On the polystability of dynamical systems (Q1804772)

A partitioned system \(dx_ i/dt = f_ i(t,x_ 1, \dots, x_ s)\) with \(f_ i (t,x_ 1, \dots, x_ s) = 0\) for all \(t\) if and only if \(x_ 1 = \cdots = x_ n = 0\), \(x_ i \in \mathbb{R}^{n_ i}\), \(i = 1, \dots, s\), \(n_ 1 + \cdots + n_ s = n\), is said to be polystable if its zero solution \(x = 0 \in \mathbb{R}^ n\) is stable and attractive with respect to certain groups of variables \((x_ 1, \dots, x_ \ell)\), \(\ell \leq s\). The author gives sufficient conditions for polystability in terms of a Lyapunov function in the particular case \(s = 2\).