On the integrability of some three-dimensional Lotka-Volterra equations with rank-$1$ resonances
We investigate the local integrability in C3 of some three-dimensional Lotka–Volterra equations at the origin with (p : q : r)-resonance, x˙ = P = x(p + ax + by + cz), y˙ = Q = y(q + dx + ey + fz), z˙ = R = z(r + gx + hy + kz). Recent work on this problem has centered on the case where the resonance is of “rank-2”. That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ /∈ iR (after a scaling, this is just the case p + q = 0 with q/r /∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0. Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x αy βz γ(1 + O(x, y, z). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.
Darboux First Integral Conditions and Integrability of the 3D Lotka-Volterra System
Abstract We apply the Darboux theory of integrability to polynomial ODE’s of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3–dimensional Lotka–Volterra systems with polynomial invariant algebraic solutions linear and quadratic and determine numerous cases of integrability.
Control of the homopause level
Abstract This paper reviews the hypothesis that internal gravity wave breaking in the factor responsible for upper atmosphere turbulence generation and control of the number density at the homopause level. Available data from Mars and Earth support the hypothesis, originally introduced by Lindzen, that breaking thermal tides are responsible for the necessary turbulence. The lowest-order, equatorially trapped, westward-propagating diurnal tidal mode is the primary cause on the Earth, and lower-order westward-propagating semidiurnal modes are the primary cause on Mars. The frequencies and vertical wavenumbers of the responsible modes determine the eddy diffusion coefficients; energy density is only indirectly involved by determining which modes can break. A general scale relationship between tidal heating and velocity amplitude can be used to assess the breaking potential of tidal modes, and is used here as the basis for arguing that the number density at the Martian homopause is likely to have been stable over much of geologic time. Difficulties with this approach arise in the cases of Venus and Titan because of their peculiar zonal winds. For Jupiter and Saturn, internal gravity waves generated by nontidal mechanisms probably control the homopause.