  In this paper, the gentlest ascent dynamics (GAD) developed in [W. E and X.
Zhou, Nonlinearity, 24 (2011), pp. 1831--1842] is extended to a constrained
gentlest ascent dynamics (CGAD) to find constrained saddle points with any
specified Morse indices. It is proved that the linearly stable steady state of
the proposed CGAD is exactly a nondegenerate constrained saddle point with a
corresponding Morse index. Meanwhile, the locally exponential convergence of an
idealized CGAD near nondegenerate constrained saddle points with corresponding
indices is also verified. The CGAD is then applied to find excited states of
single-component Bose--Einstein condensates (BECs) in the order of their Morse
indices via computing constrained saddle points of the corresponding
Gross--Pitaevskii energy functional under the normalization constraint. In
addition, properties of the excited states of BECs in the linear/nonlinear
cases are mathematically/numerically studied. Extensive numerical results are
reported to show the effectiveness and robustness of our method and demonstrate
some interesting physics.
