Global attraction in a system of delay differential equations via compact and convex sets

Sebastián Buedo-Fernández

doi:10.3934/dcdsb.2020056

Article Contents

2020, Volume 25, Issue 8: 3171-3181. Doi: 10.3934/dcdsb.2020056

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Global attraction in a system of delay differential equations via compact and convex sets

Sebastián Buedo-Fernández

Departamento de Estatística, Análise Matemática e Optimización and Instituto de Matemáticas, Universidade de Santiago de Compostela, Facultade de Matemáticas, Campus Vida, 15782 Santiago de Compostela, Spain

Received: May 2019

Early access: February 2020

Published: August 2020

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Abstract

We provide sufficient conditions for a concrete type of systems of delay differential equations (DDEs) to have a global attractor. The principal idea is based on a particular type of global attraction in difference equations in terms of nested, convex and compact sets. We prove that the solutions of the system of DDEs inherit the convergence to the equilibrium from an associated discrete dynamical system.

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Mathematics Subject Classification: Primary: 34K20, 37C70, 52A20.

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Figure 1. The origin is not a strong attractor

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Figure 2. A possible set $ Q_{x, v, \varepsilon} $ is represented in gray. Distances are pointed out with dashed green lines. A particular $ v^* $ satisfying the hypotheses of the last assertion of Lemma 3.1 is also depicted (color figure online)

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Figure 3. Possible behaviour of $ x(t, \phi) $ (blue). The boundary of the set $ K_2 $ is represented in black. The boundaries of $ K_{2, \mu} $, for some values $ \mu>1 $ are represented in grey. The blue arrow represents $ x'(t, \phi) $, which "points to the interior" of a $ K_{2, \mu} $. $ f(K_2) $ is represented in red. The equilibrium $ z_* $ is depicted as a point inside $ f(K_2) $ (color figure online)

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