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Linear Difference Equations

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2002)

Abstract

Already in an analysis of nonlinear systems, linear problems frequently occur in form of variational equations (cf. Corollary 2.3.11) when linearizing along a given reference solution. Provided this solution does lack a specific time-dependence (e.g., (almost) periodicity, or being convergent), then the resulting variational equations are nonautonomous in the general sense. A further reason for the importance of linear equations is that the difference of two solutions to a nonlinear problem always solves a linear homogeneous difference equation, as follows from an easy application of the mean value theorem (cf. [295, p. 341, Theorem 4.2]). Finally, a solid linear theory opens the door to utilize appropriate perturbation results and to generalize a global geometric theory to semilinear equations. In this chapter, we present the corresponding theory.

Keywords

  • Vector Bundle
  • Evolution Operator
  • Invariant Projector
  • Exponential Dichotomy
  • Floquet Multiplier

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Correspondence to Christian Pötzsche .

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Pötzsche, C. (2010). Linear Difference Equations. In: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics(), vol 2002. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14258-1_3

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