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

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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

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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), 2024. Published by Science Publishing Group

Keywords

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

References

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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

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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

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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

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@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}
}

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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  -

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Author Information

Chunhua Feng

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

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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

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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

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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  -

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