Chaotic Vibration of the Wave Equation with Nonlinear Feedback Boundary Control: Progress and Open Questions
نویسندگان
چکیده
The study of chaotic phenomena in partial differential equations is a challenging subject. In this paper, we survey the recent progress in the study of chaotic vibration of the linear wave equation with nonlinear boundary feedback control law. We show that when there is linear energy injection at one end of the boundary and the self-regulating or van der Pol nonlinearity at the other end of the boundary, chaos occurs as a reconciliation between linear instability and nonlinear self-regulation when the parameters enter a certain regime. A list of open problems is also posed.
منابع مشابه
Analyzing Displacement Term's Memory Effect in a van der Pol Type Boundary Condition to Prove Chaotic Vibration of the Wave equation
Consider the one-dimensional wave equation on a unit interval, where the left-end boundary condition is linear, pumping energy into the system, while the right-end boundary condition is self-regulating of the van der Pol type with a cubic nonlinearity. Then for a certain parameter range it is now known that chaotic vibration occurs. However, if the right-end van der Pol boundary condition conta...
متن کاملBoundary Feedback Stabilization of a Nonlinear Flexible Gantry Manipulator Using Disturbance Observer
This paper aims to develop a boundary control solution for a single-link gantry robot manipulator with one axis of rotation. The control procedure is considered with link’s transverse vibrations while system undergoes rigid body nonlinear large rotation and translation. Initially, based on Hamilton principle, governing equations of hybrid motions as a set of partial differential equations...
متن کاملNonisotropic Spatiotemporal Chaotic Vibration of the Wave equation due to Mixing Energy Transport and a van der Pol Boundary Condition
The imbalance of the boundary energy flow due to energy injection at one end and a nonlinear van der Pol boundary condition at the other end of the spatial one-dimensional interval can cause chaotic vibration of the linear wave equation [Chen et al., 1998b, 1998c]. However, such chaotic vibration is isotropic with respect to space and time because the two associated families of characteristics ...
متن کاملChaotic Vibrations of the Wave Equation Due to a Van Der Pol Type Nonlinear Boundary Condition
Chaotic dynamics has been hailed as the third great scientific revolution of the 20th century, along with relativity and quantum mechanics [1]. In this paper, we consider the one-dimensional wave equation with the initial-boundary value problem on the unit interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition ...
متن کاملInvestigation and Control of Unstable Chaotic Behavior Using of Chaos Theory in Two Electrical Power Systems: 1-Buck Converter2- Power Transformer
This paper consist of two sections: control and stabilizing approach for chaotic behaviour of converter is introduced in first section of this paper for the removal of harmonic caused by the chaotic behaviour in current converter. For this work, a Time- Delayed Feedback Controller (TDFC) control method for stability chaotic behaviour of buck converter for switching courses in current control mo...
متن کامل