Dynamics And Control Of Some Higher Order Systems With Time Delay

It is well known that the characteristic equation of higher order delayed systems istranscendental equation which has very high orders. Normally, for the system with multipledelays, it is impossible to get the stability condition that all the eigenvaues have negative realparts; it is also difficult to give the stability boundary conditions for such systems. Existingtheories of nonlinear dynamics are effective for low-dimensional systems, but many of themare difficult to apply to higher order systems, especially higher order delayed systems. In fact,an effective dimension reduction method can make the analysis of stability, bifurcation andchaos of complex system easier. How to reduce the dimension of large, complexhigh-dimensional systems has become one of the hot research of nonlinear dynamics.In addition, whether the dynamics control of the higher order delayed systems has newproperties? If the displacement feedback control and velocity feedback control methodswhich is used in the engineering practice can effectively control the higher order delayedsystems? These are new problems worthy to be discussed. For the above problems, thispaper analyzes the dynamics and control of several kind of higher order delayed systems,including the dynamics and control of the small-world networks, the small world networkwith memory, dynamics and control of the viscoelastic vibration system, dynamics andcontrol of the vibration system with delay dependent parameters and dynamics of two neuronsystem with delay dependent parameters. This dissertation focuses on the followingproblems.(1) A detailed analysis on the collective dynamics and delayed state feedback control ofa three-dimensional delayed small-world network is presented. The trivial equilibrium of themodel is first investigated and shows that uncontrolled model exhibits unbounded behavior.Then presente three control strategies, namely a position feedback control, a velocityfeedback control, and a hybrid control combined velocity with acceleration feedback. Itshows in these three control schemes only the hybrid control can easily stabilize the3-Dnetwork system. And with properly chosen delay and gain in the delayed feedback path, thehybrid controlled model may have stable equilibrium, or periodic solutions resulted from theHopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Theresults of this paper are further extended to any “d” dimensional network. It shows that tostabilize a “d” dimensional delayed small world network, at least a “d-1” order completeddifferential feedback is needed. This work provides a new thought for the high dimensionaldelayed systems.(2) A nonlinear small-world network model with memory is presented to investigate the effects of the distribution delays on the dynamical properties of small-world networks.Stability analysis shows that the total infected volume of this model are unbounded whichresult in the exponential growth of an infection in networks. We also discuss the effect offinite size on the network dynamics, and give the saturate time which all the nodes areinfected by the initial infection. The dynamical control is also investigated by introducing thedelayed feedback to the small-world networks. For the appropriate feedback gain and timedelay, controlled model may have stable equilibrium, periodic solution, quasi-periodicsolution, or chaos from a sequence of period-doubling bifurcations. It shows delayedfeedback control may find important applications in the management and dynamical control.This shall be the motivations of some further studies of the dynamics of real networks.(3) The dynamics and delayed state feedback control of oscillator with generalizedviscoelastic item is discussed. It shows that uncontrolled model exhibits complicatedunbounded behavior. It shows that with properly chosen delay and gain in the delayedfeedback path, the velocity controlled model may have stable equilibrium, or periodicsolutions resulted from the Hopf bifurcation, codimension2bifurcation or complex strangerattractor.(4) The dynamics of a delayed oscillator with delay-dependent parameter is presented.On the basis of stability switch criteria, the equilibrium is studied, and the stabilityconditions and the periodic solutions bifurcating from the equilibrium are determined. Theresults show that as the delay varies from zero to infinity, the oscillator may undergo anumber of stability switches, it could be eventual stable or unstable. This is very differentfrom that in the system without delay-dependent parameters. In addition, the numericalstudies show there are two routes to chaos for this oscillator even with the very simplehyperbolic tangent nonlinear function: period-doubling bifurcations process to Chaos andquasi-periodic solutions bifurcation to Chaos. It shows that appropriate memory function canbe used as a promising scheme to control the system dynamics.(5) A detailed analysis on the stability switches of an inertial two neurons model withdelay-dependent parameter are presented. It shows that system will undergo a finite numberof stability switches with an increase of time delay. The system finally will be stable orunstable. Moreover, the paper also analyzed the unique dynamics of the system by numericalsimulation, including complex periodic motions and chaotic solutions.Finally, a summary of this dissertation is given. The further studies on sytems with timedelays are discussed.The innovations of this dissertation are as follows:(1) A collective model of asmall-world network with memory is proposed, and a dynamical control scheme is given forthe effective control.(2) An effective delayed control scheme is proposed for the higher order system control.(3) For the sysems with delay-dependent parameters, memory functioncould be one of the promising schemes for system control.