2020/07/01 by Wei Zhang, Zhang, Wei, Mingjun Wei +1
Physics and Astronomy · Engineering · #Model Reduction and Neural Networks #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis
paper · pdf · doi:10.48550/arxiv.2007.00249
Local Koopman spectral problem is studied to resolve all dynamics for a\nnonlinear system. The proposed spectral problem is compatible with the linear\nspectral theory for various linear systems, and several properties of local\nKoopman spectrums are discovered. Firstly, proliferation rule is discovered for\nnonlinear observables and it applies to nonlinear systems recursively.\nSecondly, the hierarchy structure of Koopman eigenspace of nonlinear dynamics\nis revealed since dynamics can be decomposed into the base and perturbation\nparts, where the former can be analyzed analytically or numerically and the\nlatter is further divided into linear and nonlinear parts. The linear part can\nbe analyzed by the linear spectrums theory. They are then recursively\nproliferated to infinite numbers for the nonlinear part. Thirdly, local Koopman\nspectrums and eigenfunctions change continuously and analytically in the whole\nmanifold under suitable conditions, derived from operator perturbation theory.\nTwo cases of fluid dynamics are numerically studied. One is the two-dimensional\nflow past cylinder at the Hopf-bifurcation near the critical Reynolds number.\nTwo asymptotic stages, flow systems around an unstable fixed point and a stable\nlimit cycle were studied separately by the DMD algorithm. The triad-chain and\nthe lattice distribution of Koopman spectrums confirmed the proliferation rule\nand hierarchy structure of Koopman eigenspace. Another example is the\nthree-dimensional secondary instability of flow past a fixed cylinder, where\nthe Fourier modes, Floquet modes, and high-order Koopman modes characterizing\nthe main structure of the flow are discovered.\n