Cluster Oscillation Mechanism And Control Of Multi-scale Coupled Systems

With the development of microelectronics technology,most current circuit systems have nonlinear or time-varying characteristics.Research and analysis of non-linear circuit systems will help people better explore,understand and transform the world.Therefore,the study of non-linear circuit systems has become increasingly important.In this paper,the dynamics modeling of a clustering and bifurcation model for a class of multi-scale fast and slow coupling systems is performed.The four-step variable-length Runge-Kutta method is used to simulate the system cluster distribution and bifurcation modes.The system's bifurcation diagrams,partial maps,system phase diagrams,time history diagrams,maximum Lyapunov exponent(TLE),bifurcation trajectories of fast-changing systems,superposition diagrams of fast-change system balance points and system phase diagrams,etc.The effects of system parameter changes on the clustering mechanism and bifurcation mode of the fast and slow coupling circuit system,the stability of the equilibrium point of the fast-changing system and the bifurcation trajectory of the fast-changing system under the coexistence of multi-attraction domain are analyzed.The main research content of this paper is as follows:By introducing a periodic excitation alternating current source,a dynamic model of a three-dimensional smooth coupled cluster circuit system is established,and the system differential motion equation is written and dimensionlessly processed.By adjusting the external excitation frequency,there is a huge difference between the natural frequency and the external excitation frequency of the system,and two time-scale effects are presented,that is,the fast variable,the slow variable,and the fast variable system composed of fast variables.The slow variable plays an auxiliary regulatory role.The frequency and amplitude of the current source were adjusted.Numerical simulations were performed using C language programming under multiple sets of parameters.The superposition of the balance point and plane phase diagrams of the fast changing system indicates that the system has different speeds and amplitudes under different excitation frequencies and amplitudes.At the same time,the interactions between fast and slow variables in the coexistence of multi-attraction domains are analyzed,which results in FHHF clustering mechanism and Fold-Hopf bifurcation mode in the fast-changing system.The combination of numerical simulation and theoretical analysis shows that if the natural frequency of the system differs from the magnitude of the external excitation frequency,the fast and slow coupling phenomenon exists in the fast-changing system.The smaller the external excitation frequency is,the faster and slower the system is when the amplitude is slightly changed or unchanged.The more obvious is the cluster discovery.Finally,using the Riemann-Liouville type fractional differential operator,the Laplace transformation and time-frequency conversion are used to performperiodic orbital control of the dynamic behavior evolution of the fractional-order three-dimensional smooth fast-slow coupled cluster circuit system.Utilizing voltage-controlled current-type resistors' volt-ampere characteristic curve piece-wise linear-introducing non-smooth factors to establish a four-dimensional non-smooth coupling tufting circuit system,writing system motion differential equations and performing non-dimensionalized processing,adjusting system parameters to make four-dimensional non-smooth coupling tufting circuit system exhibits two time scale effects.Using the Routh-Hurwitz decision theorem and the generalized Jacobian matrix,the stability and non-smooth bifurcation of the fast-varying system are calculated.Through numerical simulation,the FFFHF clustering mechanism and Fold-Hopf bifurcation mode generated when the fast-changing system traverses the interface are analyzed,the parameters of the system are adjusted,and the FF clustering mechanism generated when the fast-changing system crosses the interface is studied again.The Fold-Fold bifurcation model and the FFHF clustering mechanism and the Fold-Hopf bifurcation model have revealed the formation mechanism of different clusters when the multi-attraction domains coexist and the effect of non-smooth bifurcations on cluster finding.Finally,the four-dimensional non-smooth coupling system with fast and slow couplings is transformed by fractional order using Riemann-Liouville type fractional differential operator and Laplace transform.By comparing the above two models and their clustering mechanism and bifurcation mode,although the smooth and non-smooth models are achieved through the interchange of current sources and linear inductors and resistors,the clustering mechanism and bifurcation of the two models are achieved.There are huge differences in the patterns.It can be seen that the introduction of non-smooth factors causes the system's cluster distribution to become more and more complex,which makes the system produce different clustering mechanisms and delivery patterns,and reveals the formation mechanism of different cluster distributions when there is more attraction coexistence.And the effect of non-smooth bifurcation on the clustering mechanism.