Papers by Keyword: Phase Portrait

Abstract: The complex dynamics characters of a third-order circuit system with nonlinear negative capacitance are studied. The dynamical equation and the state equation of the system are established. By the phase portraits, the motions of the system are studied under the definite parameters. And by bifurcation diagram, the route from periodic motion to chaos is studied under the presented system parameters. Two feedback methods are used to control the chaos of the circuit system. One is an adaptive method and the other is the feedback of states and parameter adjustment method. The phase plane portraits and bifurcation diagram of the controlled system are obtained. The advantages of the two controlled methods are that the collect of the control signals are simple and can put on any time and the chaotic system can be asymptotically stabilized to equilibriums with small control. The orbits of the system can be controlled by these two methods according to our target.

Abstract: A third-order circuit system with nonlinear negative capacitance is studied. The dynamical equation and state equation of the system are established. By the phase portraits, the motions of the system are studied under the definite parameters. And by bifurcation diagram, the route from periodic motion to chaos is studied under the presented system parameters. Two anticontrol methods is used to control the periodic motion of the system to chaos. The phase plane portraits and bifurcation diagram of the anticontrolled system are obtained. The threshold values of the anticontrol values of the two control method are obtained. The advantages of the two anticontrolled methods are that the collect of the control signals are simple and can put on any time and the periodic system can be asymptotically chaotic with small control. The orbits of the system can be anticontrolled by these two methods according to our target

Abstract: A third-order circuit system with nonlinear negative capacitance is studied. The dynamical equation and state equation of the system are established. By the phase portraits, the motions of the system are studied under the definite parameters. And by bifurcation diagram, the route from periodic motion to chaos is studied under the presented system parameters. Two controllers are constructed to control the chaos of a third-order circuit system with nonlinear negative capacitance. One controller is nonlinear and the other is linear. The phase plane portraits and bifurcation diagram of the controlled system are obtained. The effect of the nonlinear controller is better than the linear one. The threshold values of the control values of the two control method are obtained. The advantages of the two controlled methods are that the collect of the control signals are simple and can put on any time and the chaotic system can be asymptotically stabilized to equilibriums with small control. The orbits of the system can be controlled by these two methods according to our target.

Abstract: This paper presents a method for the vibration of a beam with a breathing crack under harmonic excitation. The infinitely thin crack is characterised by a parameter that takes into account the shape and the depth of the crack. The closed- and open-crack states are both modelled by a modal approach: two sets of equations of motion cast in the modal coordinates of their individual mode shapes. The state change (from closed to open or vice versa) involves the calculation of the modal coordinates associated with the new state from the modal coordinates of the previous state. By imposing the continuity of displacement and velocity the beam at the instant of the state change, the matrix that transforms the modal coordinates from one state to the other is determined and proved to be the Modal Scale Factor matrix. This analytical approach takes advantage of exact nature and mathematical convenience of beam modes and is time-efficient. Forced vibration at various values of crack parameter is determined. It is found that as decreases (crack length increases) the vibration becomes increasingly erratic and finally chaotic.