  Successive phase transitions in a rare-earth monoaxial chiral magnet
DyNi$_3$Ga$_{9}$ have been investigated by resonant x-ray diffraction. Magnetic
dipole and electric quadrupole degrees of freedom arising from the large
angular moment of $J=15/2$, in combination with the symmetric and antisymmetric
exchange interactions and the crystal field anisotropy, give rise to competing
ordered phases. We show that the antiferromagnetically coupled Dy moments in
the $ab$-plane form an incommensurate helimagnetic order with $q\sim(0, 0,
0.43)$ just below $T_{\text{N}}=10$ K, which further exhibits successive
first-order transitions to the commensurate helimagnetic order with
$q=(0,0,0.5)$ at $T_{\text{N}}^{\;\prime}=9.0$ K, and to the canted
antiferromagnetic order with $q=(0,0,0)$ at $T_{\text{N}}^{\;\prime\prime}=8.5$
K, both with large coexistence regions. The relation of the magnetic helicity
and the crystal chirality in DyNi$_3$Ga$_{9}$ is also uniquely determined.
Splitting of the $(6,0,0)$ Bragg peak is observed below
$T_{\text{N}}^{\;\prime\prime}$, reflecting the lattice distortion due to the
ferroquadrupole order. In the canted antiferromagnetic phase, a spin-flop
transition takes place at 5 K when the temperature is swept in a weak magnetic
field. We discuss these transitions from the viewpoint of competing energies
described above.
