## 2 Preliminary Theory and Related Work
### Koopman Theory  
Due to the complete representation of the evolution information of a linear dynamical system by the eigenvalues and eigenvectors of the matrix, globally linearizing a nonlinear dynamical system is a popular research direction. This idea of global linearization originates from Koopman theory. For a measure-preserving dynamical system with a compact invariant set \(M\), the space of all square-integrable observable functions with respect to the measure \(\mu\) on \(M\) is defined as:  
\[\mathcal{H}=\bigg{\{}f:M\to\mathbb{R}\bigg{|}\int_{M}\ \ \bigg{|}f\bigg{|}^{2}d\mu< \infty\bigg{\}}. \tag{4}\]  
The Koopman operator \(\mathcal{K}\) acts on an observable function, shifting the analysis from state space to the space of observable functions. Given an observable function \(g:M\to\mathbb{R}\), the Koopman operator \(\mathcal{K}\) is defined by:  
\[\mathcal{K}\phi=\lambda\phi. \tag{5}\]  
Here, the operator \(\mathcal{K}\) maps the observable function \(g\) to a new function \(g\circ\mathbf{T}\), representing the change in the observable function \(g\) under the action of \(\mathbf{T}\). In the new space, the evolution is linearized, i.e., for any observable functions \(g_{1}\) and \(g_{2}\) and constants \(c_{1}\) and \(c_{2}\), we have:  
\[\mathcal{K}(c_{1}g_{1}+c_{2}g_{2})=c_{1}\mathcal{K}g_{1}+c_{2}\mathcal{K}g_{2}. \tag{6}\]  
This linearity of the Koopman operator allows the application of linear system analysis tools to study the dynamics of the original nonlinear system. If a set of observable functions remains within the space generated by this set after the action of the Koopman operator, such a space is called a Koopman invariant subspace. The eigenvalues, eigenfunctions, and corresponding Koopman modes of the Koopman operator encode a significant amount of global information about the underlying system. If there exists a function \(\phi\) such that they satisfy Equation (5), then \(\phi\) is called an eigenfunction of the Koopman operator, and \(\lambda\) is the corresponding eigenvalue. Koopman mode decomposition theory indicates that the evolution of observable functions can be expanded in terms of the Koopman eigenfunctions \(\phi_{j}\) and corresponding eigenvalues \(\lambda_{j},\ j=0,\ 1,\ldots\):  
\[U^{n}f(\mathbf{X}_{0})=f\circ\mathbf{T}^{n}(\mathbf{X}_{0})=\sum_{j=1}^{\infty }v_{j}\phi_{j}(\mathbf{X}_{0})\lambda_{j}^{n}. \tag{7}\]  
If  
\[\widetilde{f}_{m}(\mathbf{X}_{0})=\Big{[}f(\mathbf{X}_{0}),f\circ\mathbf{T}( \mathbf{X}_{0}),\ldots,f\circ\mathbf{T}^{m-1}(\mathbf{X}_{0})\Big{]}, \tag{8a}\] \[\widetilde{g}_{m}(\mathbf{X}_{0})=\Big{[}g(\mathbf{X}_{0}),g\circ\mathbf{T}( \mathbf{X}_{0}),\ldots,g\circ\mathbf{T}^{m-1}(\mathbf{X}_{0})\Big{]}, \tag{8b}\]  
represent the sequence of m continuous observables of \(f,g\in\mathcal{H}\) starting from \(\mathbf{X}_{0}\). The inner product of any two observable functions in \(\mathcal{H}\) can be approximated through the time series of observables:  
\[\lim_{m\to\infty}\frac{1}{m}\Big{\langle}\widetilde{f}_{m}(\mathbf{X}_{0}), \widetilde{g}_{m}(\mathbf{X}_{0})\Big{\rangle}=\lim_{m\to\infty}\frac{1}{m} \sum_{k=0}^{m-1}\left(fg^{*}\right)\circ\mathbf{T}^{k}(\mathbf{X}_{0})=\int_{ A}\ fg^{*}d\mu=\langle f,g\rangle_{\mathcal{H}}. \tag{9}\]  
This implies that using data vectors (such as \(\widetilde{f}_{m}(\mathbf{z}_{0})\)), one can approximate the projection between observable functions. Considering a longer observation sequence:  
\[\widetilde{f}_{m+n}=\Big{[}f(\mathbf{X}_{0}),f\circ\mathbf{T}(\mathbf{X}_{0} ),\ldots,f\circ\mathbf{T}^{m+n-1}(\mathbf{X}_{0})\Big{]}, \tag{10}\]  
construct the Hankel matrix:  
\[\tilde{\mathbf{H}}=