Document Type : Original Research Paper

Authors

1 Electrical Engineering Department, Engineering Faculty, Islamic Azad University, Arak Branch, Arak, Iran.

2 Engineering Faculty, Arak University of Technology, Arak, Iran.

Abstract

Background and Objectives: The Differential transform method (DTM) is used in the analysis of ordinary, partial, and high-order differential equations. Recently, the DTM is used in the nonlinear analysis of physical nonlinear dynamic systems.
Methods: The DTM method is used to analyze and analytically solve the nonlinear mathematical model of bias current-controlled Colpitts oscillator with variable coefficients.  Intervals of the validity of the proposed method are evaluated by using the fourth order Runge-Kutta method (RK4M).  In this note, the Lyapunov exponent (LE) can be used to analyze the Colpitts oscillator. By using DTM, the LEs are calculated analytically with unknown parameters in a short interval of time t[0, 3 Sec].
Results: In this paper, intervals of the validity of the proposed method are evaluated using RK4M. In addition, LEs are calculated using analytical and numerical methods based on DTM technique and Wolf method, respectively.  LEs of the proposed system are presented as a function of the control parameter to confirm the applied technique’s usefulness.
 
Conclusion: By comparing these two methods, the proposed DTM analytical technique is relatively more precise. Simulation results confirmed the impact of different parameters on LEs with two different initial conditions. The results show good accuracy of the DTM in short time intervals t[0, 3 Sec].




Background and Objectives: The Differential transform method (DTM) is used in the analysis of ordinary, partial, and high-order differential equations. Recently, the DTM is used in the nonlinear analysis of physical nonlinear dynamic systems.
Methods: The DTM method is used to analyze and analytically solve the nonlinear mathematical model of bias current-controlled Colpitts oscillator with variable coefficients.  Intervals of the validity of the proposed method are evaluated by using the fourth order Runge-Kutta method (RK4M).  In this note, the Lyapunov exponent (LE) can be used to analyze the Colpitts oscillator. By using DTM, the LEs are calculated analytically with unknown parameters in a short interval of time t[0, 3 Sec].
Results: In this paper, intervals of the validity of the proposed method are evaluated using RK4M. In addition, LEs are calculated using analytical and numerical methods based on DTM technique and Wolf method, respectively.  LEs of the proposed system are presented as a function of the control parameter to confirm the applied technique’s usefulness.
Conclusion: By comparing these two methods, the proposed DTM analytical technique is relatively more precise. Simulation results confirmed the impact of different parameters on LEs with two different initial conditions. The results show good accuracy of the DTM in short time intervals t[0, 3 Sec].
 
 

======================================================================================================
Copyrights
©2021 The author(s). This is an open access article distributed under the terms of the Creative Commons Attribution (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, as long as the original authors and source are cited. No permission is required from the authors or the publishers.
======================================================================================================



 

Keywords

Main Subjects

[1] M.J. Feigenbaum, “Universal behavior in nonlinear systems,” Physica D, 7(1-3): 16-39, 1983.

[2] H. Lin, S. Yim, “Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors,” J. Appl. Mech. Jun, 63(2): 509-516, 1996.

[3] M.S. Soliman, “Global stability properties of equilibria, periodic, and chaotic solutions,” Appl. Math. Modell., 20(7): 488-500, 1996.

[4] B. Radziszewski, A. Sławiński, “Estimation of stability in non-linear systems,” Mach. Dyn. Probl., 24(3): 113-125, 2000.

[5] E.A. Best, "Stability assessment of nonlinear systems using the lyapunov exponent," Ohio University, 2003.

[6] M. Balcerzak, D. Pikunov, A. Dabrowski, "The fastest, simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems," Nonlinear Dyn., 94(4): 3053-3065, 2018.

[7] T. Zeren, M. ÖZBEK, N. Kutlu, M. Akilli, "Significance of using a nonlinear analysis technique, the Lyapunov exponent, on the understanding of the dynamics of the cardiorespiratory system in rats," Turk. J. Med. Sci., 46(1): 159-165, 2016.

[8] H. Kantz, T. Schreiber, Nonlinear Time Series Analysis, Cambridge university press, 2004.

[9] J.B. Dingwell, L.C. Marin, “Kinematic variability and local dynamic stability of upper body motions when walking at different speeds,” J. Biomech., 39(3): 444-452, 2006.

[10] J.B. Dingwell, Lyapunov Exponents, Wiley encyclopedia of biomedical engineering, 2006.

[11] C. Yang, Q. Wu, “On stability analysis via Lyapunov exponents calculated from a time series using nonlinear mapping—a case study,” Nonlinear Dyn., 59(1-2): 239, 2010.

[12] M.A. Fuentes, Y. Sato, C. Tsallis, "Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos," Phys. Lett. A, 375(33): 2988-2991, 2011.

[13] E. Glasner, B. Weiss, "Sensitive dependence on initial conditions,” Nonlinearity, 6(6): 1067-1075, 1993.

[14] I. Khovanov, N. Khovanova, V. Anishchenko, P. McClintock, "Sensitivity to initial conditions and lyapunov exponent of a quasiperiodic system," Tech. Phys., 45(5): 633-635, 2000.

[15] C. Abraham, G. Biau, B. Cadre, "On Lyapunov exponent and sensitivity," J. Math. Anal. Appl., 290(2): 395-404, 2004.

[16] G.-C. Wu, D. Baleanu, “Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps,” Commun. Nonlinear Sci. Numer. Simul., 22(1-3): 95-100, 2015.

[17] A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, 16(3): 285-317, 1985.

[18] H.D. Abarbanel, R. Brown, M.B. Kennel, "Local Lyapunov exponents computed from observed data," J. Nonlinear Sci., 2(3): 343-365, 1992.

[19] R. Gencay, W.D. Dechert, "An algorithm for the N-Lyapunov exponents of an N-dimensional unknown dynamical system," Physica D, 59(1-3): 142-157, 1992.

[20] X. Zeng, R. Eykholt, R. Pielke, "Estimating the Lyapunov-exponent spectrum from short time series of low precision," Phys. Rev. Lett., 66(25): 3229, 1991.

[21] M.B. Tayel, E.I. AlSaba, "Robust and sensitive method of lyapunov exponent for heart rate variability," arXiv preprint arXiv:1508.00996, 2015.

[22] L. G. de la Fraga, E. Tlelo-Cuautle, "Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators," Nonlinear Dyn. 76(2): 1503-1515, 2014.

[23] Y. Liu, X.C. Zhang, "A new method based on Lyapunov exponent to determine the threshold of chaotic systems," Appl. Mech. Mater., 511-512, 2014.

[24] H. Liao, "Novel gradient calculation method for the largest Lyapunov exponent of chaotic systems," Nonlinear Dyn. 85(3): 1377-1392, 2016.

[25] F. Ayaz, "Solutions of the system of differential equations by differential transform method," Appl. Math. Comput., 147(2):  547-567, 2004.

[26] K. Al Ahmad, Z. Zainuddin, F.A. Abdullah, "Solving non-autonomous nonlinear systems of ordinary differential equations using multi-stage differential transform method," Aust. J. Math. Anal. Appl., 17(2): 1-18, 2020.

[27] Y. Do, B. Jang, "Enhanced multistage differential transform method: application to the population models," Abstr. Appl. Anal., 2012: 1-14, 2012.

[28] E.A. Az-Zo’bi, K. Al-Khaled, A. Darweesh, "Numeric-Analytic solutions for nonlinear oscillators via the modified multi-stage decomposition method," Mathematics, 7(6): 550, 2019.

[29] T.M. Elzaki, "Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method," in International Mathematical Forum, 7(13): 631-638, 2012.

[30] T.M. Elzaki, E.M. Hilal, J.-S. Arabia, J.-S. Arabia, "Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations," Math. Theor. Model., 2(3): 33-42, 2012.

[31] A.-M. Wazwaz, "A reliable modification of Adomian decomposition method," Appl. Math. Comput., 102(1): 77-86, 1999.

[32] S. Abbasbandy, "Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method," Appl. Math. Comput., 145(2-3): 887-893, 2003.

[33] A. Arikoglu, I. Ozkol, "Solution of boundary value problems for integro-differential equations by using differential transform method," Appl. Math. Comput., 168(2): 1145-1158, 2005.

[34] C. Huang, J. Li, F. Lin, "A new algorithm based on differential transform method for solving partial differential equation system with initial and boundary conditions," Adv. Pure. Math., 10(5): 337, 2020.

[35] S. Al-Ahmad, I. M. Sulaiman, M. Mamat, K. Kamfa, "Solutions of classes of differential equations using modified differential transform method," J. Math. Comput. Sci., 10(6): 2360-2382, 2020.

[36] H. Abbasi, A. Javed, "Implementation of Differential Transform Method (DTM) for large deformation analysis of cantilever beam," in Proc. IOP Conference Series: Materials Science and Engineering, 899(1): 012003, 2020.

[37] L.-j. Xie, C.-l. Zhou, S. Xu, "Solving the systems of equations of Lane-Emden type by differential transform method coupled with adomian polynomials," Mathematics, 7(4): 377, 2019.

[38] S. Mukherjee, B. Roy, S. Dutta, "Solution of the Duffing–van der Pol oscillator equation by a differential transform method," Phys. Scripta, 83(1): 015006, 2010.

[39] I. Khatami, E. Zahedi, M. Zahedi, "Efficient solution of nonlinear Duffing oscillator," J. Comput. Appl. Mech., 6(2): 219-234, 2020.

[40] A.J. Christopher, N. Magesh, G.T. Preethi, Dynamical Analysis of Corona-virus (COVID− 19) Epidemic Model by Differential Transform Method, 2020.

[41] R.M. Kouayep, A.F. Talla, J.H. T. Mbé, P. Woafo, "Bursting oscillations in Colpitts oscillator and application in optoelectronics for the generation of complex optical signals," Opt. Quantum Electron, 2(291), 2020.                

[42] S. Sarkar, S. Sarkar, B.C. Sarkar, "On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals," Commun. Nonlinear Sci. Numer. Simul., 19(8): 2883-2896, 2014.

[43] Z. Wei, W. Zhang, "Hidden hyperchaotic attractors in a modified Lorenz–Stenflo system with only one stable equilibrium," Int. J. Bifurcation Chaos, 24(10): 1450127, 2014.

[44] M. Valsakumar, S. Satyanarayana, V. Sridhar, "Signature of chaos in power spectrum," Pramana, 48(1): 69-85, 1997.


LETTERS TO EDITOR

Journal of Electrical and Computer Engineering Innovations (JECEI) welcomes letters to the editor for the post-publication discussions and corrections which allows debate post publication on its site, through the Letters to Editor. Letters pertaining to manuscript published in JECEI should be sent to the editorial office of JECEI within three months of either online publication or before printed publication, except for critiques of original research. Following points are to be considering before sending the letters (comments) to the editor.


[1] Letters that include statements of statistics, facts, research, or theories should include appropriate references, although more than three are discouraged.

[2] Letters that are personal attacks on an author rather than thoughtful criticism of the author’s ideas will not be considered for publication.

[3] Letters can be no more than 300 words in length.

[4] Letter writers should include a statement at the beginning of the letter stating that it is being submitted either for publication or not.

[5] Anonymous letters will not be considered.

[6] Letter writers must include their city and state of residence or work.

[7] Letters will be edited for clarity and length.

CAPTCHA Image