A note on LaSalle's problems (Q2724117)

This paper deals with the question, when is the origin a global attractor of a discrete dynamical system defined on \(\mathbb{R}^n\). Indeed let \(T:\mathbb{R}^n\to \mathbb{R}^n\) be a \(C^1\) map and consider the dynamics of iterations of \(T\): NEWLINE\[NEWLINEx_{k+1}= T(x_k). \tag{1}NEWLINE\]NEWLINE Assume that 0 is a fixed point of \(T\). In his book ``The stability of dynamical systems'' (SIAM 1976; Zbl 0364.93002), \textit{J. P. LaSalle} gives sufficient conditions to have a global attractor for \(n=1\). In this note the authors present four conditions and show that just one of these implies that the origin is a global attractor in \(\mathbb{R}^n\) for polynomial maps. This condition reads as follows: \(|\lambda |<1\) for each \(\lambda\in \sigma(|T'(x)|)\) and for all \(x\in \mathbb{R}^n\), where \(T'(x)\) is the Jacobian matrix of \(T\) at \(x\in \mathbb{R}^n\), \(\sigma(T'(x))\) is the spectrum of \(T'(x)\). The authors show that further two of these conditions have a natural extension to NEWLINE\[NEWLINE\dot x= T(x). \tag{2}NEWLINE\]NEWLINE They also show that the last condition does not imply the origin to be a global attractor.