• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1241-1252
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• 2015

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.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1335-1364
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• 2023

The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the 2-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in 1 dimension [13]. As compared to [18], showing that the stability of the rolls in the 2-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.257-279
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• 2023

Applying the Lyapunov-Schmidt reduction, we consider spectral stability of small amplitude stationary periodic solutions bifurcating from an equilibrium of the generalized Swift-Hohenberg equation. We follow the mathematical framework developed in [15, 16, 19, 23] to construct such periodic solutions and to determine regions in the parameter space for which they are stable by investigating the movement of the spectrum near zero as parameters vary.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.923-937
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• 2012

In this paper we study the dynamic bifurcation of the Swift-Hohenberg equation on a periodic cell ${\Omega}=[-L,L]$. It is shown that the equations bifurcates from the trivial solution to an attractor $\mathcal{A}_{\lambda}$ when th control parameter ${\lambda}$ crosses the critical value. In the odd periodic case $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$ and consists of eight singular points and thei connecting orbits. In the periodic case, $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$, an contains a torus and two circles which consist of singular points.