Study On Chaotic Dynamics Of The Non-linear Dynamics System

The mathematical models and dynamic behaviors of many problems in real life can be described by non-linear systems.The main role of non-linear processes is to make the system to produce various complex behaviors,such as oscillation,chaos and bifurcation etc.Therefore,studying the complexity of these non-linear systems is still of great significance and value.The current research on the complexity of non-linear systems is mainly based on theoretical analysis and numerical simulation,and a large number of non-linear phenomena such as bifurcation or chaos have been found in numerical simulation.However,there is still lack of reasonable explanation for the causes of these complex dynamic phenomena in non-linear systems.Chua proposed that local activity is the origin of the complexity of non-linear systems.This paper first generalizes this idea to n dimensional non-linear dynamical systems,introduces the general form of n dimensional non-linear dynamical systems under the periodic forcing,and then calculates the complexity function or complexity matrix corresponding to its equilibrium point.The local passivity theory,local activity theory,and the edge of chaos criteria of the equilibrium point properties of systems are discriminated,and the parameter space near the equilibrium point is divided into a local passivity region and a local active region.For non-linear dynamical systems which satisfy the edge of chaos criteria,the equilibrium point is locally asymptotically stable without periodic forcing,that is,all trajectories converge to the equilibrium point of asymptotic stability.However,if a periodic forcing is applied,periodic or chaotic motion of the non-linear dynamic system may occur.That is to say,the edge of chaos criteria can be used to design the phase transition from ordered behavior to chaotic behavior of non-linear dynamical systems.This paper is based on the logic from simple to complex,from classical equations to engineering models,from theoretical analysis to practical application.The main study contents of this paper include:(1)Starting from the classical forced Brusselator equation,which can bifurcate from a stable equilibrium point to a chaotic state when satisfies the edge of chaos criteria.In addition,even if the edge of chaos criteria is not satisfied and the parameters are selected in the locally active region,the forced Brusselator equation can also exhibit a chaotic motion.(2)The local activity analysis is further extended to the generalized Duffing equation,that is,the forced vibration equation containing both quadratic and cubic nonlinear terms,which can bifurcate from a stable equilibrium point to a chaotic state when satisfies the edge of chaos criteria.It was found in the numerical simulation through the Poincaré section diagram that the parameters were selected at the edge of the chaos,and the system appeared highly complex chaotic behavior,such as tight compressing dynamics.(3)The cantilever beam model is widely existed in practical engineering.The local activity theory is used to analyze the local active area at the equilibrium point,and the appropriate parameters are selected in the local activity area.It is found that the cantilever beam model occurs different types of chaotic motion.