  In a dynamical systems description of spatiotemporally chaotic PDEs including
those describing turbulence, chaos is viewed as a trajectory evolving within a
network of non-chaotic, dynamically unstable, time-invariant solutions embedded
in the chaotic attractor of the system. While equilibria, periodic orbits and
invariant tori can be constructed using existing methods, computations of
heteroclinic and homoclinic connections mediating the evolution between the
former invariant solutions remain challenging. We propose a robust matrix-free
variational method for computing connecting orbits between equilibrium
solutions of a dynamical system that can be applied to high-dimensional
problems. Instead of a common shooting-based approach, we define a minimization
problem in the space of smooth state space curves that connect the two
equilibria with a cost function measuring the deviation of a connecting curve
from an integral curve of the vector field. Minimization deforms a trial curve
until, at a global minimum, a connecting orbit is obtained. The method is
robust, has no limitation on the dimension of the unstable manifold at the
origin equilibrium, and does not suffer from exponential error amplification
associated with time-marching a chaotic system. Owing to adjoint-based
minimization techniques, no Jacobian matrices need to be constructed and the
memory requirement scales linearly with the size of the problem. The robustness
of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky
equation.
