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Unformatted text preview: of t. For details
of the proof see Meiss. A linear system is asymptotically linearly stable if all of its solutions approach 0 as
Theorem 2.10 (Asymptotic Linear Stability).
eigenvalues of have negative real parts. for all . if and only if all Proof. If all eigenvalues have negative real part, lemma 2.9 implies
eigenvalue has positive real part, then there is a straight line solution,
an eigenvector of , that grows without bound. If there is an eigenvalue
with zero real
part, the solutions in this subspace have terms of the form
that do not go to zero.□ 12
Chapter 2 part B A system with no center subspace is hyperbolic. Lemma 2.9 and Theorem 2.10 describe
properties of these systems that depend only on the signs of their eigenvalues.
In contrast, the stability of systems with a center subspace can be affected by the nilpotent part of
The proof of theorem 2.10 suggests why these systems cannot be asymptotically stable.
Solutions of the 2D center described in section 2.2 are bounded. However, center systems with
nonzero nilpotent parts have solutions that a...
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- Fall '14