Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion

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

Figure 3.  A visual of the partition. $ \mathcal I_1 $, $ \mathcal I_2 $, $ \mathcal I_3 $, and $ \mathcal I_4 $ are encoded by the different shades

Figure 4.  Phase diagram for $ k = 2 $

Figure 5.  Phase diagram for $ k = 6 $

Figure 6.  The phase diagram at $ \ell = 2\pi $

Figure 7.  Forward in time trajectories (left) tending towards the stable periodic orbit, and backward in time trajectories (right) tending towards the unstable periodic orbit

Figure 8.  The blue line is the numerical approximations of the radius of the limit cycles as a function of $ \lambda $, and the lighter line is the analytical limiting behavior as $ \lambda \to 0 $