Some Issues In Nonlinear Dynamical Systems With Different Scales

In science and engineering,the dynamical behaviors of nonlinear system which is often composed of a certain number of coupled subsystems may be no longer a simple superposition of the behaviors of each subsystem,but involve the dynamical evolution from the subsystem level to the system level,leading to the coupled phenomena of systems with multiple scales.As one of the important branches in nonlinear science,two-timescales system which can exhibit many special nonlinear dynamical phenomena and be widely used in engineering has been paid much attention by scholars at home and abroad.It is one of the leading subjects and hot issues in nonlinear dynamical systems at present.The factors which cause the nonlinear system to contain different timescales not only exist in real time,such as the different velocities between the different subsystems,but also in the different characteristics of subsystems,such as the mass,size and stiffness,etc.Considering the dimensionless mathematical model,this kind of systems can be roughly divided into two categories.The one is in the time domain,in which there exists an order gap between the coupled vector fields,the other is in the frequency domain,in which there exists an order gap between the frequencies.Furthermore,these two kinds of coupled systems are essentially equivalent.Though a lot of results have been reported in different dynamical systems with two timescales,the related research is still in the stage of development because of the particularity of these systems,and many problems need to be further studied.In this dissertation,based on the fast-slow analysis,effective methods adopted for the concrete systems are proposed to divide the whole system into fast subsystem and slow subsystem.And then,the modern analytical methods in nonlinear science,such as bifurcation theory and numerical simulation,are used for exploring the mechanism and the transition process of different bursting pattern according to the four key problems existing in the current research.The basic contents and conclusions of this dissertation are given as following:In the frequency domain,when an order gap exists between the exciting frequency and the natural frequency in a nonlinear dynamical system with two timescales,bursting oscillations behaving in the combinations between large amplitudes oscillations andalmost quiescent states may appear.For the case when the exciting frequency is much smaller than the natural frequency,the whole exciting term can be considered as a slowvarying parameter,leading to the generalized autonomous system.To investigate the mechanism of bursting oscillations,the fast-slow analysis and the transformed phase portraits can be employed to account for the movements of the trajectories with the variation of the slow-varying parameter.After calculating the local bifurcations of the fast subsystem with respect to the slow variable,including stability of equilibrium branches,fold bifurcations and Hopf bifurcations,the typical periodic oscillations as well as the mechanism,such as symmetric fold/fold bursting,Hopf/Hopf bursting and Hopf/Hopf bursting via hysteresis loop,are revealed.A non-autonomous system with piece-wise linear function is established to explore the mechanism of nonsmooth system with two timescales.By introducing the definition of nominal equilibrium orbits,critical conditions corresponding to regular and nonsmooth bifurcations are derived based on the bifurcation analysis.For the suitable parameter values,there exists an order gap between the exciting frequency and the natural frequency.By discussing the dynamical evolution of the fast subsystem with varying amplitude of the external excitation,the trajectories pass across different numbers of nonsmooth boundaries are investigated,resulting in different forms of bursting oscillations.Finally,it is pointed out that the transition between the quiescent state and the spiking state can be caused by the nonsmooth bifurcations.A nonlinear system with sinusoidal varying term,in which there coexists multiple equilibria,is introduced to display the dynamical phenomena of the system with two timescales.Periodically changed nonlinear terms with respect to the state variable,may cause the equilibrium points as well as the related bifurcation sets to exist periodically,which can lead to the coexistence of the multiple fold bifurcation points as well as the same number of scrolls.After discussing the mechanism of different type of the bursting oscillations and the fine structure of the spiking state,the main influence of multiple coexistence equilibria on the complex behaviors of the system is revealed.The number of fold bifurcation points will increase with the increase of the amplitude of periodic external excitation;besides,the distribution of fold bifurcation points has a direct influence on the structure of the bursting attractor.To display the phenomena of slow subsystem with two slow variables,a nonlinear coupled system with one parametric excitation and one external excitation is discussed.It is proposed that the ‘slowest' timescale,which is a slow-varying parameter determined by the product of the minimum exciting frequency and the real time,can be considered as the bifurcation parameter.Another form of generalized autonomous system is defined,so that the complex dynamical behaviors of the system can be analyzed directly by use of the fast-slow analysis.After giving the formula for calculating the period of equilibrium branches,it is pointed out that the number of timescales in the original nonlinear system may change when the frequency ratio is large enough or small enough.