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On completeness of quadratic systems. (English) Zbl 1238.37005
A non-autonomous differential equation \(y'=f(t,y)\), \(y\in{\mathbb R}^k\), is called complete if every solution exists for all \(t\in{\mathbb R}\). This paper studies the completeness of non-autonomous quadratic systems of the form \(y'=f_0(t)+f_1(t)y+f_2(t,y)\), \(y\in{\mathbb R}^k\), where \(f_0(t)\) is a column vector, \(f_1(t)\) is a \(k\times k\) matrix and \(f_2(t,y)_n=(y^{t} f_{2,n}(t) y)\), with each \(f_{2,n}(t)\) a low triangular matrix, being all the involved functions of \(t\) continuous and bounded.
The main result of this work is that when \(y^tf_2(t,y)\equiv0\) the differential equation is complete. Two different proofs are provided, one using the Gronwall lemma and another one using a suitable compactification. In particular, the well-know autonomous 3-dimensional Lorenz system belongs to this class of complete quadratic systems, for all values of its parameters.
It can be seen that the dimension of the space of all the quadratic differential equations is \(K=k(k+1)(k+2)/2\) and as a consequence of the above result it holds that the dimension of the complete ones is at least \(2K/3\). The authors also study some sufficient conditions for a differential equation to be incomplete.

MSC:
37C10 Dynamics induced by flows and semiflows
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
37C60 Nonautonomous smooth dynamical systems
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