  The construction of effective and informative landscapes for stochastic
dynamical systems has proven a long-standing and complex problem. In many
situations, the dynamics may be described by a Langevin equation while
constructing a landscape comes down to obtaining the quasi-potential, a scalar
function that quantifies the likelihood of reaching each point in the
state-space. In this work we provide a novel method for constructing such
landscapes by extending a tool from control theory: the Sum-of-Squares method
for generating Lyapunov functions. Applicable to any system described by
polynomials, this method provides an analytical polynomial expression for the
potential landscape, in which the coefficients of the polynomial are obtained
via a convex optimization problem. The resulting landscapes are based upon a
decomposition of the deterministic dynamics of the original system, formed in
terms of the gradient of the potential and a remaining "curl" component. By
satisfying the condition that the inner product of the gradient of the
potential and the remaining dynamics is everywhere negative, our derived
landscapes provide both upper and lower bounds on the true quasi-potential;
these bounds becoming tight if the decomposition is orthogonal. The method is
demonstrated to correctly compute the quasi-potential for high-dimensional
linear systems and also for a number of nonlinear examples.
