Abstract
We investigate the dynamics of the parametrically-excited P.D.E.(1)∂2u∂t2-c2(∂2u∂x2+∂2u∂y2)+εβ∂u∂t+(∂+εγcost)u=εαu3 with Neumann boundary conditions on a rectangular region:∂u∂x=0forx=0,π and ∂u∂y=0fory=0,πμ where 0 < μ ≤ 1. Our approach involves expanding u(x, y, t) in a 3-term Fourier series truncation:(2)u=f0(t)+f1(t)cosx+f2(t)cosμy By substituting (2) into (1) we obtain a system of 3 coupled nonlinear Mathieu equations which we analyze using averaging in the neighborhood of 2 : 1 resonance.
By varying the parameters c and δ we obtain bifurcation curves which divide the cδ-plane into more than forty regions, each containing a distinct slow flow. Individual regions are found to differ from one another with respect to such features as the number and character of slow flow equilibria, and the presence or absence of a limit cycle. When interpreted in the original variable u, these regions account for a variety of patterns which may be classified as stationary, traveling or rotating.
This type of behavior is comparable to various experimental observations made by other investigators on vertically driven fluids or sand.
Dune field pattern formation and recent transporting winds in the Olympia Undae Dune Field, north polar region of Mars
