缩率。两项调整至关重要：预测步与检测到的轨道周期同步，且候选斜率仅在多个暂态长度上形成稳定共识时才被接受。在logistic映射上，该方法从112个负指数参数值中恢复了92个，平均绝对误差为0.0253，$R^2=0.886$。在无不动点的二维映射上，基于$x_n$、$y_n$和$z_n$三个可观测量的独立标量管道给出的平均绝对误差为0.00879至0.01145，$R^2$为0.983至0.986。由于估计阶段仅使用观测到的轨迹，该框架为重复松弛实验提供了基础，此类实验中仅能获得短传感器响应，但控制方程和解析雅可比矩阵未知。实验验证仍为未来工作。
英文摘要
Estimating positive largest Lyapunov exponents from data is comparatively natural because neighboring trajectories separate, whereas stable dynamics require resolving contraction before measurement noise or finite precision erases the signal. We introduce a period-aware forecast-error contraction procedure for estimating a dominant negative Lyapunov exponent from ensembles of short scalar trajectories without using governing equations or an analytical Jacobian. A k-nearest-neighbor predictor is trained on trajectory histories, the geometric-mean absolute forecast error is evaluated at phase-consistent horizons, and the exponent is obtained from the slope of the logarithmic error profile. Unlike data-driven approaches that reconstruct local evolution matrices or differentiate a learned surrogate, the proposed method extracts the contraction rate directly from out-of-sample forecast errors. Two adaptations are essential: the forecast step is synchronized with the detected orbit period, and candidate slopes are accepted only when they form a stable consensus across several transient lengths. On the logistic map, the method recovers 92 of 112 negative-exponent parameter values with a mean absolute error of 0.0253 and $R^2=0.886$. On a two-dimensional map without fixed points, independent scalar pipelines based on the three observables $x_n$, $y_n$, and $z_n$ give mean absolute errors of 0.00879--0.01145 and $R^2=0.983$--$0.986$. Because the estimation stage uses only observed trajectories, the framework provides a basis for repeated-relaxation experiments in which short sensor responses are available but the governing equations and analytical Jacobian are unknown. Experimental validation remains a subject of future work.
发表机构
- Institute of Physics and Technology, Petrozavodsk State University（彼得罗扎沃茨克国立大学物理与技术学院）
- School of Science and Engineering, International University of Grand-Bassam（格兰德巴萨姆国际大学科学与工程学院）
- Faculty of Electronics Technology, Industrial University of Ho Chi Minh City（胡志明市工业大学电子技术学院）
机构由 AI 辅助整理，请以论文原文为准。
