Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
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Yélomè Judicaël Fernando Kpomahou
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Clément Hodévèwan Miwadinou
Abstract
In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.
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Author contribution: All authors have contributed to the development andformulation of this work.
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Research funding: No funding.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] M. P. Kennedy, “Three steps to chaos: Part II. A Chua’s circuit primer,” IEEE Trans. Circuits Syst. Fund. Theory Appl., vol. 40, no. 10, pp. 657–674, 1993. https://doi.org/10.1109/81.246141.Search in Google Scholar
[2] O. O. Ajide and T. A. O. Salau, “Bifurcation diagrams of nonlinear RLC electrical circuits,” Int. J. Sci. Technol., vol. 3, no. 1, pp. 136–139, 2011.Search in Google Scholar
[3] A. Buscarino, L. Fortuna, M. Frasca, and G. Sciuto, A Concise Guide to Chaos Electronic Circuits, New York Dordrecht London, Springer Cham Heidelberg, 2014.10.1007/978-3-319-05900-6Search in Google Scholar
[4] J. Bienstman, R. Puers, and J. Vandewalle, “Periodic and chaotic behaviour of the autonomous impact resonator,” in Proc. MEMS 98, IEEE, 11th Annual Int. Workshop on Micro Electro Mechanical Systems. An Investigation of Micro Structures, Sensors, Actuators, Machines and Systems, Cat.No.98CH36176, 1998, pp. 562–567.10.1109/MEMSYS.1998.659819Search in Google Scholar
[5] E. Tamaseviciute, A. Tamasevicius, G. Mykolaitis, S. Bumeliene, and E. Lindberg, “Analogue electrical circuit for simulation of the Duffing–Holmes equation,” Nonlinear Anal. Model., vol. 2, no. 13, pp. 241–252, 2008. https://doi.org/10.15388/NA.2008.13.2.14582.Search in Google Scholar
[6] T. Reis, Large-Scale Networks in Engineering Life Sciences, New York Dordrecht London, Springer Cham Heidelberg, 2014.Search in Google Scholar
[7] H. Kaufman and G. E. Roberts, “A note on non-linear series RLC circuits,” J. Electron. Control, vol. 17, no. 6, pp. 679–682, 1964. https://doi.org/10.1080/00207216408937738.Search in Google Scholar
[8] G-F. Li, “The third superharmonic resonance analysis of RLC circuit with sensor electronic components,” Advances in Engineering Research, vol. 105, pp. p826–831, 2016. https://doi.org/10.2991/mme-16.2017.115.Search in Google Scholar
[9] A. Oksasoglu and D. Vavriv, “Interaction of low-and high-frequency oscillations in nonlinear RLC circuit,” IEEE Trans. Circ. Syst. Fund. Theory Appl., vol. 41, pp. 669–672, 1994. https://doi.org/10.1109/81.329728.Search in Google Scholar
[10] Z-A. Yang and Y-H. Cui, “Primary Resonance analysis of RLC series circuit with resistance and inductance nonlinearity,” J. Tianjin Univ. Sci. Technol., vol. 40, no. 5, pp. 579–583, 2007.Search in Google Scholar
[11] I. Dumitrescu, S. Bachir, D. Cordeau, J. M. Paillot, and M. Iordache, “Modeling and characterization of oscillator circuits by van der Pol model using parameter estimation,” J. Circuit Syst. Comp., vol. 21, no. 5, pp. 1–15, 2012. https://doi.org/10.1142/s0218126612500430.Search in Google Scholar
[12] R. E. Mickens, Oscillations in Planar Dynamic Systems, Singapore, World Scientific Publishing Co. Pte. Ltd, 1996.10.1142/2778Search in Google Scholar
[13] M. Lakshmanan and V. K. Chandrasekar, “Generating finite dimensional integrable nonlinear dynamical systems,” Eur. Phys. J. Spec. Top., vol. 222, pp. 665–688, 2013. https://doi.org/10.1140/epjst/e2013-01871-6.Search in Google Scholar
[14] I. A. Viorel, L. Strete, and I. F. Soran, “Analytical flux linkage model of switched reluctance motor,” Rev. Roum. Sci. Techn. Electrotechn. Energy, vol. 54, no. 2, pp. 139–146, 2009.Search in Google Scholar
[15] N. Chiesa and H. K. Hoidalen, “Modeling of nonlinear and hysteretic iron-coreinductors in ATP,” in EEUG Meeting, European EMTP-ATP Conf., Leon, Spain, 2007.Search in Google Scholar
[16] N. V. Zorn, “A study of current-dependent resistors in nonlinear circuits,” BSEE, vol. 1–66, Unversity of Pittsburgh, 2000.Search in Google Scholar
[17] B. Bagchi, S. Das, S. Ghosh, and S. Poria, “Nonlinear dynamics of a position-dependent-mass-driven Duffing-type oscillator,” J. Phys. Math. Theor., vol. 46, pp. 1–6, 2013. https://doi.org/10.1088/1751-8113/46/3/032001.Search in Google Scholar
[18] B. Bagchi, S. Ghosh, B. Pal, and S. Poria, “Qualitative analysis of certain generalized classes of quadratic oscillator systems,” J. Math. Phys., vol. 57, pp. 022701-1–022701-8, 2016. https://doi.org/10.1063/1.4939486.Search in Google Scholar
[19] D. K. K. Adjaé, L. H. Koudahoun, J. Akande, Y. J. F. Kpomahou, and M. D. Monsia, “Solutions of the duffing and Painlevé–Gambier equations by generalized sundman transformation,” J. Math. Stat., vol. 14, pp. 241–252, 2018. https://doi.org/10.3844/jmssp.2018.241.252.Search in Google Scholar
[20] A. Venkatesan and M. Lakshmanan, “Nonlinear dynamics of damped and driven velocity-dependent systems,” Phys. Rev. E, vol. 55, no. 5, pp. 5134–5146, 1997. https://doi.org/10.1103/physreve.55.5134.Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- Numerical analysis of geometrical nonlinear aeroelasticity with CFD/CSD method
- Application of moving least squares algorithm for solving systems of Volterra integral equations
- Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative
- Stochastic models with multiplicative noise for economic inequality and mobility
- Fast interaction of Cu2+ with S2O3 2− in aqueous solution
- Rotorcraft fuselage/main rotor coupling dynamics modelling and analysis
- Modeling and fault identification of the gear tooth surface wear failure system
- Wavelet-optimized compact finite difference method for convection–diffusion equations
- A methodology for dynamic behavior analysis of the slider-crank mechanism considering clearance joint
- Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps
- The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations
- Distributed consensus of multi-agent systems with increased convergence rate
- Studies on population balance equation involving aggregation and growth terms via symmetries
- Bifurcation, routes to chaos, and synchronized chaos of electromagnetic valve train in camless engines
- Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation
- Nonlinear dynamics of a RLC series circuit modeled by a generalized Van der Pol oscillator
