Bifurcation of a feed forward neural network with delay and application in image contrast enhancement

Wenlong Wang; Chunrui Zhang; Wenlong Wang; Chunrui Zhang

doi:10.3934/mbe.2020021

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 1: 387-403. doi: 10.3934/mbe.2020021

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Research article

Bifurcation of a feed forward neural network with delay and application in image contrast enhancement

Wenlong Wang ,
Chunrui Zhang ^,

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received: 29 December 2018 Accepted: 05 June 2019 Published: 10 October 2019

This paper is concerned with how the singularity and delay in a feed forward neural network affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem, the mode interactions in two parameters bifurcations with a single zero and a pair of imaginary roots are considered. The codimension two normal form with Hopf-pitchfork bifurcations are given. Then, the bifurcation diagrams and phase portraits are obtained by computing the normal form. Furthermore, we find some interesting dynamical behaviors of the original system, such as the coexistence of two unstable nontrivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically. Through numerical simulation, we also find that this model has a special signal enhancement property in Hopf bifurcation state. Using this feed-forward neural network, we show that the gray scale picture contrast is strongly enhanced even if this one is initially very small.
- feed-forward,
- neural network,
- delay,
- Hopf-pitchfork Bifurcation,
- Hopf Bifurcation,
- image contrast enhancement
Citation: Wenlong Wang, Chunrui Zhang. Bifurcation of a feed forward neural network with delay and application in image contrast enhancement[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 387-403. doi: 10.3934/mbe.2020021

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Abstract

This paper is concerned with how the singularity and delay in a feed forward neural network affect generic dynamics and bifurcations. By computation of Hopf-pitchfork point in a two-parameter nonlinear problem, the mode interactions in two parameters bifurcations with a single zero and a pair of imaginary roots are considered. The codimension two normal form with Hopf-pitchfork bifurcations are given. Then, the bifurcation diagrams and phase portraits are obtained by computing the normal form. Furthermore, we find some interesting dynamical behaviors of the original system, such as the coexistence of two unstable nontrivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically. Through numerical simulation, we also find that this model has a special signal enhancement property in Hopf bifurcation state. Using this feed-forward neural network, we show that the gray scale picture contrast is strongly enhanced even if this one is initially very small.

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