  In this work, retrograde mean motion resonances (MMRs) are investigated by
means of analytical and numerical approaches. Initially, we define a new
resonant angle to describe the retrograde MMRs and then perform a series of
canonical transformations to formulate the resonant model, in which the phase
portrait, resonant centre and resonant width can be analytically determined. To
validate the analytical developments, the non-perturbative analysis is made by
taking advantage of Poincar\'e surfaces of section. Some modifications are
introduced in the production of Poincar\'e sections and, in particular, it
becomes possible to make direct comparisons between the analytical and
numerical results. It is found that there exists an excellent correspondence
between the phase portraits and the associated Poincar\'e sections, and the
analytical results agree well with the numerical results in terms of the
resonant width and the location of resonant centre. Finally, the numerical
approach is utilized to determine the resonant widths and resonant centres over
the full range of eccentricity. In particular, seven known examples of
retrograde asteroids including 2015 BZ509, 2008 SO218, 1999 LE31, 2000 DG8,
2014 AT28, 2016 LS and 2016 JK24 are found inside the libration zones of
retrograde MMRs with Jupiter. The results obtained in this work may be helpful
for understanding the dynamical evolution for asteroids inside retrograde MMRs.
