Lorenz系统平衡点分析：新系统与原系统比较

From the perspective of equilibrium points, the Lorenz system's equilibrium points can be obtained by setting dx/dt=0, dy/dt=0, and dz/dt=0. For the initial values of a=10, b=8/3, c=28, and d=0.4, three equilibrium points can be derived, which are S1=(0,0,0), S2=(10.12,10.12,41), and S3=(-10.12,-10.12,41). It can be observed that the equilibrium points of the new system are identical to those of the original Lorenz chaotic system.

For the origin S, the characteristic roots are determined to be λ1=-25.2425, λ2=16.2425, and λ3=-2.5. Since not all the characteristic roots are positive or negative, S is an unstable saddle point. For the other two equilibrium points S2 and S3, the characteristic roots are determined to be λ1=-13.4929 and λ2=0.9965±12.2805i, respectively. Therefore, both non-zero equilibrium points are unstable foci with an exponent of 2 and can form two vortices that lead to chaotic attractors.

Comparing the equilibrium points of the system, it can be seen that the new system is similar to the original Lorenz system, indicating that the new system belongs to the Lorenz system family. However, there are significant differences in the equilibrium points of the original Lorenz chaotic system. Therefore, the improved system has certain advantages over the original Lorenz system.