Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours

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 is 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 is 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.