ON BOUNDEDNESS AND GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF DUFFING EQUATION
Keywords:
Boundedness, Duffing equation, Global asymptotic stability
Abstract
This paper investigated boundedness and global asymptotic stability of solutions of a class of certain second order nonlinear differential equation with damping using the Lyapunov second order and eigenvalue approach. Through the exploits of Schwartz inequality and assumptions on the inhomogeneous part, solutions of the Duffing equation were bounded and the equilibrium point was global asymptotically stable. Response to damping revealed that the damping effect was not negligible thereby reducing oscillations. Application of our results can be seen in the construction of door net where the trajectory returns to equilibrium as fast as possible. Furthermore, Mathcad software was used to analyze the behavior of the system which extends some results in literature.
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References
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[21] Tian, D., Ain, Q. T., Anjum, N.: Fractal N/MEMS from Pull-in Instability to Pull-in Stability. Journal of Fractals, 29(2):2150030 (2020)
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[23] Ortega, R., Alonso, J. M.: Boundedness and Asymptotic Stability of a Forced and Damped Oscillator. Nonlinear Analysis Theory, Method and Applications. 25(3): 297 - 309 (1995)
[24] Tung, C., Tung, E.: On the Asymptotic Behaviour of Solutions of Certain Second Order Differential Equations. Journal of the Franklin Institute. 344(5):391 - 398 (2007).
[25] Eze, E. O., Ajah, R. O.: Application of Lyapunov and Yoshizawa’s theorems on Stability, Asymptotic Stability, Boundaries and periodicity of solutions Duffing Equation: Asian Journal of Applied Science, 2(6):970 - 975 (2014)
[26] Eze, E. O., Ezeora, J. N., Obasi, U. E.: Astudy of Periodic Solution of a Duffing Equation by Implicit Function Theorem. Open Journal of Applied Sciences, 8(10): 459 - 464 (2016).
[27] Ding, T.: Boundedness of Solutions of Duffing Equations. Journal of Differential Equations, 61(2):178 - 207 (1986)
[28] Zheng, J.: Boundedness and Global asymptotic stability of constant equilibrium in a fully parabolic chemotaxis system with nonlinear logistic source. Journal of Mathematical Analysis and Applications. 450(2):1047 - 1061 (2017)
[29] Valdes, J. E. N.: On the Boundedness and Global Asymptotic Stability of the Lienard Equation with Resorting terms. Revista de la, Union Mathematica Argentina, 41(4):47 – 59.
[30] Laser, A. C., Mekenna, P. J.: On the Existence of Stable Periodic Solutions of Differential Equations of Duffing Type: Proceeding of American Mathematical Society, 110:125 - 133 (1990).
[31] Ortega, R.: The first Interval of Stability of a Periodic Equation of Duffing Type. Proceedings of American Mathematical Society, 115:1061 - 1067 (1992).
[32] Levin, J. J., Nohel, J. A.: Global Asymptotic Stability for Nonlinear Systems of Differential Equation and Application to Reactor Dynamics. Archs ration, Mechanical Analysis: 5; 194 - 211 1960).
[33] Thurtson, L. H., Wong, J. S. W.: On Global Asymptotic Stability of certain second order of certain Second order Differential Equation with Integrable forcing terms. SIAM Journal on Applied Mathematics, 24(1): 50 - 61 (1973)
[34] Baillieu, J., Willems, J. C.: Mathematical Control Theory: Published by Springer-Verlag New York In. Ist Edition
[35] Eze, E. O., Ukeje, E., Hilary, M. O.: Atable Bounded and Periodic Solutions in a Nonlinear Second Order Differential Equations. American Journal of Engineering Research 4(8):14 - 18 (2018)
[2] Nafeh, A. H. , Mook D. T., Nonlinear Oscillation, Wiley, New York (1979)
[3] Yang H., Guo H., He, M.: The prediction of Beinjing urban Distribution Amount using Grey Model and Industrial Structure Analysis. Fifth International Joint Conference on Computational Science and Optimization IEEE. DOI: 10.1109/CSO.2012.47
[4] He, C., Amer, T. S., Tian D., Abolila A. F., Galai, A. A., Controlling the Kinematics of a Spring-Pendulum system using an energy harvesting device, Journal of Low Frequency, Noise, Vibration and Active Control. 41(3), 1231 - 1257 (2022).
[5] Yang, L., Li, Y.: Existence and Global Exponential Stability of Almost Periodic solution for a class of Delay Duffing Equation on Time Scale, Abstract and Applied Analysis, Hindawi Publishing Corporation, 857 - 861 (2014)
[6] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical system and Bifucation of vector field. Journal of Applied Mathematical Sciences, Springer –Verlag New York (1983)
[7] Jordan, D. W., Smith, P.: Nonlinear Analysis and Differential Equations, Clavendon Press Oxford (1977)
[8] Zeeman, E.: Duffing Equation in Brain Modelling, Bull IMA 12:207 - 214
[9] Oyesanya, M. O..: Duffing Oscillator as a model for predicting earthquake occurrence. Journal of Nigerian Association of Mathematical Physics, 12: 133 - 142 (2008)
[10] Wang, G., Zheng, W.: Estimate of Amplitude and Phase of a Weak Signal by Using the Property of Sensitive Dependency on initial Condition of a Non-linear Oscillator Signal Process 82: 103-115 (2002).
[11] Chen, G., Han, F., Yu, X., Feng, Y.: A further study of Nonlinear feedback system with chaotic oscillation, 1:83-86 (2004)
[12] Fan, J., Zhang, Y. R., Liu, Y., Wang, F., Cao, Q., Yang, T.: Explanation of the Cell Orientation in a Nonofiber Membrane by the Geometric Potential Theory. Results in Physics.
[13] Zi, Y. L., Guo, H. Y., Wen, Z., Yeh, M. H., Hu, Z., Wang, Z. L.: Harvesting Low Frequency (<5HZ) Irregular Mechanical energy: A Possible Killer Application of Triboelectric Nano generator. ACS Nano, 10(4):4797 - 4805.
[14] Li, H. D., Tian, C., Deng, Z. D.: Energy Harvesting fro Low Frequency Application using piezoelectric materials. Journal of Applied Physics Reviews. 1(4): 041301. DOI:10.1063/1.4900845
[15] Njoku F. I., Omari, P.: Stability properties of periodic solutions of a Duffing Equation in the presence of lower and upper solution. Journal of Applied Mathematics and Computer. 135:471 - 490.
[16] Oyesanya, M. O., Nwamba, J. I.: Stability analysis of cubic –quintic Duffing Oscillator: World Journal of Mechanics. 3:299 - 307 (2013)
[17] Tung, C., Tung, O.: A note of Stability and Boundedness of Solutions to Nonlinear Differential Systems of Second Order, Journal of the Association of Arab Universities For Basic and Applied Sciences 24: 169 - 175 (2017).
[18] Jordan D. W., Smith, P.: Nonlinear Analysis and Differential Equations. Clavendon Press Oxford, (1977).
[19] Thomson, J. M. T., Stewart, H. B.: Nonlinear Dynamics and Chaos. John Wiley and Sons, New York (1986)
[20] Nnaji, D. U., Makuochukwu, F. O., Ezenwobodo, S. S., Stguegboh, N., Onyiaji, N. E.: On the Stability of Duffing Type Fractional Differential Equation with Cubic Nonlineariry. Journal of Scientific Research. 7(5): 1 - 14.
[21] Tian, D., Ain, Q. T., Anjum, N.: Fractal N/MEMS from Pull-in Instability to Pull-in Stability. Journal of Fractals, 29(2):2150030 (2020)
[22] Tian, D., He, C. H., He, J. H.: Eractal Pull-in Stabilitytheory for microelectro-mechanical systems. Front Phys, 9:606011 (2020).
[23] Ortega, R., Alonso, J. M.: Boundedness and Asymptotic Stability of a Forced and Damped Oscillator. Nonlinear Analysis Theory, Method and Applications. 25(3): 297 - 309 (1995)
[24] Tung, C., Tung, E.: On the Asymptotic Behaviour of Solutions of Certain Second Order Differential Equations. Journal of the Franklin Institute. 344(5):391 - 398 (2007).
[25] Eze, E. O., Ajah, R. O.: Application of Lyapunov and Yoshizawa’s theorems on Stability, Asymptotic Stability, Boundaries and periodicity of solutions Duffing Equation: Asian Journal of Applied Science, 2(6):970 - 975 (2014)
[26] Eze, E. O., Ezeora, J. N., Obasi, U. E.: Astudy of Periodic Solution of a Duffing Equation by Implicit Function Theorem. Open Journal of Applied Sciences, 8(10): 459 - 464 (2016).
[27] Ding, T.: Boundedness of Solutions of Duffing Equations. Journal of Differential Equations, 61(2):178 - 207 (1986)
[28] Zheng, J.: Boundedness and Global asymptotic stability of constant equilibrium in a fully parabolic chemotaxis system with nonlinear logistic source. Journal of Mathematical Analysis and Applications. 450(2):1047 - 1061 (2017)
[29] Valdes, J. E. N.: On the Boundedness and Global Asymptotic Stability of the Lienard Equation with Resorting terms. Revista de la, Union Mathematica Argentina, 41(4):47 – 59.
[30] Laser, A. C., Mekenna, P. J.: On the Existence of Stable Periodic Solutions of Differential Equations of Duffing Type: Proceeding of American Mathematical Society, 110:125 - 133 (1990).
[31] Ortega, R.: The first Interval of Stability of a Periodic Equation of Duffing Type. Proceedings of American Mathematical Society, 115:1061 - 1067 (1992).
[32] Levin, J. J., Nohel, J. A.: Global Asymptotic Stability for Nonlinear Systems of Differential Equation and Application to Reactor Dynamics. Archs ration, Mechanical Analysis: 5; 194 - 211 1960).
[33] Thurtson, L. H., Wong, J. S. W.: On Global Asymptotic Stability of certain second order of certain Second order Differential Equation with Integrable forcing terms. SIAM Journal on Applied Mathematics, 24(1): 50 - 61 (1973)
[34] Baillieu, J., Willems, J. C.: Mathematical Control Theory: Published by Springer-Verlag New York In. Ist Edition
[35] Eze, E. O., Ukeje, E., Hilary, M. O.: Atable Bounded and Periodic Solutions in a Nonlinear Second Order Differential Equations. American Journal of Engineering Research 4(8):14 - 18 (2018)
Published
2024-11-30
How to Cite
Ikechukwu Godwin, E. (2024). ON BOUNDEDNESS AND GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF DUFFING EQUATION. GPH-International Journal of Mathematics, 7(11), 1-13. https://doi.org/10.5281/zenodo.16936158
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Copyright (c) 2025 Ezugorie Ikechukwu Godwin
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