Bifurcation Analysis And Control For Several Kinds Of Delayed Systems

As an important factor in measuring the performance of dynamical systems,Hopf bifurcation not only has profound theoretical significance,but also it has widespread engineering background and potential application prospect in fields like electric systems,viscoelastic systems and biological systems.A great number of studies have found that the evolution of most dynamical networks is not only subjected to the current states but also the past ones.In fact,Time delays happen frequently in practical systems and they have the capability of destabilizing the systems and generate bifurcation,chaos and other complex dynamical phenomena.Therefore,scholars nowadays are highly focused on the dynamic analysis of different types of delayed systems.Based on the previous work,I will attempt to investigate the dynamics of three types of complex networks,namely the fractional controlled neuron system,hybrid controlled neuron system with the leakage delay,and fractional-ordered genetic regulatory network.The main work is as follows:(1)The problem of the delayed neuron model and its fractional PD control is discussed.With the time delay chosen as the bifurcation parameter,the local stability of the controlled system and its Hopf bifurcation performance are then discussed.The research finds that the order of the fractional PD controller and the its parameters are closely related to the bifurcation point of the system.And with the increase of the controller parameters,the bifurcation point will decrease while with the increase of the order of the controller,the bifurcation point increases as well.(2)The neuron system with the leakage delay and the distributed delay described by the strong kernel function will firstly be proposed.Then the hybrid control strategy will be applied to the proposed neuron system.With the leakage delay of the neuron system as the bifurcation parameter,the local stability of the controlled system and its Hopf bifurcation performance are then discussed.Results show that with controlling parameter of the hybrid controller increasing,the Hopf bifurcation point of the neuron system will also increase.(2)The issue of the genetic regulatory network with the leakage delay and the discrete delay is studied.With the delay of the leakage term chosen as the bifurcation parameter,the local dynamics of the controlled system are then discussed.The relationship between the fractional order and the Hopf bifurcation point will be further investigated.Results also show that with the increase of the fractional order,the bifurcation point of the system will be decreased.Results from this paper indicate that the dynamics of the system is closely related to the system order,the time delay,the system parameter and the feedback gain parameter of the controller.Firstly,the order of the system will not only affect the stability of the fractional order system,but will delay or advance the onset of the Hopf bifurcation phenomena.Therefore,the local stability will be improved by altering the system parameters like the system order.Moreover,the application of the controller to the system can also make a huge difference to the system dynamics.By changing the feedback gain parameter of the controller,the Hopf bifurcation will thereby be adjusted.The results of this paper will extend the theory of the nonlinear dynamics and also,it will provide solid theoretical background for the practical application of the dynamical system control in the engineering field.