The Lorenz chaotic systems as nonlinear oscillators with memory

摘要

Nonlinear dynamical systems (systems of 1st order ordinary differential equations) capable of generating chaos are analytically nonintegrable. Despite of this fact, analytical tools can be used to extract useful information. In this paper the original Lorenz system and its modifications are reduced to single oscillatory type integral-differential equations with delayed argument. This yields to appearance of an "endogenous" term interpreted as memory for the past. Moreover, the equations are valid far from the initial instant (theoretically at t→∞), when the system eventually evolves on its attractor set. This corresponds to the numerical solutions when an appropriate initial part of the iterates is usually discarded to eliminate the transients. Besides, the form of the equations allows statistical treatment.