Invariant foliations and stability in critical cases
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We construct invariant foliations of the extended state space for nonautonomous semilinear dynamic equations on measure chains (time scales). These equations allow a specific parameter dependence which is the key to obtain perturbation results necessary for applications to an analytical discretization theory of ODEs. Using these invariant foliations we deduce a version of the Pliss reduction principle. Copyright © 2006 Christian P¨otzsche. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction We begin with the motivation for this paper which has its origin in the classical theory of discrete dynamical systems. For this purpose, consider a C 1 -mapping f : U → ᐄ from an open neighborhood U ⊆ ᐄ of 0 into a Banach space ᐄ, which leaves the origin fixed ( f (0) = 0). It is a well-established result and can be traced back to the work of Perron in the early 1930s (to be more precise, it is due to his student Li, cf. [11]) that the origin is an asymptotically stable solution of the autonomous difference equation
xk+1 = f xk ,
(1.1)
if the spectrum Σ(D f (0)) is contained in the open unit circle of the complex plane. Similar results also hold for continuous dynamical systems (replace the open unit disc by the negative half-plane) or nonautonomous equations (replace the assumption on the spectrum by uniform asymptotic stability of the linearization). In a time scales setting of dynamic equations these questions are addressed in [4] (for scalar equations), [9] (equations in Banach spaces), and easily follow from a localized version of Theorem 2.3(a) below. Such considerations are usually summarized under the phrase principle of linearized stability, since the stability properties of the linear part dominate the nonlinear equation locally. Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 57043, Pages 1–19 DOI 10.1155/ADE/2006/57043
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Stability in critical cases
Significantly more interesting is the generalized situation when Σ(D f (0)) allows a decomposition into disjoint spectral sets Σs , Σc , where Σs is contained in the open unit disc, but Σc lies on its boundary. Then nonlinear effects enter the game and the center manifold theorem applies (cf., e.g., [8]): there exists a locally invariant submanifold R0 ⊆ ᐄ which is graph of a C 1 -mapping r0 over an open neighborhood of 0 in (P), where P ∈ ᏸ(ᐄ) is the spectral projector associated with Σc . Beyond that, the stability properties of the trivial solution to (1.1) are fully determined by those of
pk+1 = P f pk + r0 pk .
(1.2)
The advantage we obtained from this is that (1.2) is an equation in the lower-dimensional subspace (P) ⊆ ᐄ. This is known as the reduction principle. From the immense literature we only cite [13]—the pathbreaking paper in the framework of finite-dimensional ODEs. The paper at hand has two primary goals. (1) It can be c
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