Dynamical Bifurcation of the One Dimensional Modified Swift-Hohenberg Equation

Abstract

Abstract. In this paper, we study the dynamical bifurcation of the modified Swift-Hohenberg equation on a periodic interval as the system control parameter crosses through a critical number. This critical number depends on the period. We show that there happens the pitchfork bifurcation under the spatially even periodic condition. We also prove that in the general periodic condition the equation bifurcates to an attractor which is homeomorphic to a circle and consists of steady states solutions.

Bifurcation and final patterns of a modified Swift-Hohenberg equation

“…The main result of this paper is to verify the dynamic bifurcation of the MSHE defined in H. For the case (2.4), the MSHE has one dimensional center manifold when α is slightly bigger than α N . In this case, the bifurcation phenomena of the MSHE for the case (2.4) was well established in [3]. It turns out that the MSHE has a pitchfork bifurcation.…”

“…Then, by the second equation of (3O(β) with 0 < β = α − 1 1. So, we can write the first equation of(3Multiplying the first equation of (3.11) by y 0 , we get 4 O(β 5/2 ) = 0. (3.15)…”

Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation

“…There have been lots of research on the subject of MSHEs. Roughly speaking, these works mainly include three aspects: attractors ( [22, 36, 54, 56]) and the regularity ( [41]), bifurcations of solutions ( [5,6,55]) and optimal control ( [12,42]). What is more, for the nonautonomous MSHE, Wang, Yang and Duan presented in [49] a lower number of recurrent solutions by topological methods (see more in [34,[46][47][48]); Wang, Zhang and Zhao studied the existence of invariant measures and statistical solutions in [50].…”

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

“…Later, it has also played a valuable role extensively in the study of plasma confinement in toroidal devices, 5 viscous film flow, lasers, 6 and pattern formation. 7 In the previous work, most attention was paid to the existence of attractors (global attractor, 8,9 uniform attractor, 10 pullback attractor, 11,12 and random attractor [12][13][14] ), bifurcations (dynamical bifurcations 15,16 and nontrivial-solution bifurcations 17 ), and optimal control [18][19][20][21] of different types of modified Swift-Hohenberg equations. Wang et al presented in their work 22 a lower number of recurrent solutions for the nonautonomous case by topological methods (see more in other works [23][24][25][26] ).…”