Interactions of (m,n) and (m+1,n) modes with real eigenvalues: A dynamic transition approach

Abstract

In this work, we consider the multiplicity two dynamic transitions of a broad class of problems. The first main assumption is the existence of two critical eigenmodes of the linear operator which depend on at least two wave indices one of which are consecutive m, m+1 and the other identical n. The second main assumption is an orthogonality condition on the nonlinear interactions of the basis vectors of the phase space which is typical in many applications. Under this assumption we obtain a reduced system of ODE\"s which describe the first dynamic transitions. We make a careful analysis of this reduced system to classify all possible transition behavior. We then apply our main theoretical findings to the 2D Rayleigh–Bénard convection with free-slip boundary conditions and show that this problem displays an S1 attractor bifurcation.