  We introduce a generalized numerical algorithm to construct the solution
landscape, which is a pathway map consisting of all stationary points and their
connections. Based on the high-index optimization-based shrinking dimer (HiOSD)
method for gradient systems, a generalized high-index saddle dynamics (GHiSD)
is proposed to compute any-index saddles of dynamical systems. Linear stability
of the index-$k$ saddle point can be proved for the GHiSD system. A combination
of the downward search algorithm and the upward search algorithm is applied to
systematically construct the solution landscape, which not only provides a
powerful and efficient way to compute multiple solutions without tuning initial
guesses, but also reveals the relationships between different solutions.
Numerical examples, including a three-dimensional example and the phase field
model, demonstrate the novel concept of the solution landscape by showing the
connected pathway maps.
