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Chaos in a class of first-order partial difference equations with delay controllers

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Abstract

In this paper, a class of first-order partial difference equations with general delay controllers \(x(n+1,m) = f(x(n,m),x(n,m+1)) + \alpha g(\beta x(n-\ell ,m))\), \(m=0,1,\cdots , k\), is considered. This equation is equivalent to a \((k+1)(\ell +1)\)-dimensional system. To pursue the existence of chaos, the equation is investigated in the cases of \(|\beta |>1\), \(|\beta |=1\), \(0<|\beta |<1\), and in each case, there are five subcases according to different values of \(\ell \) and k, respectively. The equation is proved to be chaotic in all the cases, respectively, under some weak conditions. To illustrative the theoretical results, three examples are provided.

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Acknowledgements

This research was supported by Foundation of Henan Educational Committee (Grant 23A110008).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, Henan, China

    Yongjun Zhang, Wei Liang & Xiaolin Lv

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  1. Yongjun Zhang
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  2. Wei Liang
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  3. Xiaolin Lv
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Contributions

YZ was involved in formal analysis, writing the original draft, and writing and reviewing; WL was responsible for investigation, methodology, formal analysis, writing the original draft, and writing and reviewing; XL contributed to investigation and writing the original draft.

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Correspondence to Wei Liang.

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Zhang, Y., Liang, W. & Lv, X. Chaos in a class of first-order partial difference equations with delay controllers. Nonlinear Dyn 111, 10573–10582 (2023). https://doi.org/10.1007/s11071-023-08342-9

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