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Finding Periodic Solutions of Differential Equations

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Periodic solutions of differential-algebraic equations

a, , Siyu Liu b, and Yong Li a,c,,

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

,

and

are

-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: Yingjie Bi, Siyu Liu, Yong Li. Periodic solutions of differential-algebraic equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. Hosseini, An efficient index reduction method for differential-algebraic equations, Global Journal of Pure and Applied Mathematics, 3 (2007), 113-124.   Google Scholar

[12]

M. A. Krasnosel'skii, Translation Along Trajectories of Differential Equations, American Mathematics Society, Providence, 1938. Google Scholar

[13]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Mawhin, Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. Hosseini, An efficient index reduction method for differential-algebraic equations, Global Journal of Pure and Applied Mathematics, 3 (2007), 113-124.   Google Scholar

[12]

M. A. Krasnosel'skii, Translation Along Trajectories of Differential Equations, American Mathematics Society, Providence, 1938. Google Scholar

[13]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer, Berlin, 1984. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Mawhin, Topological Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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Finding Periodic Solutions of Differential Equations

Source: https://www.aimsciences.org/article/doi/10.3934/dcdsb.2019232