Introduction
In view of the basic approximation theory in Chapter 2, nonlinear dynamical systems can be approximated uniformly on compact intervals by linear, time-varying systems. It is therefore important to study the general questions of existence, uniqueness, etc. for dynamical systems of this type. In this chapter we shall consider the general theory of linear time-varying dynamical systems first from the point of view of existence and uniqueness, and then we shall determine a number of explicit solutions, based on the theory of Lie algebras.
The remainder of the chapter is concerned, essentially with stability theory. After considering the classical theory, we shall introduce the ideas of Lyapunov numbers and describe Oseledec’s theorem on decomposition of the state-space into invariant subbundles, which generalises the hyperbolic splitting of the state-space for timeinvariant systems. Finally we shall consider the theory of exponential dichotomies and its generalisation to invariant subbundles.
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Tomás-Rodríguez, M., Banks, S.P. (2010). The Structure and Stability of Linear, Time-varying Systems. In: Linear, Time-varying Approximations to Nonlinear Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 411. Springer, London. https://doi.org/10.1007/978-1-84996-101-1_3
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DOI: https://doi.org/10.1007/978-1-84996-101-1_3
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