Homotopy Zeroing Neural Networks Method For Solving Systems Of Time-Varying Nonlinear Equations

The time-varying nonlinear equations often arise in the fields of mathematics,physical engineering,robot kinematics,real-time online and so on.For solving this problem,a new recurrent neural networks named as zeroing neural networks(ZNN)was first proposed by Zhang et al.in 2001.The ZNN method has been received extensive attention and many excellent improved methods have been developed in recent years.Its convergence theorems were proposed by Zhang et al.in 2012.Based on them,we mainly study a new method named as homotopy zeroing neural networks(HZNN)and propose a bidirectional iterative algorithm for convergence during the whole time.First of all,we establish a connection between the ZNN method and the Newton method,analyse the original convergence theorem in the related literatures,and put forward a more perfect convergence theorem.Then we give a more complete error analysis for the single step discrete ZNN method,which is verified by the numerical results.Secondly,two examples of the inapplicability of the ZNN method are given.In order to extend the ZNN method to more time-varying problems with a larger convergence range,we combine the ZNN method with the globally convergent homotopy continuation method,and propose a homotopy zeroing neural networks(HZNN)method.Then its convergence theorem is proved and its error is analysed for the single step discrete HZNN method.The numerical results show that the new method can effectively avoid the divergence of the ZNN method and its final error accuracy is no less than the ZNN method's in the case that the ZNN method also converges.Finally,we further improve the ZNN method and the HZNN method.A reverse ZNN(or HZNN)method is constructed to solve the time-varying problems for t ?(-?,0].Then,a bidirectional ZNN(or HZNN)method is proposed to approach the exact solution for original problem over the whole time.The numerical experiments demonstrate the effectiveness of these bidirectional methods.