Dynamic Analysis And Application Of Generalized Kuramoto Models

The asymptotic behavior of generalized Kuramoto models is one of the most important topics in the study of multi-agent coupled systems and its nonlinear dynamics has a wide range of applications in image recognition and power grid.Two kinds of generalized Kuramoto models are studied,which are the binary pattern recognition model and power grid models.The details are as follows:1.A generalized Kuramoto model with mutually orthogonal standard patterns is studied.As a gradient system with periodicity,any solution of the model will converge to some equilibrium point.By establishing the corresponding relationship between binary patterns and equilibria of the system,the storage problem of binary patterns is transformed into the stability problem of equilibria,so as to recognize a standard pattern from a defect pattern.Most of the existing methods are based on eigenvalues of matrix,which is not conducive to solving the case of large dimension.In this thesis,the concept of stability of binary patterns independent of parameters is proposed.With the help of the minimum energy principle of the gradient system,a sufficient and necessary condition for judging the stability of equilibria corresponding to binary patterns is given,which is not only simple in structure and easy to calculate,but also can explain the meaning of stability from the geometric point of view.At the same time,a lower bound of the critical strength is given for the binary pattern whose stablity is parameter dependent.Finally,several numerical examples are given to show the effectiveness and practicability of the methods.2.A generalized Kuramoto model with non-mutually orthogonal standard patterns is studied.Based on the theoretical results of mutually orthogonal standard patterns,the stability of equilibria corresponding to non-mutually orthogonal standard patterns is further considered.To stabilize equilibria corresponding to standard patterns,the existing method is to adjust a parameter of the system,however it not only increases the number of stable equilibria,but also is not conducive to the successful recognition of defective patterns.With the help of the criterion of the stability of binary patterns under the mutually orthogonal standard patterns,this thesis will provide two new strategies.By looking for the mutually orthogonal memorized patterns,the stability results of the non-mutually orthogonal standard patterns independent of the parameters are given,and the feasibility and effectiveness of the results are verified by numerical experiments.3.A Kuramoto model with an adaptive power grid is studied.There are many research results on the first order and second order power grid models with constant voltage.However,for some complex systems showing synchronization,the voltage changes with time to a certain extent.In this thesis,the theoretical analysis of a second order power grid system with dynamic voltage is carried out,which is helpful to make a quick judgment on the transient stability of power system.Most of the existing methods are to directly estimate the phase and frequency,and the connection coupling between rotor angles is set to be positive,but the voltage makes the connection coupling be positive or negative,so this method fails.In this thesis,numerical experiments are used to show that phase synchronization and voltage stability will not occur under certain conditions.Then,by constructing a energy function including phase,frequency and voltage,the region of attraction of a stable equilibrium point corresponding to synchronization is given.It is proved that the system finally achieves phase synchronization and voltage stability,and the convergence rate of the solution is estimated.4.A Kuramoto-Like model with frustration is studied.The analysis of the asymptotic behavior of the system is more complicated due to frustration and the nonlinear structure of phase and frequency.In this thesis,the synchronization of phase and frequency is discussed in two cases: identical natural frequency without frustration and nonidentical natural frequency with frustration.In the former case,the main task is to generalize the domain of attraction in the existing literature and prove that the system finally achieves phase and frequency synchronization.In the latter case,the domain of attraction of a stable equilibrium point is estimated for the first time,and it is proved that the system finally achieves phase and frequency synchronization.Finally,the rationality of the results is verified by numerical examples.