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Stability and bifurcation analysis of a fractional order delay differential equation involving cubic nonlinearity

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  • Bhalekar, Sachin
  • Gupta, Deepa

Abstract

Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation Dαx(t)=δx(t−τ)−ϵx(t−τ)3−px(t)2+qx(t).We provide linearization of this system in a neighborhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters δ, ϵ, p, q and τ. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the qδ−plane for any positive ϵ and p. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter.

Suggested Citation

  • Bhalekar, Sachin & Gupta, Deepa, 2022. "Stability and bifurcation analysis of a fractional order delay differential equation involving cubic nonlinearity," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    • Handle: RePEc:eee:chsofr:v:162:y:2022:i:c:s0960077922006890
    DOI: 10.1016/j.chaos.2022.112483
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    References listed on IDEAS

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    1. Jinxing Lai & Sheng Mao & Junling Qiu & Haobo Fan & Qian Zhang & Zhinan Hu & Jianxun Chen, 2016. "Investigation Progresses and Applications of Fractional Derivative Model in Geotechnical Engineering," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-15, May.
    2. Lokenath Debnath, 2003. "Recent applications of fractional calculus to science and engineering," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2003, pages 1-30, January.
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