Periodic solutions of differential-algebraic equations

Yingjie Bi; Siyu Liu; Yong Li

doi:10.3934/dcdsb.2019232

Article Contents

2020, Volume 25, Issue 4: 1383-1395. Doi: 10.3934/dcdsb.2019232

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Periodic solutions of differential-algebraic equations

a.
School of Mathematics, Jilin University, Changchun 130012, China
b.
School of Public Health, Jilin University, Changchun 130021, China
c.
School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

^* Corresponding author: Yong Li
^* Corresponding author: Yong Li

Received: June 30, 2018

Revised: April 30, 2019

Early access: November 2019

Published: April 2020

This work was completed with the support by National Basic Research Program of China Grant 2013CB834100, NSFC Grant 11571065, NSFC Grant 11171132 and NSFC Grant 11201173

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Abstract

In this paper, we study the existence of periodic solutions for a class of differential-algebraic equation

$ \begin{equation} \nonumber h'(t, x) = f(t, x), \; \; ' = \dfrac{d}{{dt}}, \end{equation} $

where $ h(t, x) = A(t)x(t) $, $ h(t, x) $ and $ f(t, x) $ are $ T $-periodic in first variable. Via the topological degree theory, and the method of guiding functions, some existence theorems are presented. To our knowledge, this is the first approach to periodic solutions of differential-algebraic equations. Some examples and numerical simulations are given to illustrate our results.

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Mathematics Subject Classification: Primary: 65L80, 34B15; Secondary: 55M25.

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Figure 1. (a) The periodic solution of system (41). (b) The trajectory of particle motion of system (41)

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Figure 2. (a) The periodic solution of system (45). (b) The trajectory of particle motion of system (45)

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Figure 3. (a) The periodic solution of system (47) with $ z = x^{2}-y^{2} $. (b) The trajectory of particle motion of system (47) with $ z = x^{2}-y^{2} $

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Figure 4. (a) The periodic solution of system (47) with $ z = x^{2}+y^{2} $. (b) The trajectory of particle motion of system (47) with $ z = x^{2}+y^{2} $

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