Abstract
We use Groebner basis methods to extract all stationary solutions for the nine-mode shear flow model described in Moehlis et al. (New J Phys 6:56, 2004). Using rational approximations to irrational wave numbers and algebraic manipulation techniques we reduce the problem of determining all stationary states to finding roots of a polynomial of order 30. The coefficients differ by 30 powers of 10, so that algorithms for extended precision are needed to extract the roots reliably. We find that there are eight stationary solutions consisting of two distinct states, each of which appears in four symmetry-related phases. We discuss extensions of these results for other flows.
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Acknowledgments
This work was partly supported by the Deutsche Forschungsgemeinschaft via Forschergruppe 1182. VR acknowledges the support by the Slovenian Research Agency. The work was also supported by the Transnational Access Programme at RISC-Linz of the European Commission Framework 6 Programme for Integrated Infrastructures Initiatives under the Project SCIEnce (Contract 026133).
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Communicated by P. Newton.
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Pausch, M., Grossmann, F., Eckhardt, B. et al. Groebner Basis Methods for Stationary Solutions of a Low-Dimensional Model for a Shear Flow. J Nonlinear Sci 24, 935–948 (2014). https://doi.org/10.1007/s00332-014-9208-7
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DOI: https://doi.org/10.1007/s00332-014-9208-7