Bifurcation analysis of a modified Swift–Hohenberg equation

Xiao, Qingkun; Gao, Hongjun

doi:10.1016/j.nonrwa.2010.05.028

Cited by 14 publications

(13 citation statements)

References 23 publications

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“…The additional term α|∇u| 2 , reminiscent of the Kuramoto-Sivashinsky equation, which arises in the study of various pattern formation phenomena involving some kind of phase turbulence or phase transition [17,26], breaks the symmetry u → −u. There are some wonderful works about the modified Swift-Hohenberg including the bifurcation analysis [31] and the proof of the existence of attractors. For example, [27] and [32] prove the existence of the global attractor, [23] shows the existence of the pullback attractor, [21] presents the existence of the uniform attractors, and [34] gives the numerical solution for this equation by Fourier spectral method.…”

Section: Yixia Shi and Maoan Hanmentioning

confidence: 99%

Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation

Shi¹,

Han²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper, we investigate the modified steady Swift-Hohenberg equation kuxxxx + 2kuxx + αu 2 x − εu + u 3 = 0, (1) where k > 0, α and ε are constants. We obtain a homoclinic solution about the dominant system which will be proved to deform a reversible homoclinic solution approaching to a periodic solution of the whole equation with the aid of the Fourier series expansion method, the fixed point theorem, the reversibility and adjusting the phase shift. And the homoclinic solution approaching to a periodic solution of the equation are called generalized homoclinic solution.

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Section: Yixia Shi and Maoan Hanmentioning

confidence: 99%

Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation

Shi¹,

Han²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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“…Moreover, it was proved in [20,21] that a global attractor exists in the class H k per for any k ≥ 2. In particular, a bifurcation analysis with respect to α was given in [24] for two dimensional MSHE, where the authors characterized the bifurcation by using Lyapunov-Schmidt reduction method on (0, 2π) × (0, 2π). In this paper, we carry out the bifurcation analysis of one dimensional problem (2.1) in detail by using center manifold reduction on an interval (−λ, λ).…”

Section: Statement Of Main Theorem Let Us Rewrite the One Dimensionamentioning

confidence: 99%

Bifurcation and final patterns of a modified Swift-Hohenberg equation

Choi

Han³

et al. 2017

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an S 1-attractor as the control parameter α passes through a critical numberα. Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of finite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.

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“…In the previous work, most attention was paid to the existence of attractors (global attractor [16,23], pullback attractor [15,27], uniform attractor [29] and random attractor [9,27]), bifurcations (dynamical bifurcations [5,6], nontrivial-solution bifurcations [28]) and optimal control ( [8,24,30,31]) of different types of modified Swift-Hohenberg equations. Xiao and Gao in [28] gave specific nontrivial bifurcation solutions that bifurcate from the trivial solution for the modified Swift-Hohenberg equations in rectangular domain in R 2 with periodic boundary value. Nevertheless, special solutions of (1.1), such as (almost) periodic, recurrent solutions, have been sparsely discussed until now.…”

Section: Introductionmentioning

confidence: 99%