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A robust gray image encryption scheme using chaotic logistic map and artificial neural network (ANN) is introduced. In the proposed method, an external secret key is used to derive the initial conditions for the logistic chaotic maps which are employed to generate weights and biases matrices of the multilayer perceptron (MLP). During the learning process with the backpropagation algorithm, ANN determines the weight matrix of the connections. The plain image is divided into four subimages which are used for the first diffusion stage. The subimages obtained previously are divided into the square subimage blocks. In the next stage, different initial conditions are employed to generate a key stream which will be used for permutation and diffusion of the subimage blocks. Some security analyses such as entropy analysis, statistical analysis, and key sensitivity analysis are given to demonstrate the key space of the proposed algorithm which is large enough to make brute force attacks infeasible. Computing validation using experimental data with several gray images has been carried out with detailed numerical analysis, in order to validate the high security of the proposed encryption scheme.

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This work proposes a new chaotic jerk system with septic nonlinearity. The new system presents odd symmetry and undergoes typical behaviors including period doubling, merging crisis, spontaneous symmetry breaking, coexisting attractors and coexisting bubbles of bifurcations as well. The most gratifying feature discovered in this article, is the occurrence of up to eight coexisting attractors for appropriate sets of parameters. This latter feature is uncommon for a chaotic system as simple as the model proposed in this work (e.g. not reported in cubic, quintic or hyperbolic sine models). Multistability control is achieved by following the linear augmentation approach. We numerically prove that the multistable septic chaotic system can be adjusted to develop a monostable behavior when smoothly monitoring the coupling strength. More interestingly, it is found that the coupling breaks the symmetry of the chaotic jerk system and thus induces new patterns including asymmetric Hopf bifurcations; coexisting non-symmetric bubbles, critical phenomena, coexisting multiple asymmetric attractors, just to name a few. On this line, the linear augmentation scheme can be regarded as a simple means for inducing new features in odd symmetric chaotic systems. PSPICE simulation results captured from an electronic analog of the proposed septic jerk system are consistent with the theoretical investigations.