# American Institute of Mathematical Sciences

• Previous Article
Global Hopf bifurcation of a population model with stage structure and strong Allee effect
• DCDS-S Home
• This Issue
• Next Article
Mathematical modeling about nonlinear delayed hydraulic cylinder system and its analysis on dynamical behaviors
October  2017, 10(5): 959-971. doi: 10.3934/dcdss.2017050

## Existence of periodic solutions of dynamic equations on time scales by averaging

 a. College of Mathematics, Jilin University, Changchun, 130012, China b. School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China c. State Key Laboratory of Automotive Simulation and control, Jilin University, Changchun, 130012, China

Received  December 2016 Revised  January 2017 Published  June 2017

Fund Project: The first author was supposed by NSFC (grant No. 11301541). The second author was supposed by National Basic Research Program of China (grant No. 2013CB834100), NSFC (grant No. 11571065), NSFC (grant No. 11171132). The fourth author was supposed by NSFC (grant No. 11201173)

In this paper, we study the existence of periodic solutions for perturbed dynamic equations on time scales. Our approach is based on the averaging method. Further, we extend some averaging theorem to periodic solutions of dynamic equations on time scales to $k-$th order in $\varepsilon$. More precisely, results of higher order averaging for finding periodic solutions are given via the topological degree theory.

Citation: Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050
##### References:
 [1] M. Adivar and Y. N. Raffoul, Existence results for periodic solutions of intego-dynamic equations on time scales, Ann. Mat. Pura. Appl., 188 (2009), 543-559. doi: 10.1007/s10231-008-0088-z. Google Scholar [2] N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945. Google Scholar [3] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Acad. Sci. , Kiev, 1934.Google Scholar [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications Birkh$ä$user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. Google Scholar [5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003.Google Scholar [6] M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379. Google Scholar [7] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22. doi: 10.1016/j.bulsci.2003.09.002. Google Scholar [8] P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139. Google Scholar [9] S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988.Google Scholar [10] Y. Li and C. Wang, Almost periodic functions on time scales and applications Discrete Dyn. Nat. Soc., 2011 (2011), Art. ID 727068, 20 pp. doi: 10.1155/2011/727068. Google Scholar [11] C. Lizama and J. G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal., 265 (2013), 2267-2311. doi: 10.1016/j.jfa.2013.06.013. Google Scholar [12] J. Llibre, D. D Novaes and M. A Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearly, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563. Google Scholar [13] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244. doi: 10.1016/j.bulsci.2014.08.011. Google Scholar [14] A. Slavík, Averaging dynamic equations on time scales, J. Math. Anal. Appl., 388 (2012), 996-1012. doi: 10.1016/j.jmaa.2011.10.043. Google Scholar [15] C. Wang and Y. Li, Affine-periodic solutions for nonlinear differential equations on time scales, Adv. Differ. Equ., 2015 (2015), 286-302. doi: 10.1186/s13662-015-0634-0. Google Scholar

show all references

##### References:
 [1] M. Adivar and Y. N. Raffoul, Existence results for periodic solutions of intego-dynamic equations on time scales, Ann. Mat. Pura. Appl., 188 (2009), 543-559. doi: 10.1007/s10231-008-0088-z. Google Scholar [2] N. N. Bogoliubov, On some Statistical Methods in Mathematical Physics Lzv. Akad. Nauk Ukr. SSR, Kiev, 1945. Google Scholar [3] N. N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Acad. Sci. , Kiev, 1934.Google Scholar [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications Birkh$ä$user, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. Google Scholar [5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh$ä$user, Boston, 2003.Google Scholar [6] M. Bohner and G. Sh. Guseinov, Partial differentiation on time scales, Dynam. Syst. and Appl., 13 (2004), 351-379. Google Scholar [7] A. Buica and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22. doi: 10.1016/j.bulsci.2003.09.002. Google Scholar [8] P. Fatou, Sur le movement d'un systáme soumis á Des forces á courte période, Bull. Soc. Math. Fance., 56 (1928), 98-139. Google Scholar [9] S. Hilger, Ein Ma$β$kettenkalk$ü$ mit Anwendung auf Zentrumsmanningfaltigkeiten PhD thesis, Universit$ä$t W$ü$rzburg, 1988.Google Scholar [10] Y. Li and C. Wang, Almost periodic functions on time scales and applications Discrete Dyn. Nat. Soc., 2011 (2011), Art. ID 727068, 20 pp. doi: 10.1155/2011/727068. Google Scholar [11] C. Lizama and J. G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal., 265 (2013), 2267-2311. doi: 10.1016/j.jfa.2013.06.013. Google Scholar [12] J. Llibre, D. D Novaes and M. A Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearly, 27 (2014), 563-583. doi: 10.1088/0951-7715/27/3/563. Google Scholar [13] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244. doi: 10.1016/j.bulsci.2014.08.011. Google Scholar [14] A. Slavík, Averaging dynamic equations on time scales, J. Math. Anal. Appl., 388 (2012), 996-1012. doi: 10.1016/j.jmaa.2011.10.043. Google Scholar [15] C. Wang and Y. Li, Affine-periodic solutions for nonlinear differential equations on time scales, Adv. Differ. Equ., 2015 (2015), 286-302. doi: 10.1186/s13662-015-0634-0. Google Scholar
 [1] Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609 [2] Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022 [3] Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161 [4] Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553 [5] B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558 [6] Paul Fife, Joseph Klewicki, Tie Wei. Time averaging in turbulence settings may reveal an infinite hierarchy of length scales. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 781-807. doi: 10.3934/dcds.2009.24.781 [7] Saroj Panigrahi. Liapunov-type integral inequalities for higher order dynamic equations on time scales. Conference Publications, 2013, 2013 (special) : 629-641. doi: 10.3934/proc.2013.2013.629 [8] Y. Gong, X. Xiang. A class of optimal control problems of systems governed by the first order linear dynamic equations on time scales. Journal of Industrial & Management Optimization, 2009, 5 (1) : 1-10. doi: 10.3934/jimo.2009.5.1 [9] Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 [10] Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267 [11] Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753 [12] Xiaoyan Lin, Xianhua Tang. Solutions of nonlinear periodic Dirac equations with periodic potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2051-2061. doi: 10.3934/dcdss.2019132 [13] Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765 [14] Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861 [15] Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661 [16] Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51 [17] Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 [18] Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002 [19] Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137 [20] Madalina Petcu, Roger Temam, Djoko Wirosoetisno. Averaging method applied to the three-dimensional primitive equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5681-5707. doi: 10.3934/dcds.2016049

2018 Impact Factor: 0.545