Transient quasi period in Lorenz system

Abstract

Although traditionally the study of nonlinear dynamical systems mainly deals with the long-time stable behavior, nowadays it is known that the transient process from initial state to the attractor can demonstrate a lot of interesting phenomena. In this paper, a new transient quasi periodic behavior in the initial stage of the system is found in the classical Lorenz system. When the parameters change very little,the system dynamic behavior changes suddenly and the system state is easy to distinguish. By selecting 10000 groups of data from the front,middle and rear parts of the system time series as the state judgment index of the basin diagram, the influence of the initial point on the transient behavior is analyzed. Through the bifurcation diagram of the system, the dynamic behavior of the system under the influence of system parameters is analyzed in detail. It is found that the amplitude of quasi periodic state and transient quasi periodic state will change with the increase of parameter c, and a conjecture is put forward that the special phenomenon of quasi periodic state is caused by the gradual decrease of the absolute value of the real part of the eigenvalue of system equilibrium points S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> and S <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , the decrease of the trajectory convergence speed of the neighborhood of the equilibrium point and the decrease of the attraction ability of the equilibrium point to the trajectory around the neighborhood.