Dynamic transitions of generalized Kuramoto-Sivashinsky equation

In this article, we study the dynamic transition for the one dimensional generalized Kuramoto-Sivashinsky equation with periodic condition. It is shown that if the value of the dispersive parameter $\nu$ is strictly greater than $\nu^{\ast}$, then the transition is Type-I (continuous) and the bifurcated periodic orbit is an attractor as the control parameter $\lambda$ crosses the critical value $\lambda_0$. In the case where $\nu$ is strictly less than $\nu^{\ast}$, then the transition is Type-II (jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter $\lambda$ crosses the critical value $\lambda_0$. The value of $\nu^{\ast}$ is also calculated in this paper.