Figure 1.
(a) The periodic solution of system (41). (b) The trajectory of particle motion of system (41)
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Ergodicity of non-autonomous discrete systems with non-uniform expansion
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 |
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) $ |
| $ f(t, x) $ |
| $ T $ |
Mathematics Subject Classification:
Primary: 65L80, 34B15; Secondary: 55M25.
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. |
show all references
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. |
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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