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By combining the Chebyshev polynomials and the Sigmoid function into a new non-monotonic excitation function, a new transient chaotic neural network model is constructed. The dynamic characteristics of the single neuron are analyzed by the time evolution graph and the inverted bifurcation graph of the largest Lyapunov exponent. Verify the rationality of the additional energy term of the constructed model. The effectiveness of the model is verified by nonlinear function optimization and traveling salesman problem. The simulation results show that the newly constructed model can effectively avoid the problem of converging to a local minimum in the optimization process, and can effectively solve the combinatorial optimization problem.

A new chaotic neural network is constructed by introducing trigonometric function into the self-feedback term of the transient chaotic neural network; By analyzing the inverse bifurcation graph and the maximum Lyapunov exponent of the single neuron, the chaotic dynamic characteristics of the network are analyzed; The energy function of the chaotic neural network is constructed. Its stability is analyzed and the condition of gradual stability of the network is given; The network parameters for solving the traveling salesman problem are set. The simulation results show that the new model can avoid local minima effectively and is effective for solving combinatorial optimization problems.

Chaotic Neural Network has the dynamics property of gradient descent and chaotic at the same time by introducing self-feedback item and simulated annealing strategy. The trigonometric function disturbance was introduced into chaotic neural network and chaotic neural units with function disturbance were studied. The reversed bifurcation and the maximum Lyapunov exponent were given and the dynamic system was analyzed. Energy function was constructed by applying the unified framework theory and was proven to be asymptotically stable. The wavelet chaotic neural network was applied in the function and combinatorial optimization problems. The simulation results show that the chaotic neural network with disturbance of trigonometric function can solve function and combinatorial optimization problems effectively.

Chaotic neural networks have been proved to be strong tools to solve the optimization problems. In order to escape the local minima, a new function is introduced into the self-feedback item of the transiently chaotic neural network, this paper proposes chaotic neural network with nonlinear function self-feedback by improving the self-feedback item of Chen and Aihara's network making that the network has new character which is different from linear self-feedback network. The figures of the reversed bifurcation and the maximal Lyapunov exponents of single neural unit were given, We constructed the energy function of chaotic neural network, and analyzed the sufficient condition for the networks to reach asymptotical stability and set the parameters of the networks for solving traveling salesman problem (TSP). The simulation results in solving 10-city TSP show that the new model is valid in solving optimization problems.

A cooperation model for the balance of one-shot cooperation in the two-echelon supply chain was derived following the principal-agents mechanism with the factors influencing profit distribution being known based on of supply chain coordination model. The results of analysis showed that the effort level is lower when a noncooperative strategy is adopted than when a cooperative strategy is adopted. At the same time, the optimal effort level and distribution coefficient were got by using of the model.

In the practical applications and hardware realization of chaotic neural networks, because of the instability of hardware components and impacts of the surrounding environment, the system is subject to noise interference inevitably. In this paper, in order to study the effect of white noises on chaotic neural network, the optimization performance of chaotic neural network interrupted by white noises is deeply researched and analyzed. The simulation result of chaotic neural network interrupted by white noises illustrates the influence of white noises on the system. The simulation results show that the system still has good optimization ability under the white noises with suitable signal to noise.

A chaotic neural network with self-feedback of trigonometric function is presented by introducing non-linear trigonometric function as self-feedback of chaotic neural network. The reversed bifurcation and the maximum Lyapunov exponent of the chaotic neuron are given. This chaotic neural network is used to the 10-city traveling salesman problem (TSP), and the influence of trigonometric function self-feedback on TSP is analyzed. The simulation on TSP indicates that the chaotic neural network with self-feedback of trigonometric function can realize the global optimization with faster speed and that the optimization performance of the proposed chaotic neural network is superior to that of the network with linear self-feedback.

Wavelet chaotic neural networks have successfully solved function and combinatorial optimization problems. Gauss wavelet chaotic neural units with the annealing function of subparagraph index were studied. The reversed bifurcation and Lyapunov exponent figures were respectively given. On the basis of Gauss wavelet chaotic neural network, the annealing function of subparagraph index was introduced into network, a new reformative wavelet chaotic neural network was presented. Then it was applied to function and combinatorial optimization problems. The simulation results show that the search-optimization capacity of wavelet chaotic neural network has been improved and the reformative wavelet chaotic neural network is superior to the primary wavelet chaotic neural networks.

The fuzzy integral equations are important part of the fuzzy analysis theory. The integro-integral equations are a class of important integral equations, which come from the problems on high-order differential equations or some physics, mechanics, medicine or control engineering problems. This paper studies solvability to a class of new fuzzy integro-integral equations and gives some new results.