Mixed Mode Oscillations And Limit Cycle Bifurcation In Two Types Of Nonlinear Systems

Mixed mode oscillation is a complicated nonlinear behavior,which involves small amplitude oscillations and large amplitude oscillations,and exists in many areas,such as heart beating,chemical reaction,neuronal firing,economic fluctuation,etc.Limit cycle is a closed orbit in plane dynamical system,which can be categorized as stable and unstable.Many engineering phenomena are related to limit cycle,such as wing flutter,vehicle slalom and so on.In this thesis,mixed mode oscillations in calcium oscillation system and periodic parametric perturbated unified chaotic system,and limit cycle motions in galloping of iced transmission line,are mainly focused.As an important second messenger in cell,calcium ions can regulate various cel-lular process by the way of oscillation.Variety of information or signals in cells are encoded by frequency and amplitude of calcium oscillation.Most of the existing litera-tures analyzed calcium oscillations using numerical method,and found rich dynamical behaviours,i.e.,mixed mode oscillations.In this thesis,an established three-store mod-el is considered,which contains endoplasmic reticulum?ER?,mitochondria and calcium binding proteins.Linear analysis is used to identify two Hopf bifurcation points in this model,and nonlinear analysis is applied with normal form theory to study stability of limit cycles.This study indicates that Hopf bifurcation is a source of the oscillation behaviors.Further,dimensionless process is conducted to transform the model into a slow-fast system,and then the geometrical singular perturbation method?GSPM?is employed to investigate the mechanism of generating slow-fast motions.The folded singularities and singular orbits are identified,and numerical simulations are presented to demonstrate the correct analytical predictions.To be more realistic,an influx channel,called store-operated calcium entry chan-nel?SOCE?,is taken into consideration,and a new model to describe the calcium ex-changes and interactions between different stores is proposed.By employing bifurca-tion analysis on the model,change law of the amplitude and period of oscillation with the maximum capacity of SOCE is obtained.Furthermore other four important pa-rameters are selected to perform simulation separately,the influences of which on the oscillation system are discussed.The result shows that the amplitude decreases with k_{S OCE}and k_ch,while the period of oscillation increases.For the physical meaningful value of k_Mitoutand k_Mitin,the system is always in the oscillatory state,and the oscilla-tion period slightly changes.The amplitude rises with k_Mitoutgradually,whereas sharply decreases with k_Mitin,indicating the system is more sensitive to k_Mitinthan k_Mitout.When varying k_basalfrom small to large,it is read that the amplitude decreases with k_basal,and the system does not oscillate anymore for large value of k_basal.Since the close relationship between SOCE and tumor,physiological progress sug-gests that SOCE could be selected as a therapeutic targets for tumor or cancer.Consider-ing the targeted chemotherapy,a mathematical model of tumor-immune system is con-structed.In present thesis,qualitative analysis on the mathematical model is conducted.It shows that if some conditions hold,then all solutions of the model are non-negative for non-negative initial conditions,and bounded.Moreover,two types equilibrium solu-tions are obtained.The tumor free equilibrium solution is locally asymptotically stable if some conditions are satisfied.Furthermore,to make the tumor free equilibrium so-lution to be globally stable,the condition should be compacted.Based on numerical calculations,it states that the stable range of tumor free equilibrium solution of tar-geted chemotherapy is larger than that of regular chemotherapy in the same condition,confirming that the targeted chemotherapy is benefit to kill tumor cells.Mixed mode oscillations in the periodic parametric perturbated unified chaotic system are studied,which is a nonautonomous system.The system has one equilibrium solution or three equilibrium solutions depending on different combination of parame-ters.Then,based on the methods of bifurcation analysis,the conditions of static and Hopf bifurcations are derived.It shows that there exists seven different bifurcation cas-es.Using numerical simulation,several typical bursting phenomena in the model are performed,and the underlying mechanisms are explained.Applying Melnikov method,the bifurcation condition of homoclinic orbits is derived.Numerical simulations are done to demonstrate that the analytical result is appropriate.Considering the geometrical and aerodynamical nonlinearities,a model of in-plane galloping of iced transmission line is established using Hamilton principle.After rescaling,a near Hamiltonian system is obtained,in which Melnikov function expan-sion is employed to study the limit cycles.It is demonstrated that this model can have at least 4 limit cycles in some wind velocities.Moreover,by employing numerical sim-ulation,the theoretical results are illustrated and the stability of these limit cycles are determined.The results show that under some engineering and environmental condi-tions,the iced transmission line system may behave stable limit cycle motion,which is harm to the safe operation of transmission system.