Bursting Oscillations And Its Mechanism Of The Nonsmooth Hindmarsh-Rose Model With Slow-Varying Periodic Excitations

Due to the ubiquity of non-smooth factors,the theory of non-smooth systems has always had a wide application background.As one of the most special non-smooth dynamical system,Filippov system,which has discontinuous vector filed on both sides of the non-smooth interface,may cause the trajectory to produce very complex dynamic behaviors,such as sliding when the trajectory comes into contact with the interface.Meanwhile,the existence of multi-scale structure may cause the system transform between fast and slow processes.When the coexistence of multi-scale structure and nonsmooth structure are involving in the system it may show a special non-smooth bursting oscillation phenomenon,which cannot be solved via the traditional dynamics.Therefore,it is necessary to further develop the corresponding theory system.What's more,the research about different control policy in dynamical systems has attracted more and more attention in recent years.According to the above discussion,we proposed a Filippov system of Hindmarsh-Rose neuronal model with Periodic slow-varying excitations.Based on the different control policy,such as amplitude control policy and threshold control policy.The main work of this dissertation can be divided into the following three parts:Part one: In this part,we established a two dimensional Filippov Hindmarsh-Rose model with amplitude control policy in order to investigate the mechanism of slow-fast dynamics in different scales coupled system.Firstly,we use slow-fast analysis to obtain the Generalized autonomous system i.e.fast subsystem.Through bifurcation analysis of the fast subsystem,we get the equilibrium curve with the S-shaped structure.It is also found that there is a pseudo-equilibrium point on the non-smooth interface in the Filippov system,which will cause the trajectory to slid along the interface.When controlling the excitation amplitude,the slowly varying parameters will pass through different bifurcation areas,leading to the fast subsystem involving different types of bifurcation,which leads to changes in the structure of the fast and slow motion of the entire coupled system.According to the specific influence of the bifurcation involved in the system on the fast and slow motion structure,the bifurcations involve in the system can be roughly divided into two categories: the first are the bifurcations of equilibrium which may cause the trajectory of the coupled system to reciprocate between the fast process and the slow process,such as Hopf Bifurcation and non-smooth Hopf bifurcation;the second are the bifurcations of limit cycles that lead to structural changes in the fast process,such as the switch-sliding bifurcation of limit cycles,and the gaze-sliding bifurcation of limit cycles.Part two:This part still uses the system in the first part for research.The only difference between the two part is that a different control policy called threshold control policy is used at this time.By using the fast and slow analysis method and the bifurcation analysis method to study the system,we found that the non-smooth bifurcation of the system may occur at the intersection of the interface and the trajectory.When we change the threshold,the contact position of the non-smooth interface and the trajectory may also change.This will cause the location of the bifurcation of the boundary equilibrium point in the fast system to change,such as non-smooth Hopf bifurcation.There are limit cycles exist in the fast-subsystem.When the threshold parameter is controlled,non-smooth bifurcations of the limit cycle will occur,such as the switching-sliding bifurcation of the limit cycle and the gazing-sliding bifurcation of the limit cycle.These bifurcations will affect the fast and slow motion structure of the coupled system.The change of the position of the non-smooth interface will cause the system's equilibrium point and smooth bifurcation point to switch back and forth between the real state and the virtual state,thereby controlling the movement of the trajectory or losing control of the trajectory movement.Part three: This part still adopts the threshold control policy to study the fast-slow effect of the three-dimensional Filippov-type Hindmarsh-Rose system with Parametric and external excitation.At this time,it is found that due to the existence of the parameter excitation term,the equilibrium curve of the system is no longer a smooth S-curve structure,but a hyperbolic structure with an asymptotic line.There is a subcritical Hopf bifurcation in the smooth subregion of the system,which will lead to the generation of smooth limit cycles.As the threshold changes,smooth homoclinic bifurcation,as well as non-smooth fold bifurcation and non-smooth global bifurcation appear in the system.The appearance of non-smooth global bifurcation will lead to the mutual evolution between the pseudo-equilibrium point and the non-smooth limit cycle.The appearance of these bifurcations will change the structure of the bursting oscillation of the coupled system.In addition,the subcritical Hopf bifurcation has a slow channel effect,which will cause the trajectory to maintain an oscillating motion with a gradually decreasing amplitude for a certain distance when passing through the bifurcation before finally converging to a stable equilibrium branch.