2024/04/02 by Alon Duvall, Eduardo D. Sontag, Duvall, Alon +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2404.02352
openalex publication_date 2024/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 ℓp norms with p =1 or p=∞) are equilibria. In contrast, for non-expansive systems with respect to Euclidean (ℓ2) 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 ℓp for any p\not=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.