Research On The Bursting Patterns And Bifurcation Mechanism Of Multi-time Scale Coupling System

The phenomenon of multiple time scales is widespread in daily life,quietly changing the way people observe the world.Due to its extensiveness and universality,the multi-time-scale coupling system has great value and development prospects in practical engineering applications,and is the current research focus of nonlinear dynamics.Based on this background,based on nonlinear dynamics theory and analysis methods,this paper studies the bursting phenomena and generation mechanism of multi-time-scale coupled systems under multi-frequency slow excitation disturbances.The main works are given as following:1.The dynamic behavior and mechanism of clusters in the parameter-driven Rucklidge system under a single excitation are studied.Four cluster oscillation modes are obtained,namely symmetrical delay pitchfork point-point bursting,symmetrical delay pitchfork/sup-Hopf point-cycle bursting,symmetrical delay pitchfork/homoclinic-sup Hopf point-cycle bursting and composite delay pitchforkhomoclinic/sup Hopf point-cycle-chaos bursting.The study found that the occurrence of these four bursting oscillations is closely related to the delayed pitchfork bifurcation.The change of the excitation amplitude will cause the pitchfork bifurcation delay to occur in different bifurcation areas,and the system will switch back and forth in various burst modes.We completed the actual physical circuit to simulate the dynamics of the parameter-driven Rucklidge system,and conducted experimental tests on the resulting circuit.By selecting appropriate circuit parameters,the bifurcation delay behavior in the circuit can be clearly observed,and four burst modes caused by this delay behavior under different excitation amplitudes have been verified.2.The bursting behavior and generation mechanism of the multi-time scale memristive Shimizu-Morioka system are studied.The study found that after the introduction of additional excitation,the original "Hopf/Hopf" bursting appeared in two forms of evolution: the first is that the balance curve is distorted and deformed,and new extreme points appear,causing the system to exhibit small oscillations during the silent state.That is,twisted "Hopf/Hopf" bursting hair.The other is that new bifurcation points appear on the balance curve,forming a cascading "Hopf/Hopf" bursting.The article analyzes in detail the generation mechanism and evolution of these two cluster hair modes.Research shows that the relationship between the amplitude and frequency of the two excitations has an important impact on the cluster oscillation of the system.3.Based on the analysis thought of the two-time scale memristive Shimizu-Morioka system,the Bonhoeffer-van der-Pol oscillator is studied.Under single excitation perturbation,due to the rapid transition from unstable oscillation to large-amplitude relaxation oscillation in the bursting structure,the system can observe a special duck explosion phenomenon.The distribution of the Hopf bifurcation set of the BVP oscillator is affected by the parameter B0,which means that the system can be regarded as a Hopf bifurcation controllable system,and the Hopf bifurcation point can be freely shifted by a constant controller.With the introduction of additional incentives,the system produced more complex symmetrical and twisted Hopf/Hopf burstings that traverse the duck explosion,and symmetrical cascaded Hopf/Hopf bursting that traverse the duck explosion.By changing the value of B0,the symmetrical Hopf/Hopf cluster structure can be transformed to the asymmetric Hopf/Hopf cluster structure,and a new asymmetric and twisted Hopf/Hopf bursting and crossing duck can be obtained.The asymmetric cascaded Hopf/Hopf bursting with explosive explosion.The generation mechanism and evolution principle of these cluster hair modes are analyzed in detail.