  In this article, we consider two dynamical systems: the McMillan sextupole
and octupole integrable mappings, originally proposed by Edwin McMillan. Both
represent the simplest symmetric McMillan maps, characterized by a single
intrinsic parameter. While these systems find numerous applications across
various domains of mathematics and physics, some of their dynamical properties
remain unexplored. We aim to bridge this gap by providing a comprehensive
description of all stable trajectories, including the parametrization of
invariant curves, Poincar\'e rotation numbers, and canonical action-angle
variables.
  In the second part, we establish connections between these maps and general
chaotic maps in standard form. Our investigation reveals that the McMillan
sextupole and octupole serve as first-order approximations of the dynamics
around the fixed point, akin to the linear map and quadratic invariant (known
as the Courant-Snyder invariant in accelerator physics), which represents
zeroth-order approximations (referred to as linearization). Furthermore, we
propose a novel formalism for nonlinear Twiss parameters, which accounts for
the dependence of rotation number on amplitude. This stands in contrast to
conventional betatron phase advance used in accelerator physics, which remains
independent of amplitude. Notably, in the context of accelerator physics, this
new formalism demonstrates its capability in predicting dynamical aperture
around low-order resonances for flat beams, a critical aspect in beam
injection/extraction scenarios.
