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In this paper, we study the maximum number, denoted by H(m, n), of hyperelliptic limit cycles of the Liénard systemṡwhere, respectively, f m (x) and g n (x) are real polynomials of degree m and n, g n (0) = 0. The main results of the paper are as follows: We obtain the upper bound and lower bound of H(m, n) in all the cases with n = 2m + 1. When n = 2m + 1, we derive the lower bound of H(m, n). Furthermore, these upper bound can be reached in some cases.

Abstract. The optimal design for computer control software is studied. Since computer control software is susceptible to interference during the control process, a computer control software design model based on improved PID algorithm is proposed. The PID algorithm is combined with particle swarm algorithm to calculate PID control parameters, which is viewed as evolutionary particles of the particle population, and given a certain flight speed in the search space, the speed of the particles will be adjusted iteratively and dynamically in accordance with the experience of population's evolution calculation, in order to achieve computer control software design. The simulation results show that the proposed algorithm applied for computer control software design, can improve the control precision and meet the actual needs of computer control.

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The large-scale e-commerce website transaction security detection methods are studied. For the defect that the large-scale e-commerce website transaction security detection is susceptible to be interfered by ambient noise, a large-scale e-commerce website transaction security detection method based on characteristic attribute mapping model is proposed. According to principal component analysis method to process redundant transaction data of large e-commerce websites, so as to reduce the complexity of the data. Characteristic attributes mapping model is established to achieve intrusion detection when transacting on the large-scale e-commerce website, to determine the security of transactions. Experimental results show that the use of proposed algorithm for large-scale e-commerce websites transaction security detection, can greatly improve the accuracy of detection, so that obtaining satisfactory results.