| In this thesis,by bifurcation theory and numerical simulations,we focus on the dynamic behaviors of several typically nonlinear dynamical systems,including codimension-1 bifurcation,the existence and stability of periodic solutions and chaos,codimension-2 bifurcation and nonlocal bifurcation to confirm the evolvement rule and set up some theoretical foundations for explaining and controlling certain physical phenomenons.The thesis can be mainly divided into the following four sections.Firstly a discrete predator-prey model with Crowley-Martin functional response is discussed.It is proved that this model undergoes flip bifurcation,Neimark-Sacker bifurcation and chaos in the sense of Marotto.An explicit approximate expression of the invariant curve,caused by Neimark-Sacker bifurcation,is given.Moreover numerical simulations using AUTO and MATLAB including bifurcation diagrams,phase portraits and Lyapunov exponents diagrams,are presented to verify theoretical results.Then a two-dimensional discrete model is studied to illustrate how two pieces of information interact in online social networks.According to the correlation of two pieces of information,the model can be classified into the following three types: Reverse type,Intervention type and Mutualistic type.We investigate1:2 resonance,1:3 resonance and 1:4 resonance caused by the absence of three non-degenerate conditions for Nermark-Sacker bifurcation.It is found that the model undergoes flip bifurcation,Nermark-Sacker bifurcation and heteroclinic bifurcation near 1:2 point;has a homoclinic structure formed by the stable and unstable invariant manifolds near 1:3 resonance point and four small invariant circles bifurcated by Nermark-Sacker bifurcation near 1:4 resonance point.Moreover for information security,we give two control strategies to control chaos and bifurcation respectively in the two-dimensional discrete system.Next a kind of jerk equation which has explicit physic meaning is investigated.For a continuous version jerky class,the direction of Hopf bifurcation is derived,and the expression of bifurcating periodic solution is presented.The largest Lyapunov exponent,phase portraits and power spectrum are given to support theoretical results.Besides,constant control,state feedback control and time-delayed feedback control are considered in the continuous jerk class.It is shown that the discrete version of jerk class undergoes Neimark-Sacker bifurcation by applying center manifold theorem.Periodic-12,47 orbits are obtained from numerical simulations.Dynamic theory is applied to material science in the last chapter.The dynamic and statistical analyses of the serrated-flow behavior in the nanoindentation of a high-entropy alloy,at various holding times and temperatures,are performed to reveal the hidden dynamic behavior associated with the seemingly-irregular intermittent flow.Two distinct types of dynamics are identified in the high-entropy alloy.The dynamic plastic behavior at both room temperature and 200°C exhibits a positive Lyapunov exponent,suggesting that the underlying dynamics is chaotic.On the other hand,by fractal analysis and detrended fluctuation analysis,for the indentation with the holding time of 10 s at room temperature,the slip process evolves as a self-similar random process. |
