Dynamical System in Wavelet Chaotic Neuron with Delay and Its Application in Color Image Encryption

Wavelet chaotic neuron combines Gauss wavelet function and Sigmoid function as non-monotonic excitation function. A new wavelet chaotic neuron with delay is proposed based on wavelet chaotic neuron. The effect of different parameters on chaotic dynamics of neuron are analyzed by using bifurcation diagram. The sensitivity and randomness of time series generated by wavelet chaotic neuron with delay are analyzed. A color image encryption algorithm based on wavelet chaotic neuron with delay is proposed. The security of the proposed encryption algorithm is researched. From the experimental results, we can see the chaotic sequences generated by wavelet chaotic neuron with delay are feasible and effective in encryption.

1 Introduction

Delay is often used to describe the lag phenomenon during transmission. The phenomenon of delay is objective and inevitable [1]. The flow of steam and fluid through the pipe, the transmission of signals, all have time delay. The logistic map with a delayed feedback is studied as a generic model in [2]. A kind of time-delay fractional order dynamic system is studied in [3]. In recent years, the analysis of systems with delay has become a research hotspot. The time delay improves the dynamic characteristics of the original system and can be better applied in the field of encryption. Different from chaotic neuron that use monotonically increasing Sigmoid function as excitation function. The excitation function of wavelet chaotic neuron is composed of Sigmoid function and Gauss wavelet function [4]. The excitation function is not monotonically increasing, but it has an increasing tendency in general. Wavelet function can not only make the excitation function non-monotone, but also make the excitation function have the advantages of wavelet function, which can effectively improve the efficiency of chaotic search. The research on wavelet chaotic neurons has also made great development. In [5], wavelet function disturbance is introduced into Morlet wavelet chaotic neural network and chaotic neural units with wavelet function disturbance were studied. An image encryption algorithm combining wavelet chaotic neurons with Cubic mapping is proposed in [6]. However, this model does not consider the delay of signal transmission in chaotic neuron. In this paper, the output of wavelet chaotic neuron at time () is delay to time , and the wavelet chaotic neuron with delay is constructed. The dynamical characteristics of wavelet chaotic neuron with delay are analysed. The sensitivity of wavelet delay neuron to initial values is analysed. NIST is used to analyse the randomness of time series generated by wavelet chaotic neuron with delay. The results show that wavelet chaotic neuron with delay can be used in image encryption. Therefore, a color image encryption algorithm is proposed by combining wavelet chaotic neuron with delay with classical permutation-diffusion [7] structure. The performance index of the encryption result is analysed.