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# Periodic solutions of differential-algebraic equations

• * Corresponding author: Yong Li

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

• 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.

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

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

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

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 $z = x^{2}-y^{2}$

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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