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Periodic Oscillatory Solutions for a Nonlinear Model with Multiple Delays

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  • Chunhua Feng

    (Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36104, USA)

Abstract

For systems such as the van der Pol and van der Pol–Duffing oscillators, the study of their oscillation is currently a very active area of research. Many authors have used the bifurcation method to try to determine oscillatory behavior. But when the system involves n separate delays, the equations for bifurcation become quite complex and difficult to deal with. In this paper, the existence of periodic oscillatory behavior was studied for a system consisting of n coupled equations with multiple delays. The method begins by rewriting the second-order system of differential equations as a larger first-order system. Then, the nonlinear system of first-order equations is linearized by disregarding higher-degree terms that are locally small. The instability of the trivial solution to the linearized equations implies the instability of the nonlinear equations. Periodic behavior often occurs when the system is unstable and bounded, so this paper also studied the boundedness here. It follows from previous work on the subject that the conditions here did result in periodic oscillatory behavior, and this is illustrated in the graphs of computer simulations.

Suggested Citation

  • Chunhua Feng, 2025. "Periodic Oscillatory Solutions for a Nonlinear Model with Multiple Delays," Mathematics, MDPI, vol. 13(14), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:14:p:2275-:d:1701968
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    References listed on IDEAS

    as
    1. Ma, Xindong & Zhang, Zhao, 2024. "Symmetric and asymmetric bursting oscillations in a hybrid van der Pol-Duffing-Rayleigh system," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    2. Fotsin, H.B. & Woafo, P., 2005. "Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1363-1371.
    3. Nganso, E. Njinkeu & Mbouna, S.G. Ngueuteu & Yamapi, R. & Filatrella, G. & Kurths, J., 2023. "Two-attractor chimera and solitary states in a network of nonlocally coupled birhythmic van der Pol oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    4. Fotsin, Hilaire & Bowong, Samuel, 2006. "Adaptive control and synchronization of chaotic systems consisting of Van der Pol oscillators coupled to linear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 27(3), pages 822-835.
    5. Liqin Liu & Chunrui Zhang, 2023. "Multiple Hopf Bifurcations of Four Coupled van der Pol Oscillators with Delay," Mathematics, MDPI, vol. 11(23), pages 1-16, November.
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