Analysis of the stability of nonlinear dynamical systems (Q762026)

For the equation (1) \(x'=A(x)x\), \(x\in R^ n\), the asymptotic estimation of the solutions and the range of asymptotic stability are investigated in terms of the eigenvalues of the matrix A(x). The main result is the following: If there are an open region \(D\subset R^ n\) and a \(C^ 1\)- continuous function \(\phi\) : \(R^ n\to R\) such that \(0\in \phi^{-1}((- \infty,0])\subset D,\) \(\phi^{-1}((\)-\(\infty,0])\) is positively invariant for (1) and \(\eta (x)<0\) on D, then the solution x(t) of (1) satisfies \(\| x(t)\| \leq \| x(0)\| e^{\eta t}\) for \(t\geq 0\) if \(x(0)\in \phi^{-1}((-\infty,0])\), where \(\eta\) (x) is a function determined by the eigenvalues of A(x) and \(\eta =\sup_{x\in \phi}- 1_{((-\infty,0])}\eta (x).\)