  The gentlest ascent dynamics (GAD) (Nonlinearity, vol. 24, no. 6, p1831,
2011) is a continuous time dynamics coupling both the position and the
direction variables to efficiently locate the saddle point with a given index.
These saddle points play important roles in the activated process of the
randomly perturbed dynamical systems. For index-1 saddle points in non-gradient
systems, the GAD requires two direction variables to approximate the
eigenvectors of the Jacobian matrix and its transpose, respectively, while in
the gradient systems, these two directions collapse to be the single min mode
of the Hessian matrix. In this note, we present a simplified GAD which only
needs one direction variable even for non-gradient systems. This new method not
only reduces computational cost for directions by half, but also can avoid
inconvenient operations on the transpose of Jacobian matrix. We prove the same
convergence property for the simplified GAD as for the original GAD. The
motivation of our simplified GAD is its formal analogy to the Hamiltonian
dynamics governing the exit dynamics when the system is perturbed by small
noise. Several non-gradient examples are presented to demonstrate our method,
including the two dimensional models and the Allen-Cahn equation in the
presence of shear flow.
