Studies of the Attractor Property of Error Equations for the Lorenz System

The simultaneous equation from the Lorenz model and its error equation without linear approximation were obtained in this study. Theoretical analysis revealed that the simultaneous equation can be transformed to a special operator equation with a property such that all the orbits obtained from the equation will converge to a finite volume. In addition, the equation demonstrates that the divergence of the flow has a negative value, which implies that the volume of the orbit in phase space approaches zero; therefore, all points will be attracted to a low-level dimensional curve surface. These two properties of the simultaneous equation make the evolution of error for the Lorenz system approach to an attractor. The mean absolute error is constant due to the certainty of probability density distribution of the error attractor; this property can be used to understand the phenomena of a small error in the Lorenz system saturating after a lengthy integration. The location and the number of attractor’s centers are obtained from the stability analysis method, and the results are validated by numerical experiments. The figure of error attractor shows that it differs from the solution’s attractor in location, number, and structure. Moreover, the authors demonstrated a method of extending this error analysis for the Lorenz equation to general ordinary differential equations (ODEs). The property of error equations for ODEs has been obtained, and the relationships between the locations of stable points and stability properties for error systems and ordinary systems are reported.