Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems

Abstract

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Grain boundaries in the Swift–Hohenberg equation

“…We can therefore apply the centre-manifold theorem (see for instance [7,Section 2.2]) and conclude that there exist three neighbourhoods of the origin U ⊂ X c , V ⊂ ( id − P c )Y, W ⊂ R and a map Ψ : U × W → V of class C m , for an arbitrary, but fixed m 1, such that the following hold. First, for any µ ∈ W, the bounded solutions of (2.3) with U (x) ∈ U × V, for all x ∈ R, lie on the centre manifold given by the graph of Ψ(·, µ), that is,…”

“…These tools have been extensively used to solve concrete problems arising in physics and natural sciences (e.g., see [7,11,15,16,20] and the references therein). Spatial dynamics goes back to the work of Kirchgässner [12].…”

“…It consists of studying a stationary problem by writing it as an evolutionary system in which an unbounded spatial coordinate plays the role of the time-like variable. The resulting evolutionary problem is typically ill-posed, but its small bounded solutions can be found by a centre-manifold reduction, which shows that this system is locally equivalent to a finite-dimensional reduced dynamical system (e.g., see [7,10,15,24]). Normal form theory provides an efficient way of studying this reduced system, by transforming it to a simpler system containing a minimal number of nonlinear terms (e.g., see [7,10,25]).…”

“…The resulting evolutionary problem is typically ill-posed, but its small bounded solutions can be found by a centre-manifold reduction, which shows that this system is locally equivalent to a finite-dimensional reduced dynamical system (e.g., see [7,10,15,24]). Normal form theory provides an efficient way of studying this reduced system, by transforming it to a simpler system containing a minimal number of nonlinear terms (e.g., see [7,10,25]). We emphasize that we do not rely on variational structure or the reflection symmetry u → −u: we exploit variational structure only at leading order, an artifact of the leading-order expansion, and we use the fact that quadratic terms of a centre-manifold expansion vanish.…”

Dislocations in an Anisotropic Swift-Hohenberg Equation

“…We are now ready to apply the center manifold theorem as stated for instance in [6]. We find three neighborhoods of the origin, U ⊂ R 4 , V * ⊂ ( id − P * )Y e , W ⊂ R 2 , and a map h * : U × W → V * of class C k , for an arbitrary, but fixed k 1, with estimates…”