American Institute of Mathematical Sciences

April  2020, 25(4): 1383-1395. doi: 10.3934/dcdsb.2019232

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

Received  July 2018 Revised  May 2019 Published  April 2020 Early access  November 2019

Fund Project: 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
 $$$\nonumber h'(t, x) = f(t, x), \; \; ' = \dfrac{d}{{dt}},$$$
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.
Citation: Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1383-1395. doi: 10.3934/dcdsb.2019232
References:
 [1] G. Ali, A. Bartel and N. Rotundo, Index-2 elliptic partial differential-algebraic models for circuits and devices, Journal of Mathemtical Analysis and Applications, 423 (2015), 1348-1369.  doi: 10.1016/j.jmaa.2014.10.065. [2] R. Altmann, Index reduction for operator differential-algebraic equations in elastodynamics, Zeitschrift f$\ddot{u}$r Angewandte Mathematik und Mechanik, 93 (2013), 648-664.  doi: 10.1002/zamm.201200125. [3] U. M. Ascher and P. Lin, Sequential regularization methods for higher DAEs with constraint singularities: the linear index-$2$ case, SIAM Journal on Numerical Analysis, 33 (1996), 1921-1940.  doi: 10.1137/S0036142993253254. [4] P. Benner, P. Losse and V. Mehrmann, Numerical Linear Algebra Methods for Linear Differential-Algebraic Equations, Surveys in Differential-algebraic Equations, 3. Springer, Cham, 2015. [5] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics in Applied Mathematics, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971224. [6] S. L. Campbell and P. Kunkel, On the numerical treatment of linear-quadratic optimal control problems for general linear time-varying differential-algebraic equations, Journal of Computational and Applied Mathematics, 242 (2013), 213-231.  doi: 10.1016/j.cam.2012.10.011. [7] S. Campbell and P. Kunkel, Solving higher index DAE optimal control problems, Numer. Algebra Control Optim., 6 (2016), 447-472.  doi: 10.3934/naco.2016020. [8] R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, Journal of Differential Equations, 26 (1977), 200-222.  doi: 10.1016/0022-0396(77)90191-7. [9] J. K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, Journal of Differential Equations, 15 (1974), 295-307.  doi: 10.1016/0022-0396(74)90081-3. [10] M. M. Hosseini, Numerical solution of linear high-index DAEs, Computational Science and its Applications—ICCSA 2004, Part III, Lecture Notes in Comput. Sci., Springer, Berlin, 3045 (2004), 676-685.  doi: 10.1007/978-3-540-24767-8_71. [11] M. Hosseini, An efficient index reduction method for differential-algebraic equations, Global Journal of Pure and Applied Mathematics, 3 (2007), 113-124. [12] M. A. Krasnosel'skii, Translation Along Trajectories of Differential Equations, American Mathematics Society, Providence, 1938. [13] M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. [14] Y. Li, X. G. Lu and Y. Su, A homotopy method of finding periodic solutions for ordinary differential equations from the upper and lower solutions, Nonlinear Analysis. Theory, Methods & Applications, 24 (1995), 1027-1038.  doi: 10.1016/0362-546X(94)00129-6. [15] Y. Li and X. R. Lü, Continuation theorems for boundary value problems, Journal of Mathematical Analysis and Applications, 190 (1995), 32-49.  doi: 10.1006/jmaa.1995.1063. [16] J. Mawhin, Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1979. [17] J. Mawhin and J. R. Ward, Guiding-like functions for periodic or bounded solutions of ordinary differential equations, Discrete and Continuous Dynamical Systems. Series A, 8 (2002), 39-54.  doi: 10.3934/dcds.2002.8.39. [18] R. N. Methekar, V. Ramadesigan, J. C. Pirkle Jr. and V. R. Subramanian, A perturbation approach for consistent initialization of index-1 explicit differential-algebraic equations arising from battery model simulations, Computers and Chemical Engineering, 35 (2011), 2227-2234.  doi: 10.1016/j.compchemeng.2011.01.003. [19] D. L. Michels and M. Desbrun, A semi-analytical approach to molecular dynamics, Journal of Computational Physics, 303 (2015), 336-354.  doi: 10.1016/j.jcp.2015.10.009. [20] C. Pöll and I. Hafner, Index reduction and regularisation methods for multibody systems, IFAC-Papers OnLine, 48 (2015), 306-311.  doi: 10.1016/j.ifacol.2015.05.150. [21] Y. Pomeau, On the self-similar solution to the Euler equations for an incompressible fluid in three dimensions, Comptes Redus Mécanique, 346 (2018), 184-197.  doi: 10.1016/j.crme.2017.12.004. [22] P. Stechlinski, M. Patrascu and P. I. Barton, Nonsmooth differential-algebraic equations in chemical engineering, Computers and Chemical Engineerig, 114 (2018), 52-68.  doi: 10.1016/j.compchemeng.2017.10.031. [23] M. Takamatsu and S. Iwata, Index reduction for differential-algebraic equations by substitution method, Linear Algebra and Its Applications, 429 (2008), 2268-2277.  doi: 10.1016/j.laa.2008.06.025.

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References:
 [1] G. Ali, A. Bartel and N. Rotundo, Index-2 elliptic partial differential-algebraic models for circuits and devices, Journal of Mathemtical Analysis and Applications, 423 (2015), 1348-1369.  doi: 10.1016/j.jmaa.2014.10.065. [2] R. Altmann, Index reduction for operator differential-algebraic equations in elastodynamics, Zeitschrift f$\ddot{u}$r Angewandte Mathematik und Mechanik, 93 (2013), 648-664.  doi: 10.1002/zamm.201200125. [3] U. M. Ascher and P. Lin, Sequential regularization methods for higher DAEs with constraint singularities: the linear index-$2$ case, SIAM Journal on Numerical Analysis, 33 (1996), 1921-1940.  doi: 10.1137/S0036142993253254. [4] P. Benner, P. Losse and V. Mehrmann, Numerical Linear Algebra Methods for Linear Differential-Algebraic Equations, Surveys in Differential-algebraic Equations, 3. Springer, Cham, 2015. [5] K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Classics in Applied Mathematics, 14. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971224. [6] S. L. Campbell and P. Kunkel, On the numerical treatment of linear-quadratic optimal control problems for general linear time-varying differential-algebraic equations, Journal of Computational and Applied Mathematics, 242 (2013), 213-231.  doi: 10.1016/j.cam.2012.10.011. [7] S. Campbell and P. Kunkel, Solving higher index DAE optimal control problems, Numer. Algebra Control Optim., 6 (2016), 447-472.  doi: 10.3934/naco.2016020. [8] R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, Journal of Differential Equations, 26 (1977), 200-222.  doi: 10.1016/0022-0396(77)90191-7. [9] J. K. Hale and J. Mawhin, Coincidence degree and periodic solutions of neutral equations, Journal of Differential Equations, 15 (1974), 295-307.  doi: 10.1016/0022-0396(74)90081-3. [10] M. M. Hosseini, Numerical solution of linear high-index DAEs, Computational Science and its Applications—ICCSA 2004, Part III, Lecture Notes in Comput. Sci., Springer, Berlin, 3045 (2004), 676-685.  doi: 10.1007/978-3-540-24767-8_71. [11] M. Hosseini, An efficient index reduction method for differential-algebraic equations, Global Journal of Pure and Applied Mathematics, 3 (2007), 113-124. [12] M. A. Krasnosel'skii, Translation Along Trajectories of Differential Equations, American Mathematics Society, Providence, 1938. [13] M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. [14] Y. Li, X. G. Lu and Y. Su, A homotopy method of finding periodic solutions for ordinary differential equations from the upper and lower solutions, Nonlinear Analysis. Theory, Methods & Applications, 24 (1995), 1027-1038.  doi: 10.1016/0362-546X(94)00129-6. [15] Y. Li and X. R. Lü, Continuation theorems for boundary value problems, Journal of Mathematical Analysis and Applications, 190 (1995), 32-49.  doi: 10.1006/jmaa.1995.1063. [16] J. Mawhin, Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1979. [17] J. Mawhin and J. R. Ward, Guiding-like functions for periodic or bounded solutions of ordinary differential equations, Discrete and Continuous Dynamical Systems. Series A, 8 (2002), 39-54.  doi: 10.3934/dcds.2002.8.39. [18] R. N. Methekar, V. Ramadesigan, J. C. Pirkle Jr. and V. R. Subramanian, A perturbation approach for consistent initialization of index-1 explicit differential-algebraic equations arising from battery model simulations, Computers and Chemical Engineering, 35 (2011), 2227-2234.  doi: 10.1016/j.compchemeng.2011.01.003. [19] D. L. Michels and M. Desbrun, A semi-analytical approach to molecular dynamics, Journal of Computational Physics, 303 (2015), 336-354.  doi: 10.1016/j.jcp.2015.10.009. [20] C. Pöll and I. Hafner, Index reduction and regularisation methods for multibody systems, IFAC-Papers OnLine, 48 (2015), 306-311.  doi: 10.1016/j.ifacol.2015.05.150. [21] Y. Pomeau, On the self-similar solution to the Euler equations for an incompressible fluid in three dimensions, Comptes Redus Mécanique, 346 (2018), 184-197.  doi: 10.1016/j.crme.2017.12.004. [22] P. Stechlinski, M. Patrascu and P. I. Barton, Nonsmooth differential-algebraic equations in chemical engineering, Computers and Chemical Engineerig, 114 (2018), 52-68.  doi: 10.1016/j.compchemeng.2017.10.031. [23] M. Takamatsu and S. Iwata, Index reduction for differential-algebraic equations by substitution method, Linear Algebra and Its Applications, 429 (2008), 2268-2277.  doi: 10.1016/j.laa.2008.06.025.
(a) The periodic solution of system (41). (b) The trajectory of particle motion of system (41)
(a) The periodic solution of system (45). (b) The trajectory of particle motion of system (45)
(a) The periodic solution of system (47) with $z = x^{2}-y^{2}$. (b) The trajectory of particle motion of system (47) with $z = x^{2}-y^{2}$
(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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