  Dissipative mechanical systems on the torus with a friction that is
proportional to the velocity are modeled by conformally symplectic maps on the
annulus, which are maps that transport the symplectic form into a multiple of
itself (with a conformal factor smaller than 1). It is important to understand
the structure and the dynamics on the attractors. With the aid of parameters,
and under suitable non-degeneracy conditions, one can obtain that, by adjusting
parameters, there is an attractor that is an invariant torus whose internal
dynamics is conjugate to a rotation [CCdlL13]. By analogy with symplectic
dynamics, there have been some debate in establishing appropriate definitions
for twist and non-twist invariant tori (or systems). The purpose of this paper
is two-fold: (a) to establish proper definitions of twist and non-twist
invariant tori in families of conformally symplectic systems; (b) to derive
algorithms of computation of non-twist invariant tori. The last part of the
paper is devoted to implementations of the algorithms, illustrating the
definitions presented in this paper, and exploring the mechanisms of breakdown
of non-twist tori. For the sake of simplicity we have considered here 2D
systems, i.e. defined in the 2D annulus, but generalization to higher
dimensions is straightforward.
