Classification of chaotic systems by using canonical (jerk) forms: The case of Lorenz-like systems

When a d-dimensional system is investigated through one of its variables v, a natural space for characterizing its dynamics is provided by the v-induced differential embedding spanned by that variable and its (d - 1) successive derivatives. The corresponding governing equations are called the canonical form of that system. Canonical forms-often called jerk equations when, in addition, the system is three-dimensional-can be used for classifying chaotic systems not only by comparing their algebraic structures, as done in previous works, but also by considering their parameter values, helping in identifying system's parameter values for producing equivalent dynamics, that is, for producing a given type of attractor. Two types of equivalence can, therefore, be distinguished: (i) structural, when only the algebraic structures of canonical forms match, and (ii) dynamical, when the canonical parameter values coincide. Despite their potential, canonical forms have been only partially exploited for systematically relating and identifying chaotic systems. In this work, we address this gap by using the x-induced canonical form of the Lorenz system to investigate its relationships with other Lorenz-like systems. In particular, we show how canonical forms can be used to determine parameter values that reproduce a Lorenz attractor, even when the underlying algebraic structures differ. This provides a novel perspective in which the algebraic structure of the governing equations is explicitly leveraged to generate prescribed chaotic attractors, opening new ways for system identification and design.