On the Periodic Solutions of the Five-Dimensional Lorenz Equation Modeling Coupled Rosby Waves and Gravity Waves

Abstract

Lorenz studied the coupled Rosby waves and gravity waves using the differential system U = -VW + bVZ, V = UW - bUZ, W = -UV, Ẋ = -Z, Ż = bUV + X. This system has the two first integrals H1 = U2 + V2, H2 = V2 + W2 + X2 + Z2. Our main result shows that in each invariant set {H1 = h1 > 0}∩{H2 = h2 > 0} there are at least four (resp., 2) periodic solutions of the differential system with b≠0 and h2 > h1 (resp., h2 < h1).