The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory [
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Figure 1. These vector fields show the general shape of the case ${\mathop{\rm Re}\nolimits} C > 0 $ and ${\mathop{\rm Re}\nolimits} D < 0 $. The vertical axis represents $ \rho_2 $ and the horizontal represents $ \rho_1 $. The darker line is given by $ \rho_2 = m_1 \rho_1 $ and the lighter line is given by $ \rho_2 = m_2 \rho_1 $
Figure 2. Plot of ${\mathop{\rm Re}\nolimits} \beta_n(0) $ for various $ \ell $. One can see from the figure that for large $ \ell $, the maximas over $ n \in \mathbb{Z} $ are attained at one or two pairs of conjugate eigenvalues. For sufficiently small $ \ell $, $ \beta_0(0) $ will be the maximum eigenvalue
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These vector fields show the general shape of the case
Plot of
A visual of the partition.
Phase diagram for
Phase diagram for
The phase diagram at
Forward in time trajectories (left) tending towards the stable periodic orbit, and backward in time trajectories (right) tending towards the unstable periodic orbit
The blue line is the numerical approximations of the radius of the limit cycles as a function of