A kernel principal component analysis of coexisting attractors within a generalized Lorenz model

Abstract

Based on recent studies that reveal the coexistence of chaotic and non-chaotic solutions using a generalized Lorenz model (GLM), a revised view on the dual nature of weather has been proposed by Shen et al. [41,42], as follows: the entirety of weather is a superset consisting of both chaotic and non-chaotic processes. Since better predictability for non-chaotic processes can be expected, an effective detection of regular or chaotic solutions can improve our confidence in numerical weather and climate predictions. In this study, by performing a kernel principal component analysis of coexisting attractors obtained from the GLM, we illustrate that the time evolution of the first eigenvector of the kernel matrix, referred to as the first kernel principal component (K-PC), is effective for the classification of chaotic and non-chaotic orbits. The spatial distribution of the first K-PC within a two-dimensional phase space can depict the shape of a decision boundary that separates the chaotic and non-chaotic orbits. We additionally present how a large number (e.g., 128 or 256) of K-PCs can be used for the reconstruction of data in order to illustrate the different portions of the phase space occupied by chaotic and non-chaotic orbits, respectively.