Classification and characteristics of abrupt change based on the Lorenz equation

Abstract

In this paper, preliminary theoretical research on abrupt change induced by the forcing term in a dynamical system is described. Taking the Lorenz equationtrajectoryasthe research object, the trajectory response to different pulse forcing terms is studied based on the stability theorem of differential equations and numerical methods. From the perspective of a dynamical system, abrupt changecan be classified as internal or external. The former reflectstrajectory self-adjustment inside the attractor, whereasthe latter represents the bizarre behaviorof the trajectoryin its deviation from the attractor. This classification helps in understanding the physical mechanisms of different manifestations of atmospheric abrupt change. For different intensities and durations of the pulse forcing term,which are simplified to the magnitude and width of a rectangular wave, respectively, the corresponding abrupt change is analyzed quantitatively. It is established that the larger the amplitude of the pulse forcing term, the greater the deviation of thetrajectory from the attractor and the more violent theabrupt change. Moreover, the greater the width of the pulse forcing term, the longer the duration over which the trajectory deviates from the attractor. Finally, two simple but meaningful linear relationships are obtained: one between the amplitude of the pulse forcing term and the distance of trajectory deviation from the attractor, and the other between the width of the pulse forcing term and the duration over which the trajectory dwells outside of the attractor. These relationships indicate that nonlinear systems have some linear properties.