It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NP-hard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NP-hardness result, we obtain a Lyapunov-inspired technique for proving positivity of forms. We also show that for asymptotically stable homogeneous cubic vector fields in as few as two variables, the minimum degree of a polynomial Lyapunov function can be arbitrarily large. Finally, we show that there is no monotonicity in the degree of polynomial Lyapunov functions that prove AS; i.e., a homogeneous cubic vector field with no homogeneous polynomial Lyapunov function of some degree $d$ can very well have a homogeneous polynomial Lyapunov function of degree less than $d$.
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