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Spatial Dynamics of Multilayer Cellular Neural Networks

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Abstract

The purpose of this work is to study the spatial dynamics of one-dimensional multilayer cellular neural networks. We first establish the existence of rightward and leftward spreading speeds of the model. Then we show that the spreading speeds coincide with the minimum wave speeds of the traveling wave fronts in the right and left directions. Moreover, we obtain the asymptotic behavior of the traveling wave fronts when the wave speeds are positive and greater than the spreading speeds. According to the asymptotic behavior and using various kinds of comparison theorems, some front-like entire solutions are constructed by combining the rightward and leftward traveling wave fronts with different speeds and a spatially homogeneous solution of the model. Finally, various qualitative features of such entire solutions are investigated.

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Acknowledgements

We are very grateful to the anonymous referees for their careful reading and helpful suggestions which led to an improvement of our original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, People’s Republic of China

    Shi-Liang Wu

  2. Department of Mathematics, National Central University, Chungli, 32001, Taiwan

    Cheng-Hsiung Hsu

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  1. Shi-Liang Wu
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  2. Cheng-Hsiung Hsu
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Correspondence to Shi-Liang Wu.

Additional information

Communicated by Ram Ramaswamy.

S.-L. Wu: Partially supported by the NSF of China (11671315), the NSF of Shaanxi Province of China (2017JM1003) and the Fundamental Research Funds for the Central Universities (JB160714, JBG160706). C.-H. Hsu: Partially supported by the Ministry of Science and Technology of Taiwan.

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Wu, SL., Hsu, CH. Spatial Dynamics of Multilayer Cellular Neural Networks. J Nonlinear Sci 28, 3–41 (2018). https://doi.org/10.1007/s00332-017-9398-x

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