  The dynamics of many systems from physics, economics, chemistry, and biology
can be modelled through polynomial functions. In this paper, we provide a
computational means to find positively invariant sets of polynomial dynamical
systems by using semidefinite programming to solve sum-of-squares (SOS)
programmes. With the emergence of SOS programmes, it is possible to efficiently
search for Lyapunov functions that guarantee stability of polynomial systems.
Yet, SOS computations often fail to find functions, such that the conditions
hold in the entire state space. We show here that restricting the SOS
optimisation to specific domains enables us to obtain positively invariant
sets, thus facilitating the analysis of the dynamics by considering separately
each positively invariant set. In addition, we go beyond classical Lyapunov
stability analysis and use SOS decompositions to computationally implement
sufficient positivity conditions that guarantee existence, uniqueness, and
exponential stability of a limit cycle. Importantly, this approach is
applicable to systems of any dimension and, thus, goes beyond classical methods
that are restricted to two-dimensional phase space. We illustrate our different
results with applications to classical systems, such as the van der Pol
oscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.
