Stability and Bifurcation Analysis of Two-Term Fractional Differential Equation with Delay
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916189385523200 |
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| author | Bhalekar, Sachin Gupta, Deepa |
| author_facet | Bhalekar, Sachin Gupta, Deepa |
| contents | This manuscript deals with the stability and bifurcation analysis of the equation $D^{2α}x(t)+c D^αx(t)=a x(t)+b x(t-τ)$, where $0<α<1$ and $τ>0$. We sketch the boundaries of various stability regions in the parameter plane under different conditions on $α$ and $b$. First, we provide the stability analysis of this equation with $τ=0$. Change in the stability of the delayed counterpart is possible only when the characteristic roots cross the imaginary axis. This leads to various delay-independent as well as delay-dependent stability results. The stability regions are bifurcated on the basis of the following behaviors with respect to the delay $τ$ viz. stable region for all $τ>0$, unstable region, single stable region, stability switch, and instability switch. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_01824 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability and Bifurcation Analysis of Two-Term Fractional Differential Equation with Delay Bhalekar, Sachin Gupta, Deepa Dynamical Systems 34K20, 34K18, 26A33 This manuscript deals with the stability and bifurcation analysis of the equation $D^{2α}x(t)+c D^αx(t)=a x(t)+b x(t-τ)$, where $0<α<1$ and $τ>0$. We sketch the boundaries of various stability regions in the parameter plane under different conditions on $α$ and $b$. First, we provide the stability analysis of this equation with $τ=0$. Change in the stability of the delayed counterpart is possible only when the characteristic roots cross the imaginary axis. This leads to various delay-independent as well as delay-dependent stability results. The stability regions are bifurcated on the basis of the following behaviors with respect to the delay $τ$ viz. stable region for all $τ>0$, unstable region, single stable region, stability switch, and instability switch. |
| title | Stability and Bifurcation Analysis of Two-Term Fractional Differential Equation with Delay |
| topic | Dynamical Systems 34K20, 34K18, 26A33 |
| url | https://arxiv.org/abs/2404.01824 |