Multistable Analysis And Prediction Of Nonlinear Time-Delay Equations Based On Machine Learning

The dimension of linear systems has always been within a feasible range for investigation,but the dimension of nonlinear systems can continue to increase,which limits the traditional methods of calculating and visualizing attractors.The challenge of analyzing and visualizing the multiple stability of nonlinear time-delay systems in infinite dimensions is presented.To overcome this problem,this paper begins with Fourier transform orthogonal basis and customizes a pair of standard orthogonal bases to approximate the delay state of nonlinear time-delay systems,and uses basis functions and coefficients to represent the energy of signals,determining the stability of the system through energy.At the same time,the ordinary attractor is also extended to the statistical attractor,and Monte Carlo method is used to estimate the multiple stability for different initial conditions.The theoretical validation of the attractor and attractor domain,which is also the delay state of the nonlinear time-delay equation,is provided in a chapter to ensure the correctness of the theory.In the subsequent papers,the energy is used to extract system characteristics and classify different stable states.In this process,multiple sets of initial conditions are used for energy analysis after frequency domain transformation.On the one hand,the number of base functions that can best approximate the system equation can be obtained through comparison of different base function numbers.On the other hand,after obtaining the appropriate number of base functions,energy visualization and grading are required.Different equations have different numbers of attractors.There are two stable states in the cutting system: chatter and steady,so this paper only focuses on the stability of one state.For the time-delayed inertial neural dynamic system,after frequency domain analysis,there are three different attractors,so energy analysis and prediction are required for all attractors.The time-delayed nonlinear isolation system has the most attractors,five,making it more difficult to analyze and predict.Although energy analysis can be used to directly analyze the stable state of the system,there are many sets of initial conditions for different systems,and the number of base functions for different systems may not be completely applicable.Therefore,neural networks are proposed to facilitate computation.However,since the initial conditions of the system in this paper contain both delayed and non-delayed terms,the neural network model needs to be modified according to the initial conditions.Therefore,three network models are proposed for training and analysis comparison.Overall,this paper aims to analyze and predict the steady-state model of nonlinear time-delay systems,extract features based on system characteristics,and train the neural network model proposed in this paper.Different neural network models are analyzed based on the data of different system models.