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The dynamics of many systems from physics, economics, chemistry, and biologycan be modelled through polynomial functions. In this paper, we provide acomputational means to find positively invariant sets of polynomial dynamicalsystems by using semidefinite programming to solve sum-of-squares (SOS)programmes. With the emergence of SOS programmes, it is possible to efficientlysearch for Lyapunov functions that guarantee stability of polynomial systems.Yet, SOS computations often fail to find functions, such that the conditionshold in the entire state space. We show here that restricting the SOSoptimisation to specific domains enables us to obtain positively invariantsets, thus facilitating the analysis of the dynamics by considering separatelyeach positively invariant set. In addition, we go beyond classical Lyapunovstability analysis and use SOS decompositions to computationally implementsufficient positivity conditions that guarantee existence, uniqueness, andexponential stability of a limit cycle. Importantly, this approach isapplicable to systems of any dimension and, thus, goes beyond classical methodsthat are restricted to two-dimensional phase space. We illustrate our differentresults with applications to classical systems, such as the van der Poloscillator, the Fitzhugh-Nagumo neuronal equation, and the Lorenz system.

Finding positively invariant sets and proving exponential stability of limit cycles using Sum-of-Squares decompositions