Yıl 2018, Cilt 47, Sayı 3, Sayfalar 521 - 538 2018-06-01
| | | |

## Existence of periodic solutions for a mechanical system with piecewise constant forces

#### Duygu Aruğaslan [1] , Nur Cengiz [2]

##### 39 61

In this study, we consider spring-mass systems subjected to piecewise constant forces. We investigate sufficient conditions for the existence of periodic solutions of homogeneous and nonhomogeneous damped spring-mass systems with the help of the Floquet theory. In addition to determining conditions for the existence of periodic solutions, stability analysis is performed for the solutions of the homogeneous system. The
Floquet multipliers are taken into account for the stability analysis [3]. The results are stated in terms of the parameters of the systems. These results are illustrated and supported by simulations for different values of the parameters.

Generalized piecewise constant forces, spring-mass system, stability and periodic solutions
• Aftabizadeh, A.R., Wiener, J. and Xu, J.-M. Oscillatory and periodic solutions of delay differential equations with piecewise constant argument, Proc. Amer. Math. Soc. 99, 673 679, 1987.
• Akhmet, M.U. Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. 66, 367383, 2007.
• Akhmet, M.U. Nonlinear Hybrid Continuous/Discrete Time Models, Amsterdam, Paris, Atlantis Press, 2011.
• Akhmet, M.U. On the integral manifolds of the differential equations with piecewise constant argument of generalized type, Proceedings of the Conference on Dierential and Dierence Equations at the Florida Institute of Technology, August 1-5, 2005, Melbourne, Florida, Editors: R.P. Agarval and K. Perera, Hindawi Publishing Corporation, 1120, 2006.
• Akhmet, M.U. Quasilinear retarded differential equations with functional dependence on piecewise constant argument, Communications On Pure And Applied Analysis 13 (2), 929 947, 2014.
• Akhmet, M.U. Stability of differential equations with piecewise constant arguments of gen- eralized type, Nonlinear Anal. 68, 794803, 2008.
• Akhmet, M.U. and Aru§aslan, D. Lyapunov-Razumikhin method for differential equations with piecewise constant argument, Discrete and Continuous Dynamical Systems, Series A 25 (2), 457466, 2009.
• Akhmet, M.U., Aru§aslan, D. and Liu, X. Permanence of nonautonomous ratio-dependent predator-prey systems with piecewise constant argument of generalized type, Dynamics of Continuous Discrete and Impulsive Systems Series A. Mathematical Analysis 15 (1), 3751, 2008.
• Akhmet, M.U., Aru§aslan, D. and Ylmaz, E. Stability analysis of recurrent neural networks with piecewise constant argument of generalized type, Neural Networks 23, 805811, 2010.
• Akhmet, M.U., Aru§aslan, D. and Ylmaz, E. Stability in cellular neural networks with a piecewise constant argument, Journal of Computational and Applied Mathematics 233, 23652373, 2010.
• Akhmet, M.U., Öktem, H., Pickl, S.W. and Weber, G.-W. An anticipatory extension of Malthusian model, CASYS'05-Seventh International Conference, AIP Conference Proceedings 839, 260264, 2006.
• Busenberg, S. and Cooke, K.L. Models of vertically transmitted diseases with sequential- continuous dynamics, Nonlinear Phenomena in Mathematical Sciences, Academic Press, New York, 179187, 1982.
• Chiu, K.-S., Pinto, M. Periodic solutions of differential equations with a general piecewise constant argument and applications, Electron. J. Qual. Theory Dier. Equ. 46, 19 pp, 2010.
• Chiu, K.-S., Pinto, M. Variation of parameters formula and Gronwall inequality for differ- ential equations with a general piecewise constant argument, Acta Math. Appl. Sin. Engl. Ser. 27 (4), 561568, 2011.
• Cooke, K.L. and Wiener, J. Retarded differential equations with piecewise constant delays, J. Math. Anal. Appl. 99, 265297, 1984.
• Dai, L. and Singh, M.C. On oscillatory motion of spring-mass systems subjected to piecewise constant forces, Journal of Sound and Vibration 173 (2), 217231, 1994.
• Gopalsamy, K. Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, 1992.
• Györi, I. On approximation of the solutions of delay differential equations by using piecewise constant argument, Int. J., Math. Math. Sci., 14, 111126, (1991).
• Gopalsamy, K. and Liu, P. Persistence and global stability in a population model, J. Math. Anal. Appl. 224, 5980, 1998.
• Gyori, I. and Ladas, G. Oscillation Theory of Delay Differential Equations with Applications, Oxford University Press, New York, 1991.
• Matsunaga, H., Hara, T. and Sakata, S. Global attractivity for a logistic equation with piecewise constant argument, Nonlinear Dierential Equations Appl. 8, 4552, 2001.
• Muroya, Y. Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270, 602635, 2002.
• Papaschinopoulos, G. On the integral manifold for a system of differential equations with piecewise constant argument, J. Math. Anal. Appl. 201, 7590, 1996.
• Seifert, G. Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Dierential Equations 164, 451458, 2000.
• Shah, S.M. and Wiener, J. Advanced differential equations with piecewise constant argument deviations, Int. J. Math. Math. Sci. 6, 671703, 1983.
• Shen, J.H. and Stavroulakis, I.P. Oscillatory and nonoscillatory delay equation with piece- wise constant argument, J. Math. Anal. Appl. 248, 385401, 2000.
• Wang, G. Periodic solutions of a neutral differential equation with piecewise constant argu- ments, J. Math. Anal. Appl. 326, 736747, 2007.
• Wang, Z. and Wu, J. The stability in a logistic equation with piecewise constant arguments, Differential Equations Dynam. Systems 14, 179193, 2006.
• Wiener, J. Generalized Solutions of Functional Differential Equations, World Scientic, Singapore, 1993.
• Wiener, J. and Cooke, K.L. Oscillations in systems of differential equations with piecewise constant argument, J. Math. Anal. Appl. 137, 221239, 1989.
• Wiener, J. and Lakshmikantham, V. A damped oscillator with piecewise constant time delay, Nonlinear Stud. 7, 7884, 2000.
• Xia, Y., Huang, Z. and Han, M. Existence of almost periodic solutions for forced perturbed systems with piecewise constant argument, J. Math. Anal. Appl. 333, 798816, 2007.
• Yang, X. Existence and exponential stability of almost periodic solution for cellular neural networks with piecewise constant argument, Acta Math. Appl. Sin. 29, 789800, 2006.
Birincil Dil en Matematik Matematik Yazar: Duygu Aruğaslan (Sorumlu Yazar) Yazar: Nur Cengiz
 Bibtex @araştırma makalesi { hujms439928, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe Üniversitesi}, year = {2018}, volume = {47}, pages = {521 - 538}, doi = {}, title = {Existence of periodic solutions for a mechanical system with piecewise constant forces}, key = {cite}, author = {Aruğaslan, Duygu and Cengiz, Nur} } APA Aruğaslan, D , Cengiz, N . (2018). Existence of periodic solutions for a mechanical system with piecewise constant forces. Hacettepe Journal of Mathematics and Statistics, 47 (3), 521-538. Retrieved from http://dergipark.gov.tr/hujms/issue/38121/439928 MLA Aruğaslan, D , Cengiz, N . "Existence of periodic solutions for a mechanical system with piecewise constant forces". Hacettepe Journal of Mathematics and Statistics 47 (2018): 521-538 Chicago Aruğaslan, D , Cengiz, N . "Existence of periodic solutions for a mechanical system with piecewise constant forces". Hacettepe Journal of Mathematics and Statistics 47 (2018): 521-538 RIS TY - JOUR T1 - Existence of periodic solutions for a mechanical system with piecewise constant forces AU - Duygu Aruğaslan , Nur Cengiz Y1 - 2018 PY - 2018 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 521 EP - 538 VL - 47 IS - 3 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2017 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Existence of periodic solutions for a mechanical system with piecewise constant forces %A Duygu Aruğaslan , Nur Cengiz %T Existence of periodic solutions for a mechanical system with piecewise constant forces %D 2018 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 47 %N 3 %R %U ISNAD Aruğaslan, Duygu , Cengiz, Nur . "Existence of periodic solutions for a mechanical system with piecewise constant forces". Hacettepe Journal of Mathematics and Statistics 47 / 3 (Haziran 2018): 521-538.