On completeness of quadratic systems

A dynamical system is called complete if every solution of it exists for all t∈Rt∈R. Let KK be the dimension of the vector space of quadratic systems. The set of complete quadratic systems is shown to contain a vector subspace of dimension 2K/32K/3. We provide two proofs, one by the Gronwall lemma and the second by compactification that is capable of showing incompleteness as well. Characterization of a vector subspace of complete quadratic systems is provided. The celebrated Lorenz system for all real ranges of its parameters is shown to belong to this subspace. We also provide a sufficient condition for a system to be incomplete.