Abstract
We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question. Numerical examples illustrate the effectiveness of the method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bao B C, Shi G D, Xu J P, et al. Dynamic analysis of chaotic circuit with two memristors. Sci China Tech Sci, 2011, 54: 2180–2187
Besicovitch A S. Almost Periodic Functions. Cambridge: Cambridge University Press, 1932
Chartier P, Murua A, Sanz-Serna J M. Higher-order averaging, formal series and numerical integration I: B-series. Found Comp Maths, 2010, 10: 695–727
Chartier P, Murua A, Sanz-Serna J M. Higher-order averaging, formal series and numerical integration II: The quasiperiodic case. Found Comp Maths, 2012, 12: 471–508
Chedjou J C, Fotsin H B, Woafo P, et al. Analog simulation of the dynamics of a Van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans Circ Syst I: Fundam Theory Appl, 2001, 48: 748–757
Condon M, Deaño A, Iserles A. On systems of differential equations with extrinsic oscillation. Disc Cont Dynam Syst, 2010, 28: 1345–1367
E W, Engquist B. The heterogeneous multiscale methods. Commun Math Sci, 2003, 1: 87–132
Fodjouong G J, Fotsin H B, Woafo P. Synchronizing modified van der Pol-Duffing oscillators with offset terms using observer design: Application to secure communications. Phys Scr, 2007, 75: 638–644
Giannini F, Leuzzi G. Nonlinear Microwave Circuit Design. Chichester: John Wiley Sons, Ltd, 2004
Iserles A, Nørsett S P, Olver S. Highly oscillatory quadrature: The story so far. In: Proceedings of ENuMath. Berlin: Springer-Verlag, 2006, 97–118
Ramírez F, Suáarez A, Lizarraga I, et al. Stability analysis of nonlinear circuits driven with modulated signals. IEEE Trans Microwave Theory Tech, 2010, 58: 929–940
Sanz-Serna J M. Modulated Fourier expansions and heterogeneous multiscale methods. IMA J Numer Anal, 2009, 29: 595–605
Slight T J, Romeira B, Wang L Q, et al. A Lienard oscillator resonant tunnelling diode-laser diode hybrid integrated circuit: Model and experiment. IEEE J Quantum Electron, 2008, 44: 1158–1163
Verhulst F. Nonlinear Differential Equations and Dynamical Systems. Heidelberg: Springer-Verlag, 1990
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Condon, M., Deaño, A., Gao, J. et al. Asymptotic solvers for ordinary differential equations with multiple frequencies. Sci. China Math. 58, 2279–2300 (2015). https://doi.org/10.1007/s11425-015-5066-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-015-5066-5
Keywords
- highly oscillatory problems
- ordinary differential equation
- modulated Fourier expansions
- multiple frequencies
- numerical analysis