Research On Bifurcation And Chaos For A Class Inverted Pendulum System

The inverted pendulum system is an absolutely unstable system with the typical nonlinear terms and high order.Previous studies show it can produce abundant complex dynamics.By employing the theories of chaotic dynamics and bifurcation,the varying of various complex dynamic behavior of the inverted pendulum system is studied in great detail in present thesis.The research serves a theoretical result for practical application in engineer science.Furthermore,the methodology of present research,especially,the employed nonlinear theories and methods,guides the direction of studying other analogous inverted systems.Numerical simulations illustrate the bifurcation and chaos of the inverted pendulum system,which are investigated by using related nonlinear analysis.Parameter control is helpful to depict the transitions between stable and unstable regions on multi-dimensional parameter space.The main results of the thesis are listed as follows:At first,the research background,previous research results and what are concerning of the inverted pendulum system are introduced briefly.It proposes the problems to be settled in studying the nonlinear dynamics of the inverted pendulum system,and then gives a brief introduction on the bifurcation of the system.Meanwhile,the research methods to the nonlinear dynamics of inverted pendulum system are presented.The second part is devoted to modeling the dynamics of the inverted pendulum system.The dimensionless equations of the system are obtained via introducing dimensionless variables.It is necessary to translate the dimensionless motion equation to the state-space equations.This case provides convenience for studying the successive chaos and bifurcation analysis about the system.The third part discusses the stability of the inverted pendulum system at the equilibrium point.Generally in terms of the Routh-Hurwitz criterion,the parameter conditions for stabilizing the system can be found.In the case of some special critical conditions,it needs to use the center manifold theory to reduce the system dimension from four to two due to the invalidity of general methods.Hereby the reduced system is simplified by the PB Normal Form theory to get its polar form.Based on the polar form,the conditions of Hopf bifurcation in the system can be got by using the Normal Form theory to analyze the stability of the system.The existence of the chaotic attractor of the system is discussed.The divergence of the system is discussed and the statistical aspects,namely,Lyapunov exponents are calculated with Wolf algorithm.The related results numerically verify that there are some distinct chaotic attractors in the inverted pendulum.Moreover,the existence of the Hopf bifurcation is confirmed by exploring the transversality condition.Related numerical simulations are completed to validate the above theoretical results.Detecting the types of bifurcations of the system becomes very important.Fortunately,a friendly software named Matcont is quite qualified for such a hard work.Perturbation analysis of nonlinear response of the inverted pendulum system is given in part four based on multiple method of scale.The average equation and the frequency response equation of the inverted pendulum system are quantified by introducing a small disturbance quantity when it is in1:1 internal resonance.Based on the numerical simulations of the single mode frequency response equations,the changes of the system on the frequency response curve with the change of the frequency and damping coefficient are demonstrated.It really affects the frequency range and the number of the nonzero solution.In part five,the main aim is to make sure what bring about the changes of the dynamics of the inverted pendulum system by varying either single or two parameters.The bifurcation and chaos of the system can be explored via the quantitative and qualitative methods.Especially,some single parameters,such as damping term coefficient,the mass ratio term coefficient,on play an import role in changing the dynamics of the system.The stability of the system can be expressed in a qualitative way with the bifurcation diagram,phase diagram,time response and Poincaré section by varying different parameters.And the stability conditions of the system can be determined in the quantitative way by the Lyapunov exponent.Whereby,the stable interval of the inverted pendulum system is determined.Moreover,the change of the two parameters is also studied in according to the actual situation.Two-dimensional parameters of the system are controlled in combining the bifurcations in the two-dimensional parameter planes with the Lyapunov index spectrum.In the end of part five,a stable parameter region of the inverted pendulum system by means of various parameters matching is gained.