BOUNDEDNESS AND GLOBAL ASYMPTOTIC STABILITY OF A CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING
Keywords:
boundedness, stability, oscillator, damping, Lyapunov functionDOI:
https://doi.org/10.17654/2277141724006Abstract
This paper investigated boundedness and global asymptotic stability of a certain second order nonlinear differential equation with damping using Lyapunov second method and eigenvalue approach. The critical point of the system was found to have multiple equilibria due to the cubic nature of the equation. Different regions of stability and instability were observed for different equilibria. The existence of total derivative confirmed boundedness of the solution. The damping effect was negligible. Mathcad software was used to illustrate the numerical behavior of the solutions. The results show that out of the three equilibrium points obtained, only two of them were asymptotically stable. Our results improve and extend some results in literature.
Received: May 7, 2024
Accepted: August 13, 2024
References
L. Tamaevuoe, Duffing Oscillator, Wiley, New York, 2007.
A. H. Nayfeh and D. T. Mook, Nonlinear Oscillation, John Wiley and Sons Inc., New York, 1979.
A. H. S. Salas, Analytic solution to the generalized complex Duffing equation and its application in soliton theory, Applicable Analysis 100(13) (2019), 2867-2872.
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Journal of Applied Mathematical Sciences 42 (1983), 1-65.
T. A. O. Salau and O. O. Ajide, Comparative analysis of time steps distribution in Runge-Kutta algorithms, International Journal of Scientific and Engineering Research 3(1) (2012), 253-257.
E. Zeeman, Duffing equation in brain modelling, Bull. IMA 12 (1976), 207-214.
M. O. Oyesanya, Duffing oscillator as a model for predicting earthquake occurrence, Journal of Nigerian Association of Mathematical Physics 12 (2008), 133-142.
G. Wang, W. Zheng and S. He, Estimation of amplitude and phase of a weak signal by using the property of sensitive dependence on initial conditions of a nonlinear oscillator, Signal Processing 82(1) (2002), 103-115.
C. Tunc and O. Tunc, A note on stability and boundedness of solutions to non- linear differential systems of second order, Journal of the Association of Arab Universities for Basic and Applied Sciences 24 (2017), 169-175.
R. Ortega and J. M. Alonso, Boundedness and asymptotic stability of a forced and damped oscillator, Nonlinear Analysis, Theory, Methods and Applications 25(3) (1995), 297-309.
C. Tunc and E. Tunc, On the asymptotic behaviour of solutions of certain second order differential equations, Journal of the Franklin Institute 344(5) (2007), 391-398.
E. O. Eze and R. O. Ajah, Application of Lyapunov and Yoshizawa’s theorems on stability, asymptotic stability, boundaries and periodicity of solutions of Duffing equation, Asian Journal of Applied Science 2(6) (2014), 970-975.
E. O. Eze, J. N. Ezeora and U. E. Obasi, A study of periodic solution of a Duffing’s equation by implicit function theorem, Open Journal of Applied Sciences 8(10) (2018), 459-464.
C. Tunc, On the properties of solutions for a system of non-linear differential equations of second order, International Journal of Mathematics and Computer Science 14(2) (2019), 519-534.
C. Tunc and O. Tunc, On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order, Journal of Advanced Research 7(1) (2016), 165-168.
C. Tunc, Boundedness results of solutions of certain nonlinear differential equation of second order, Journal of Indonesia Mathematical Society 16(2) (2010), 115-128.
C. Tunc, Stability and boundedness of solutions of non-autonomous differential equations of second order, Journal of Computational Analysis and Applications 13(6) (2011), 1067-1074.
A. C. Laser and P. J. Mckenna, On the existence of stable periodic solutions of differential equations of Duffing type, Proceeding of American Mathematical Society 110 (1990), 125-133.
R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proceedings of American Mathematical Society 115 (1992), 1061-1067.
J. J. Levin and J. A. Nohel, Global asymptotic stability for nonlinear systems of differential equations and applications to reactor dynamics, Archsration, Mechanical Analysis 5 (1960), 194-211.
J. Baillieul and J. C. Willems, Mathematical Control Theory, 1st edition, Springer-Verlag, New York, 1999.
G. Chen, F. Han, X. Yu and Y. Feng, A further study of nonlinear feedback system with chaotic oscillation, Conference paper, Intelligent Control and Automation 1 (2004), 83-86.
E. O. Eze, E. Ukeje and M. O. Hilary, Stable, bounded and periodic solutions in a non-linear 2nd order ordinary differential equation, American Journal of Engineering Research 4(8) (2015), 14-18.
F. I. Njoku and P. Omari, Stability properties of periodic solutions of a Duffing’s equation in the presence of lower and upper solution, Journal of Applied Mathematics and Computer 135 (2003), 471-490.
M. O. Oyesanya and J. I. Nwamba, Stability analysis of cubic-quintic Duffing oscillator, World Journal of Mechanics 3 (2013), 299-307.
U. A. Osisiogu, A First Course in Mathematical Analysis and Differential Equations, Bestsoft Educational Books, 2013, p. 34.
E. O. Eze, U. E. Obasi and T. C. Urama, Stability analysis of periodic solution of Bonhoeffer-Van Der Pol system with applied impulse, International Journal of Application and Computational Mathematics 9 (2023), 62.
E. O. Eze, Existence and stability of periodic solutions for a class of second order nonlinear differential equations, Ph.D. Thesis, LAP LAMBERT Academic Publishing, 2016.
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