Double-diffusive convection andλbifurcation

Direct Resonance in Double‐Diffusive Systems

“…However, the value for v in (3.24) seems to agree exactly with its counterpart in [8]. More importantly, the value obtained here for l3 in (3.12) agrees exactly with the result in [7]. After taking account of differences in scaling, agreement with [6] is obtained, as well.…”

“…The second solution contains a generalized eigenvector of the form 1)t + t, and it must be included in order to deal with arbitrary initial conditions. (In [7], the constant multiplying this solution was set to zero, which unnecessarily restricts the results. )…”

“…This brings us finally to the topic of direct resonance, a term coined by Akylas and Benney [9] to describe the nonlinear consequences when the frequencies of two modes in a linearized problem coincide or become close. There will, of course, be a linear growth with time of the basic disturbance, but it is equally significant that the behavior of higher-order terms reveals the appropriate time scale for finite-amplitude effects to be el/ 2 t, rather than et as is usually supposed (e.g., in [6] and [7]). Hence, the basic disturbance reaches a magnitude of O( e -1/2), whereas it is usually taken to be 0(1).…”

“…(1.1) with the damping term multiplied by y absent. Magnan and Reiss [7], noting inconsistencies in the analysis of [6], reconsidered the problem by employing the Poincare-Lindstedt method to derive the Duffing equation and also investigated the stability of its periodic solutions to three-dimensional perturbations.…”

Comment on ‘‘Bifurcations from periodic solution in a simplified model of two-dimensional magnetoconvection,’’ by N. Bekki and T. Karakisawa [Phys. Plasmas 2, 2945 (1995)]

“…As explicitly stated by KP, the model uses all the modes generated from the PDEs to second order in the amplitude and no others, and hence the codimension-two reduction performed by KP using the fifth-order model is exact for the PDEs as well. This point was not understood by several authors, [13][14][15] in addition to BK. This convenient property is a consequence of the boundary conditions used in all these papers and does not carry over to other boundary conditions ͑for which one would indeed have to work directly with the PDEs in order to get the correct normal form coefficients͒.…”