Investigation Of Chaotic Behavior Of One-dimensional Linear Wave Equation Under Nonlinear Boundary Control

Since the middle of last century, nonlinear vibration caused great interest for theacademia in mechanical and electrical, to form an important research area of power system. atpresent, the chaotic behavior of dynamical systems exported by the ordinary differentialequation has been studied in great depth, and access to a wealth of results. Has been observedthat the dynamical system determined by the partial differeniatl equation has a morelycomplex dynamic behavior ,but because of the complexity of the system,making for export bythe partial differential equations dynamical systems chaotic oscillation.However, the mathematical study of chaotic oscillations of the mechanical systemcontrolled by the partial differential eqtations does not have much. In general, the study ofpartial differential equations requires deeper mathematical knowledge and skills. Appear asnonlinear, like that the solution existence and uniqueness of such a fundamental problem hasbeen difficult to solve, not to mention the chaotic behavior, this paper studies partialdifferential and representative one end with a linear boundary value condition while the otherend of the nonlinear boundary conditions of the chaotic oscillation of the vibrating stringproblem, chaos with One-dimensional wave equation behabiors. And improved the existingliterature results, Full text is divided into four chapters.As the introduction ,In chapter One, the background and development of the partialdifferential chaotic behavior. Briefly describes the main work of this paper.In Chapter Two, some fundamental knowledge, including the equivalent transformationof the one-dimensional wave equation under nonlinear boundary bifurcation control, composi-te mapping image and the nature of the chaos of the definition.In Chapter Three ,Prove with non 2-power cycle interval mapping iterative with theincrease in the number of iterations grow indefinitely, the main tool and a foothold Stefanring , this conclusion in this article to improve for the exponential growth and use theconclusion to prove the chaotic behavior of the composite map.Finally, in Chapter Four, Composite mapping on the period-doubling bifurcation theoremto be perfect,Section I,period doubling bifurcation be introduced ; section II the perod-doub-ling bifurcation theorem for Composite mapping GoFon I to be perfect introduced in SectionI ; section III the perod-Doubling bifurcation theorem for Composite mapping GoFon I to beperfect.