Stability of mappings with spectrum on a unit circle (Q1910098)

The author considers the mapping (1) \(x^*=Ax+\sum^\infty_{j=2} F^{(j)}(x)\), where \(x\in\mathbb{R}^m\), the matrix \(A\) has \(g\) pairs of non-resonant complex eigenvalues with absolute values of unity and \(F^{(j)}(x)\) are \(m\)-dimensional vector-valued forms of the \(j\)th degree. Sufficient conditions for asymptotic stability and instability of the fixed point \(x=0\) of (1) are obtained. In the main case they are necessary and sufficient conditions.