二阶耗散动力系统的降维对解长期行为的误差估计

Error Estimate for the Influence of Model Reduction of Nonlinear Dissipative Autonomous Dynamical System on the Long-Term Behaviours

基于非线性动力学理论,对一类高维二阶耗散自治动力系统的降维及其对解的长期行为的影响进行了理论分析· 该分析将方程的解投影到控制方程的线性算子的特征向量所张成的完备空间中,并在相空间中引入一距离的概念,方便地解决了缩减后系统与原始系统解之间的误差或距离的描述· 基于此距离定义,首先,分析了由于高阶模态的截取对解的长期行为的影响,并推导出了相应的误差估计,该估计表明由于降维对系统长期行为的影响不仅与系统的高阶子空间中的固有频率和阻尼比乘积的最小值有关,并且与高阶子空间中的某一最大固有频率有关· 然后,将一般的模态截取视为对原系统的解的一个扰动。

From viewpoint of nonlinear dynamics, the model reduction and its influence on the long_term behaviours of a class of nonlinear dissipative autonomous dynamical system with higher dimension are investigated theoretically under some assumptions.The system was analyzed in the state space with an introduction of a distance definition which can be used to describe the distance between the full system and the reduced system, and the solution of the full system was then projected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. As a result, the influence of mode series truncation on the long_term behaviours and the error estimate are derived, showing that the error is dependent on the first products of frequencies and damping ratios in the subspace spanned by the eigenvectors with higher modal damping. Furthermore, the fundamental understanding for the topological change of the solution due to the application of different model reduction is interpreted in a mathematically precise way, using the qualitative theory of nonlinear dynamics.