Bifurcation And Chaotic Dynamics Of Nonlinear Vibration Systems Based On Parameter Modulation

In this thesis,several kinds of nonlinear vibration systems are constructed based on the classical Duffing equation.Complex dynamical behaviors of the systems are investigated by utilizing chaos theory and numerical simulations.By modulating parameters,depicting bifurcation diagrams and calculating Lyapunov exponents,the existence of chaos in the systems is verified within a wide range of parameters.By employing control theory,velocity and displacement feedback controllers are designed to eliminate chaos of the nonlinear vibration systems.Finally,a problem on chaotic dynamics and control of a three-dimensional quadratic polynomial system with infinite equilibria or no equilibria is investigated further.Firstly,the background and significance of research on nonlinear vibration systems are briefly introduced.The research status and development trend in this field at home and abroad are analyzed.The main research contents of the thesis are briefly summarized.The theories and methods are explained.A general expression on a class of single-degree-of-freedom nonlinear vibration systems including the Duffing equation is established.The local stability of the systems near the equilibrium point is investigated under the condition of no external excitations.A class of nonlinear vibration systems with a sinusoidal excitation is constructed.The theoretical analysis and numerical simulations are carried out.The velocity and displacement feedback controllers are designed to eliminate chaos.A class of nonlinear vibration systems with sine and cosine double-frequency excitations is constructed.Dynamical behaviors of the systems are investigated in detail.Bifurcation diagrams of states on damping and frequency parameters are depicted respectively.Corresponding Lyapunov exponents of the systems with the variation of parameters are obtained.Finally,simple displacement feedback controllers are designed to eliminate chaotic behaviors of the system.A class of piecewise nonlinear vibration systems with a three-scroll attractor is constructed by adding an external excitation with a single sign function.Complex dynamical behaviors of the systems are analyzed.The displacement and velocity feedback controllers are designed to eliminate chaotic phenomenon of the systems.A class of piecewise nonlinear vibration systems with a double-sign function is constructed.A four-scroll chaotic attractor can be generated from the systems.The problems on complex dynamical behaviors and chaos control of the systems are investigated.The trigonometric function and hyperbolic tangent function are introduced as the external excitations of the systems.The designed systems have the same capability of generating a four-scroll chaotic attractor.The number of scrolls is also variable by modulating the sign function.A class of nonlinear vibration systems with infinite equilibria or no equilibria is investigated.The most remarkable characteristic is that the systems have no equilibria or infinite equilibria located a straight line.By changing the parameters,the systems can generate different chaotic attractors.Dynamical behaviors of the systems are analyzed.Finally,new systems with a symmetrical parameter are constructed by comparing with the original systems.A problem on switching control of the systems is also investigated.The design approaches on double-scroll,three-scroll and four-scroll chaotic attractors in single-degree-of-freedom nonlinear vibration systems are put forward in this thesis.Impact results on system states with the variation of parameters including damping,frequency,the highest degree of nonlinear terms and amplitude are analyzed by parameter modulation and numerical calculation.Research results and design approaches derived in this thesis are helpful to solve some problems on nonlinear vibration in practical engineering fields.