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Dynamic bifurcation of a modified Kuramoto-Sivashinsky equation with higher-order nonlinearity

Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity μ(ux)puxx are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.