A remark on omega limit sets for non-expansive dynamics
arxiv(2024)
摘要
In this paper, we study systems of time-invariant ordinary differential
equations whose flows are non-expansive with respect to a norm, meaning that
the distance between solutions may not increase. Since non-expansiveness (and
contractivity) are norm-dependent notions, the topology of ω-limit sets
of solutions may depend on the norm. For example, and at least for systems
defined by real-analytic vector fields, the only possible ω-limit sets
of systems that are non-expansive with respect to polyhedral norms (such as
ℓ^p norms with p =1 or p=∞) are equilibria. In contrast, for
non-expansive systems with respect to Euclidean (ℓ^2) norm, other limit
sets may arise (such as multi-dimensional tori): for example linear harmonic
oscillators are non-expansive (and even isometric) flows, yet have periodic
orbits as ω-limit sets. This paper shows that the Euclidean linear case
is what can be expected in general: for flows that are contractive with respect
to any strictly convex norm (such as ℓ^p for any p≠1,∞), and if
there is at least one bounded solution, then the ω-limit set of every
trajectory is also an omega limit set of a linear time-invariant system.
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