  Chirping Alfv\'{e}n modes are considered as potentially harmful in burning
Tokamak plasmas. In this paper, the nonlinear evolution of a
single-toroidal-number chirping mode is analysed by numerical particle
simulation. This analysis can be simplified if the different resonant
phase-space structures can be investigated as isolated ones. This can be done
adopting a coordinate system that includes two constants of motion. In our
simulations, we adopt as constants of motion, the magnetic momentum and the
initial particle coordinates. For each resonant structure, a density-flattening
region is formed around the respective resonance radius, with radial width that
increases as the mode amplitude grows. It is delimited by two large negative
density gradients, drifting inward and outward. With constant mode frequency,
this density flattening would be responsible for the exhausting of the drive
when large negative density gradients leave the resonance region. The frequency
chirping, however, causes the resonance radius and the resonance region to
drift inward. This drift delays the moment in which the inner density gradient
reaches the inner boundary of the resonance region. On the other side, the
island reconstitutes around the new resonance radius; as a consequence, the
large negative density gradient further moves inward. This process continues as
long as it allows to keep the large gradient within the resonance region. When
this is no longer possible, the resonant structure ceases to be effective in
driving the mode. To further grow, the mode has to tap a different resonant
structure, possibly making use of additional frequency variations.
