A note on multiple flow equilibria

Abstract

A set of ordinary differential equations describing a mechanical system subject to forcing and dissipation is considered. A topological argument is employed to show that if all time-dependent solutions of the governing equations are bounded, the equations admit N steady solutions, where N is a positive odd integer and where at least ( N −1)/2 of the steady solutions are unstable. The results are discussed in the context of atmospheric flows, and it is shown that truncated forms of the quasigeostrophic equations of dynamic meteorology and of Budyko-Sellers climate models satisfy the hypotheses of the theorem.