Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

Chunhua Feng

doi:doi:10.11648/j.mcs.20190406.12

| Peer-Reviewed

Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

Chunhua Feng

Published in Mathematics and Computer Science (Volume 4, Issue 6)

Received: 31 October 2019 Accepted: 29 November 2019 Published: 10 December 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.

Published in	Mathematics and Computer Science (Volume 4, Issue 6)
DOI	10.11648/j.mcs.20190406.12
Page(s)	104-111
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Coupled Van der Pol-Duffing Equation, Delay, Stability, Oscillation

References

[1]	G. F. Kuiate, S. T. Kingni, V. K. Tamba, and P. K. Talla, Three-dimensional chaotic autonomous van der pol-Duffing type oscillator and its fractional-order form, Chinese J. Physics, 2018, 56, 2560-2573.
[2]	M. A. Barron, Stability of a ring of coupled van der Pol oscillators with non-uniform distribution of the coupling parameter, J. Applied Research Technology, 2016, 14 (1), 62-66.
[3]	P. Kumar, A. Kumar, and S. Erlicher, A modified hybrid Van der Pol-Duffing-Rayleigh oscillator for modelling the lateral walking force on a rigid floor, Physica D, 2017, 358, 1-14.
[4]	S. F. Wen, Y. J. Shen, X. H. Li, and S. P. Yang, Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms, Inter. J. Nonlinear Mach. 2016, 84 (1), 130-138.
[5]	K. Rompala, R. Rand, and H. Howland, Dynamics of three coupled van der Pol oscillators with application to circadian rhythms, Commun Nonlinear Sci Numer Simulat, 2007, 12 (5), 794-803.
[6]	H. G. Kadji, J. B. Orou, and P. Woafo, Synchronization dynamics in a ring of four mutually coupled biological systems, Commun Nonlinear Sci Numer Simulat, 2008, 13 (7), 1361-1372.
[7]	R. Yamapi, R. M. Yonkeu, G. Filatrella, and C. Tchawoua, Effects of noise correlation on the coherence of a forced van der Pol type birhythmic system, Commun Nonlinear Sci Numer Simulat, 2018, 62, 1-17.
[8]	V. K. Chandrasekar, A. Venkatesan, and I. R. Mohamed, Duffing-van der Pol oscillator type dynamics in Murali-Lakshmanan-Chua (MLC) circuit, Chaos, Solitons and Fractals, 2016, 82, 60-71.
[9]	D. L. Wang, W. Xu, J. W. Xu, X. D. Gu, and G. D. Yang, Resonance responses in a two-degree-of-freedom viscoelastic oscillator under randomly disordered periodic excitations, Commun Nonlinear Sci Numer Simulat, 2019, 68, 302-316.
[10]	J. Brechtl, X. Xie, and P. K. Liaw, Investigation of chaos and memory effects in the Bonhoeffer-van der Pol oscillator with a non-ideal capacitor, Commun Nonlinear Sci Numer Simulat, 2019, 73, 195-216.
[11]	P. V. Kuptsov, and A. V. Kuptsova, Radial and circular synchronization clusters in extended starlike network of van der Pol oscillators, Commun Nonlinear Sci Numer Simulat, 2017, 50, 115-127.
[12]	L. Makouo, and P. Woafo, Experimental observation of bursting patterns in Van der Pol oscillators, Chaos, Solitons and Fractals, 2017, 94, 95-101.
[13]	X. Li, J. Ji, and C. H. Hansen, Dynamics of two delay coupled van der Pol oscillators, Mech. Res. Commun. 2006, 33 (5), 614-627.
[14]	J. M. Zhang, and X. S. Gu, Stability and bifurcation analysis in the delay-coupled van der Pol oscillators, Appl. Math. Model. 2010, 34 (9), 2291-2299.
[15]	S. Wirkus, and R. Rand, The dynamics of two coupled van der Pol oscillators with delay coupling, Nonlinear Dyn. 2002, 30, 205-221.
[16]	W. Y. Wang, and L. J. Chen, Weak and non-resonant double Hopf bifurcations in m coupled van der Pol oscillators with delay coupling, Appl. Math. Model. 2015, 39 (10-11), 3094-3102.
[17]	Z. Ghouli, M. Hamdi, F. Lakrad, and M. Belhaq, Quasiperiodic energy harvesting in a forced and delayed Duffing harvester device, J. Sound and Vibration, 2017, 407, 27 (l), 271-285.
[18]	P. Kumar, S. Narayanan, and S. Gupta, Investigations on the bifurcation of a noisy Duffing-Van der Pol oscillator, Probabilistic Engineering Mechanics, 2016, 45, 70-86.
[19]	C. R. Zhang, B. D. Zheng, and L. C. Wang, Multiple Hopf bifurcations of three coupled van der Pol oscillators with delay, Appl. Math. Comput. 2011, 217 (17), 7155-7166.
[20]	C. R. Zhang, W. X. Li, and K. Wang, Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling, Appl. Math. Model. 2013, 37 (7), 5394-5402.
[21]	A. Maccari, Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, Intern. J. Nonlinear Mechan. 2003, 38 (1), 123-131.
[22]	D. Ghosh, A. R. Chowdhury, and P. Saha, On the various kinds of synchronization in delayed Duffing-Van der Pol system, Commun Nonlinear Sci Numer Simulat, 2008, 13 (4), 790-803.
[23]	J. Xu, and K. W. Chung, Effects of time delayed position feedback on a van der Pol-Duffing oscillator, Physica D: Nonlinear Phenomena, 2003, 180 (1-2), 17-39.
[24]	R. Rusinek, A. Weremczuk, K. Kecik, and J. Warminski, Dynamics of a time delayed Duffing oscillator, Intern. J. Nonlinear Mechan. 2014, 65, 98-106.
[25]	N. Chafee, A bifurcation problem for a functional differential equation of finitely retarded type, J. Math. Anal. Appl. 1971, 35, 312-348.
[26]	C. Feng, and R. Plamondon, An oscillatory criterion for a time delayed neural ring network model, Neural Networks, 2012, 29, 70-79.

Cite This Article

Plain Text BibTeX RIS

APA Style

Chunhua Feng. (2019). Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Mathematics and Computer Science, 4(6), 104-111. https://doi.org/10.11648/j.mcs.20190406.12

Copy | Download

ACS Style

Chunhua Feng. Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Math. Comput. Sci. 2019, 4(6), 104-111. doi: 10.11648/j.mcs.20190406.12

Copy | Download

AMA Style

Chunhua Feng. Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Math Comput Sci. 2019;4(6):104-111. doi: 10.11648/j.mcs.20190406.12

Copy | Download

@article{10.11648/j.mcs.20190406.12,
  author = {Chunhua Feng},
  title = {Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {6},
  pages = {104-111},
  doi = {10.11648/j.mcs.20190406.12},
  url = {https://doi.org/10.11648/j.mcs.20190406.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190406.12},
  abstract = {In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays
AU  - Chunhua Feng
Y1  - 2019/12/10
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20190406.12
DO  - 10.11648/j.mcs.20190406.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 104
EP  - 111
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20190406.12
AB  - In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.
VL  - 4
IS  - 6
ER  -

Copy | Download

Author Information

Chunhua Feng

Department of Mathematics and Computer Science, Faculty of Alabama State University, Montgomery, USA

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Chunhua Feng. (2019). Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Mathematics and Computer Science, 4(6), 104-111. https://doi.org/10.11648/j.mcs.20190406.12

Copy | Download

ACS Style

Chunhua Feng. Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Math. Comput. Sci. 2019, 4(6), 104-111. doi: 10.11648/j.mcs.20190406.12

Copy | Download

AMA Style

Chunhua Feng. Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays. Math Comput Sci. 2019;4(6):104-111. doi: 10.11648/j.mcs.20190406.12

Copy | Download

@article{10.11648/j.mcs.20190406.12,
  author = {Chunhua Feng},
  title = {Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {6},
  pages = {104-111},
  doi = {10.11648/j.mcs.20190406.12},
  url = {https://doi.org/10.11648/j.mcs.20190406.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190406.12},
  abstract = {In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays
AU  - Chunhua Feng
Y1  - 2019/12/10
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20190406.12
DO  - 10.11648/j.mcs.20190406.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 104
EP  - 111
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20190406.12
AB  - In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.
VL  - 4
IS  - 6
ER  -

Copy | Download