scholarly journals Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

2020 ◽  
Vol 121 (23) ◽  
pp. 5320-5349
Author(s):  
Sridhar Chellappa ◽  
Lihong Feng ◽  
Peter Benner
Keyword(s):  
Dynamical Systems ◽  
Error Estimation ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Basis Construction

Error Estimation of Fourier Series Expansion and Implication to Solution Accuracy for Nonlinear Dynamical Systems

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Y. M. Chen ◽  
Z. R. Lv ◽  
J. K. Liu
Keyword(s):  
Fourier Series ◽  
Dynamical Systems ◽  
Frequency Domain ◽  
Error Estimation ◽  
Series Expansion ◽  
Nonlinear Dynamical Systems ◽  
Frequency Domain Analysis ◽  
Fourier Series Expansion ◽  
Nonlinear Dynamical ◽  
Time Points

Fourier series expansion (FSE) plays a pivotal role in frequency domain analysis of a wide variety of nonlinear dynamical systems. To the best of our knowledge, there are two general approaches for FSE, i.e., a collocation method (CM) previously proposed by the authors and the classical discrete FSE. Though there are huge applications of these methods, it still remains much less understood in their relationship and error estimation. In this study, we proved that they are equivalent if time points are uniformly chosen. Based on this property, more importantly, the error was analytically estimated for both discrete Fourier expansion (DFE) and CM. Furthermore, we revealed that the accuracy of frequency domain solutions cannot be improved by increasing the number of time points alone, whereas it absolutely depends upon the truncated number of harmonics. It indicates that an appropriate number of time points should be chosen in FSE if frequency domain solutions are targeted for nonlinear dynamical systems, especially those with complicated functions.

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