  This paper develops sparse moment-sum-of-squares approximations for three
problems from nonlinear dynamical systems: region of attraction, maximum
positively invariant set and global attractor. We prove general results
allowing for a decomposition of these sets provided that the vector field and
constraint set posses certain structure. We combine these decompositions with
the methods from [10], [12] and [20] based on infinite-dimensional linear
programming. For polynomial dynamics, we show that these problems admit a
sparse sum-of-squares (SOS) approximation with guaranteed convergence such that
the number of variables in the largest SOS multiplier is given by the dimension
of the largest subsystem appearing in the decomposition. The dimension of such
subsystems depends on the sparse structure of the vector field and the
constraint set and can allow for a significant reduction of the size of the
semidefinite program (SDP) relaxations, thereby allowing to address far larger
problems without compromising convergence guarantees. The method is simple to
use and based on convex optimization. Numerical examples demonstrate the
approach.
