[Paper Review] Attractor Bifurcation and Final Patterns of the N-Dimensional and Generalized Swift-Hohenberg Equations

Masoud Yari|ArXiv.org|Feb 8, 2008

Nonlinear Dynamics and Pattern Formation14 references3 citations

TL;DR

This paper investigates attractor bifurcation and final patterns in the n-dimensional and generalized Swift-Hohenberg equations under various boundary conditions using a novel attractor bifurcation theory. It proves that when the control parameter λ crosses the critical value λ_c, the system bifurcates from the trivial solution to an attractor 𝒜_λ with dimension between m−1 and m, where m is the multiplicity of the first eigenvalue of (I+Δ)², and establishes the asymptotic stability of the bifurcated attractor.

ABSTRACT

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a one- dimensional domain $(0,L)$. I will also study the bifurcation and stability of patterns in the n-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions. It is shown that each equation bifurcates from the trivial solution to an attractor, when the control parameter crosses the principal eigenvalue of the linearized equation. The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.

Motivation & Objective

• To analyze the bifurcation and asymptotic behavior of solutions in the 1D Swift-Hohenberg equation with Dirichlet and odd-periodic boundary conditions.
• To study the stability and structure of final patterns in the n-dimensional Swift-Hohenberg equation under odd-periodic and periodic boundary conditions.
• To determine the precise dimension and topological structure of the attractor 𝒜_λ bifurcated from the trivial solution at λ = λ_c.
• To establish the asymptotic stability of the bifurcated attractor in the phase space outside the stable manifold of the trivial solution.

Proposed method

• Application of the new attractor bifurcation theory developed by Ma and Wang to analyze the bifurcation from the trivial solution at λ = λ_c.
• Reduction of the equation to its center manifold via Lyapunov-Schmidt reduction to study nonlinear interactions near the critical parameter.
• Use of energy estimates and differential inequality techniques to prove global asymptotic stability of the trivial solution for λ ≤ λ_c.
• Computation of eigenvalues and eigenfunctions of the operator (I+Δ)² to determine λ_c and the multiplicity m of the first eigenvalue.
• Analysis of the bifurcation equations derived from center manifold reduction to identify the number and nature of steady-state solutions.
• Topological characterization of the attractor 𝒜_λ using homology and homeomorphism, showing it is homotopic to S^{n−1} or homeomorphic to S¹ for n=2.

Experimental results

Research questions

• RQ1 What is the precise structure and dimension of the attractor 𝒜_λ that bifurcates from the trivial solution when λ crosses λ_c in the 1D Swift-Hohenberg equation?
• RQ2 How does the attractor bifurcate under odd-periodic and periodic boundary conditions in the n-dimensional case, and what is its topological type?
• RQ3 What is the number and stability type of steady-state solutions contained in the bifurcated attractor for λ > λ_c?
• RQ4 How does the attractor 𝒜_λ behave asymptotically, and what is its global stability property in the phase space?

Key findings

• For λ > λ_c, the 1D Swift-Hohenberg equation with Dirichlet or odd-periodic boundary conditions bifurcates to exactly two steady-state solutions.
• With periodic boundary conditions in 1D, the bifurcated attractor 𝒜_λ is homeomorphic to S¹.
• In the n-dimensional case with odd-periodic boundary conditions, the attractor 𝒜_λ contains exactly 2^n regular steady-state solutions.
• For periodic boundary conditions in n-dimensional space (n ≤ 3), the attractor 𝒜_λ contains a torus 𝕋^n consisting of steady-state solutions.
• The attractor 𝒜_λ bifurcated at λ_c has dimension between m−1 and m, where m is the algebraic multiplicity of the first eigenvalue of (I+Δ)².
• The trivial solution u=0 is globally asymptotically stable for λ ≤ λ_c, and the attractor 𝒜_λ attracts all solutions outside the stable manifold of codimension m.