Simplified gentlest ascent dynamics for saddle points in non-gradient systems

The gentlest ascent dynamics (GAD) [W. E and X. Zhou, Nonlinearity 24, 1831 (2011)] is a time continuous dynamics to efficiently locate saddle points with a given index by coupling the position and direction variables together. These saddle points play important roles in the activated process of randomly perturbed dynamical systems. For index-1 saddle points in non-gradient systems, the GAD requires two direction variables to approximate, respectively, the eigenvectors of the Jacobian matrix and its transposed matrix. In the particular case of gradient systems, the two direction variables are equal to the single minimum 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 the computational cost for the direction variable by half but also avoids inconvenient transpose operation of the Jacobian matrix. We prove the same convergence property for the simplified GAD as that for the original GAD. The motivation of our simplified GAD is the formal analogy with Hamilton's equation governing the noise-induced exit dynamics. Several non-gradient examples are presented to demonstrate our method, including a two dimensional model and the Allen-Cahn equation in the presence of shear flow.