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Delayed Hopf bifurcation in time-delayed slow-fast systems

中国科学:技术科学 2010年第2期 212-212页

作者： ZHENG YuanGuang1 & WANG ZaiHua1,2 1 Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China 2 Institute of Science, PLA University of Science and Technology, Nanjing 211101, China

This paper presents an investigation on the phenomenon of delayed bifurcation in time-delayed slow-fast differential systems. Here the two delayed's have different meanings. The delayed bifurcation means that the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through some bifurcation point, but at some other point which is above the bifurcation point by an obvious distance. In a time-delayed system, the evolution of the system depends not only on the present state but also on past states. In this paper, the time-delayed slow-fast system is firstly simplified to a slow-fast system without time delay by means of the center manifold reduction, and then the so-called entry-exit function is defined to characterize the delayed bifurcation on the basis of Neishtadt's theory. It shows that delayed Hopf bifurcation exists in time-delayed slow-fast systems, and the theoretical prediction on the exit-point is in good agreement with the numerical calculation, as illustrated in the two illustrative examples.

关键词： time delay delayed bifurcation Hopf bifurcation slow-fast systems exit-point entry-exit function

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Balanced and Unbalanced Components of Moist Atmospheric Flows with Phase Changes

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Chinese Annals of Mathematics,Series B 2019年第6期40卷 1005-1038页

作者： Alfredo N.WETZEL Leslie M.SMITH Samuel N.STECHMANN Jonathan E.MARTIN Department of Mathematics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Engineering Physics University of Wisconsin-MadisonMadisonWI 53706USA Department of Mathematics and Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA Department of Atmospheric and Oceanic Sciences University of Wisconsin-MadisonMadisonWI 53706USA

Atmospheric variables(temperature, velocity, etc.) are often decomposed into balanced and unbalanced components that represent low-frequency and high-frequency waves, respectively. Such decompositions can be defined, for instance, in terms of eigenmodes of a linear operator. Traditionally these decompositions ignore phase changes of water since phase changes create a piecewise-linear operator that differs in different phases(cloudy versus non-cloudy). Here we investigate the following question: How can a balanced–unbalanced decomposition be performed in the presence of phase changes? A method is described here motivated by the case of small Froude and Rossby numbers,in which case the asymptotic limit yields precipitating quasi-geostrophic equations with phase changes. Facilitated by its zero-frequency eigenvalue, the balanced component can be found by potential vorticity(PV) inversion, by solving an elliptic partial differential equation(PDE), which includes Heaviside discontinuities due to phase changes. The method is also compared with two simpler methods: one which neglects phase changes, and one which simply treats the raw pressure data as a streamfunction. Tests are shown for both synthetic, idealized data and data from Weather Research and Forecasting(WRF) model simulations. In comparisons, the phase-change method and no-phase-change method produce substantial differences within cloudy regions, of approximately 5K in potential temperature, due to the presence of clouds and phase changes in the data. A theoretical justification is also derived in the form of a elliptic PDE for the differences in the two streamfunctions.

关键词： Potential vorticity inversion Moist atmospheric dynamics slow-fast systems Balanced-unbalanced decomposition Elliptic partial differential equations

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