DYNAMIC BIFURCATION OF THE PERIODIC SWIFT-HOHENBERG EQUATION

Abstract

In this paper we study the dynamic bifurcation of the Swift- Hohenberg equation on a periodic cell = ( L,L). It is shown that the equations bifurcates from the trivial solution to an attractor Awhen the control parametercrosses the critical value. In the odd periodic case, Ais homeomorphic to S1 and consists of eight singular points and their connecting orbits. In the periodic case, Ais homeomorphic to S1, and contains a torus and two circles which consist of singular points. Fluid motion driven by the thermal gradients is common in nature, especially in geophysical flows such as the atmosphere, the oceans, the mantle of the earth, and the interior of stars. A typical model for fluid convection is the Rayleigh- Benard convection describing a fluid placed between flat horizontal plates such that the lower plate is maintained at a temperature above the upper plate temperature. Due to the thermal expansion, the fluid near the lower plate is less dense and become unstable in the gravitational field. Eventually, we encounter an instability at a finite wave length giving a spatio-temporal pattern formation. The mathematical model for the Rayleigh-Benard convection comes from the equation of fluid dynamics in the Boussinesq approximation which involves the Navier-Stokes equations coupled with the temperature equation. In 1977, Swift and Hohenberg derived in (14) that when the Rayleigh number is near the onset of the convection, the Rayleigh-Benard convection model may be approximated by the following Swift-Hohenberg equation (SHE)