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Zero-Hopf Calculations for Neutral Differential Equations

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Abstract

In this manuscript, we consider a neutral functional differential equation. We make some assumptions and formulate conditions that ensure the existence of a zero-Hopf singularity. Using normal form theory and center manifold reduction, we derive the normal form up to third-order terms.

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Acknowledgements

The author expresses many thanks to the referee for valuable suggestions. The author would like to thanks Professor Chaouki Aouiti for his helpful remarks and support during the paper’s preparation.

Funding

This work was supported by GAMA Laboratory LR21ES10, Ministry of Higher Education and Scientific Research, Faculty of Science of Bizerte, Bizerte, Tunisia.

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  1. GAMA Laboratory LR21ES10, Department of Mathematics, Faculty of Sciences of Bizerta, University of Carthage, BP W, 7021, Zarzouna, Bizerta, Tunisia

    Houssem Achouri

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  1. Houssem Achouri
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A Appendix

A Appendix

In this part, we will define the notations:

$$\begin{aligned} H_{1}= & {} A_{1}\varphi _{1}(0)+B_{1}\varphi _{1}(-\tau ), \\ H_{2}= & {} A_{2}\varphi _{1}(0)+B_{2}\varphi _{1}(-\tau ), \\ H_{3}= & {} A_{1}\bar{\varphi }_{1}(0)+B_{1}\bar{\varphi }_{1}(-\tau ), \\ H_{4}= & {} A_{2}\bar{\varphi }_{1}(0)+B_{2}\bar{\varphi }_{1}(-\tau ), \\ H_{5}= & {} A_{1}\varphi _{2}(0)+B_{1}\varphi _{2}(-\tau ), \\ H_{6}= & {} A_{2}\varphi _{2}(0)+B_{2}\varphi _{2}(-\tau ), \\ H_{7}= & {} \displaystyle \sum _{i=1}^{n}(E_{i}\varphi _{1i}(0) \varphi _{1}(-\tau ) + F_{i}\varphi _{1i}(0)\varphi _{1}(0) + K_{i}\varphi _{1i}(-\tau )\varphi _{1}(-\tau )), \\ H_{8}= & {} \displaystyle \sum _{i=1}^{n}(E_{i}\bar{\varphi }_{1i}(0) \bar{\varphi }_{1}(-\tau ) + F_{i}\bar{\varphi }_{1i}(0) \bar{\varphi }_{1}(0) + K_{i}\bar{\varphi }_{1i}(-\tau ) \bar{\varphi }_{1}(-\tau )), \\ H_{9}= & {} \displaystyle \sum _{i=1}^{n}(E_{i}\varphi _{2i} (0)\varphi _{2}(-\tau ) + F_{i}\varphi _{2i}(0)\varphi _{2}(0) + K_{i}\varphi _{2i}(-\tau )\varphi _{2}(-\tau )), \\ H_{10}= & {} \displaystyle \sum _{i=1}^{n}\big (E_{i} (\varphi _{1i}(0)\bar{\varphi }_{1}(-\tau ) + \bar{\varphi }_{1i} (0)\varphi _{1}(-\tau )) + F_{i}(\varphi _{1i}(0)\bar{\varphi }_{1}(0) + \bar{\varphi }_{1i}(0)\varphi _{1}(0)) \\{} & {} \quad +\, K_{i}(\varphi _{1i} (-\tau )\bar{\varphi }_{1}(-\tau ) + \bar{\varphi }_{1i}(-\tau ) \varphi _{1}(-\tau ))\big ), \\ H_{11}= & {} \displaystyle \sum _{i=1}^{n}\big (E_{i}(\varphi _{1i} (0)\varphi _{2}(-\tau ) + \varphi _{2i}(0)\varphi _{1}(-\tau )) + F_{i}(\varphi _{1i}(0)\varphi _{2}(0) + \varphi _{2i}(0) \varphi _{1}(0)) \\{} & {} \quad +\, K_{i}(\varphi _{1i}(-\tau )\varphi _{2} (-\tau ) + \varphi _{2i}(-\tau )\varphi _{1}(-\tau ))\big ), \\ H_{12}= & {} \displaystyle \sum _{i=1}^{n}\big (E_{i} (\bar{\varphi }_{1i}(0)\varphi _{2}(-\tau ) + \varphi _{2i}(0) \bar{\varphi }_{1}(-\tau )) + F_{i}(\bar{\varphi }_{1i}(0) \varphi _{2}(0) + \varphi _{2i}(0)\bar{\varphi }_{1}(0))\\{} & {} \quad +\, K_{i}(\bar{\varphi }_{1i}(-\tau )\varphi _{2}(-\tau ) + \varphi _{2i}(-\tau )\bar{\varphi }_{1}(-\tau ))\big ), \\ G_{1}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} \varphi _{1i}(0)\varphi _{1}^{2}(0) + \varOmega _{i}^{2} \varphi _{1i}(0)\varphi _{1}^{2}(-\tau ) + \varOmega _{i}^{3} \varphi _{1i}(-\tau )\varphi _{1}^{2}(0) + \varOmega _{i}^{4} \varphi _{1i}(-\tau )\varphi _{1}^{2}(-\tau ) \big ], \\ G_{2}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} \bar{\varphi }_{1i}(0)\bar{\varphi }_{1}^{2}(0) + \varOmega _{i}^{2}\bar{\varphi }_{1i}(0)\bar{\varphi }_{1}^{2} (-\tau ) + \varOmega _{i}^{3}\bar{\varphi }_{1i}(-\tau ) \bar{\varphi }_{1}^{2}(0) + \varOmega _{i}^{4} \bar{\varphi }_{1i}(-\tau )\bar{\varphi }_{1}^{2}(-\tau ) \big ], \\ G_{3}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} \varphi _{2i}(0)\varphi _{2}^{2}(0) + \varOmega _{i}^{2} \varphi _{2i}(0)\varphi _{2}^{2}(-\tau ) + \varOmega _{i}^{3} \varphi _{2i}(-\tau )\varphi _{2}^{2}(0) + \varOmega _{i}^{4} \varphi _{2i}(-\tau )\varphi _{2}^{2}(-\tau ) \big ],\\ G_{4}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\bar{\varphi }_{1i}(0)\varphi _{1}^{2}(0) + 2\varphi _{1i} (0)\varphi _{1}(0)\bar{\varphi }_{1}(0)) \\{} & {} \quad +\,\varOmega _{i}^{2} (\bar{\varphi }_{1i}(0)\varphi _{1}^{2}(-\tau ) + 2\varphi _{1i}(0)\varphi _{1}(-\tau )\bar{\varphi }_{1}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\bar{\varphi }_{1i}(-\tau )\varphi _{1}^{2}(0) + 2\varphi _{1i}(-\tau )\varphi _{1}(0)\bar{\varphi }_{1}(0))\\{} & {} \quad +\,\varOmega _{i}^{4}(\bar{\varphi }_{1i}(-\tau )\varphi _{1}^{2} (-\tau ) + 2\varphi _{1i}(-\tau )\varphi _{1}(-\tau ) \bar{\varphi }_{1}(-\tau ))\big ],\\ G_{5}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\varphi _{2i}(0)\varphi _{1}^{2}(0) + 2\varphi _{1i}(0) \varphi _{1}(0)\varphi _{2}(0)) \\{} & {} \quad +\, \varOmega _{i}^{2}(\varphi _{2i}(0) \varphi _{1}^{2}(-\tau ) + 2\varphi _{1i}(0)\varphi _{1}(-\tau ) \varphi _{2}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\varphi _{2i}(-\tau )\varphi _{1}^{2}(0) + 2\varphi _{1i}(-\tau )\varphi _{1}(0)\varphi _{2}(0))\\{} & {} \quad +\,\varOmega _{i}^{4}(\varphi _{2i}(-\tau )\varphi _{1}^{2} (-\tau ) + 2\varphi _{1i}(-\tau )\varphi _{1}(-\tau ) \varphi _{2}(-\tau ))\big ],\\ G_{6}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\varphi _{1i}(0)\bar{\varphi }_{1}^{2}(0) + 2\bar{\varphi }_{1i} (0)\varphi _{1}(0)\bar{\varphi }_{1}(0)) \\{} & {} \quad +\, \varOmega _{i}^{2} (\varphi _{1i}(0)\bar{\varphi }_{1}^{2}(-\tau ) + 2\bar{\varphi }_{1i}(0)\varphi _{1}(-\tau )\bar{\varphi }_{1}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\varphi _{1i}(-\tau )\bar{\varphi }_{1}^{2} (0) + 2\bar{\varphi }_{1i}(-\tau )\varphi _{1}(0)\bar{\varphi }_{1}(0))\\{} & {} \quad +\, \varOmega _{i}^{4}(\varphi _{1i}(-\tau )\bar{\varphi }_{1}^{2}(-\tau ) + 2\bar{\varphi }_{1i}(-\tau )\varphi _{1}(-\tau )\bar{\varphi }_{1} (-\tau ))\big ], \\ G_{7}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\varphi _{2i}(0)\bar{\varphi }_{1}^{2}(0) + 2\bar{\varphi }_{1i} (0)\varphi _{2}(0)\bar{\varphi }_{1}(0)) \\{} & {} \quad +\, \varOmega _{i}^{2} (\varphi _{2i}(0)\bar{\varphi }_{1}^{2}(-\tau ) + 2\bar{\varphi }_{1i}(0)\varphi _{2}(-\tau )\bar{\varphi }_{1}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\varphi _{2i}(-\tau )\bar{\varphi }_{1}^{2}(0) + 2\bar{\varphi }_{1i}(-\tau )\varphi _{2}(0)\bar{\varphi }_{1}(0))\\{} & {} \quad +\, \varOmega _{i}^{4}(\varphi _{2i}(-\tau )\bar{\varphi }_{1}^{2}(-\tau ) + 2\bar{\varphi }_{1i}(-\tau )\varphi _{2}(-\tau )\bar{\varphi }_{1} (-\tau ))\big ], \\ G_{8}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\varphi _{1i}(0)\varphi _{2}^{2}(0) + 2\varphi _{2i}(0) \varphi _{1}(0)\varphi _{2}(0)) \\{} & {} \quad +\, \varOmega _{i}^{2} (\varphi _{1i}(0)\varphi _{2}^{2}(-\tau ) + 2\varphi _{2i}(0) \varphi _{1}(-\tau )\varphi _{2}(-\tau )) \\{} & {} \quad +\, \varOmega _{i}^{3}(\varphi _{1i}(-\tau )\varphi _{2}^{2}(0) + 2\varphi _{2i}(-\tau )\varphi _{1}(0)\varphi _{2}(0))\\{} & {} \quad +\,\varOmega _{i}^{4}(\varphi _{1i}(-\tau )\varphi _{2}^{2} (-\tau ) + 2\varphi _{2i}(-\tau )\varphi _{1}(-\tau ) \varphi _{2}(-\tau ))\big ], \\ G_{9}= & {} \displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\bar{\varphi }_{1i}(0)\varphi _{2}^{2}(0) + 2\varphi _{2i}(0) \bar{\varphi }_{1}(0)\varphi _{2}(0)) \\{} & {} \quad +\, \varOmega _{i}^{2} (\bar{\varphi }_{1i}(0)\varphi _{2}^{2} (-\tau ) + 2\varphi _{2i} (0)\bar{\varphi }_{1}(-\tau )\varphi _{2}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\bar{\varphi }_{1i}(-\tau )\varphi _{2}^{2}(0) + 2\varphi _{2i}(-\tau )\bar{\varphi }_{1}(0)\varphi _{2}(0))\\{} & {} \quad +\,\varOmega _{i}^{4}(\bar{\varphi }_{1i}(-\tau )\varphi _{2}^{2} (-\tau ) + 2\varphi _{2i}(-\tau )\bar{\varphi }_{1}(-\tau ) \varphi _{2}(-\tau ))\big ],\\ G_{10}= & {} 2\displaystyle \sum _{i=1}^{n}\big [\varOmega _{i}^{1} (\varphi _{1i}(0)\bar{\varphi }_{1}(0)\varphi _{2}(0) +\bar{\varphi }_{1i}(0)\varphi _{1}(0)\varphi _{2}(0) + \varphi _{2i}(0)\bar{\varphi }_{1}(0)\varphi _{1}(0))\\{} & {} \quad +\, \varOmega _{i}^{2}(\varphi _{1i}(0)\bar{\varphi }_{1} (-\tau )\varphi _{2}(-\tau ) + \bar{\varphi }_{1i}(0) \varphi _{1}(-\tau )\varphi _{2}(-\tau ) + \varphi _{2i} (0)\bar{\varphi }_{1}(-\tau )\varphi _{1}(-\tau ))\\{} & {} \quad +\, \varOmega _{i}^{3}(\varphi _{1i}(-\tau )\bar{\varphi }_{1} (0)\varphi _{2}(0) + \bar{\varphi }_{1i}(-\tau )\varphi _{1} (0)\varphi _{2}(0) + \varphi _{2i}(-\tau )\bar{\varphi }_{1} (0)\varphi _{1}(0))\\{} & {} \quad +\, \varOmega _{i}^{4}(\varphi _{1i}(-\tau )\bar{\varphi }_{1} (-\tau )\varphi _{2}(-\tau ) + \bar{\varphi }_{1i}(-\tau ) \varphi _{1}(-\tau )\varphi _{2}(-\tau ) \\{} & {} \quad + \varphi _{2i} (-\tau )\bar{\varphi }_{1}(-\tau )\varphi _{1}(-\tau ))\big ]. \end{aligned}$$

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Achouri, H. Zero-Hopf Calculations for Neutral Differential Equations. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10261-7

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