Criterion for orbital stability of trajectories of dynamic systems (Q1902749)

The authors introduce the notion of Zhukovskij stability which -- as they point out -- in certain situations where complicated dynamics occur is more appropriate for stability investigations than the well-known notion of orbital (Poincaré) stability. A solution \(x= x(t, x_0)\) of a system \(dx/dt= f(x)\) is said to be stable in the sense of Zhukovskij if for any \(\varepsilon> 0\) there exists a \(\delta= \delta(\varepsilon, t_0)> 0\) such that for any \(y_0\) with \(|y_0- x_0|< \delta\) one can find two parametrizations \(\tau_1, \tau_2: [t_0, +\infty)\to [t_0, +\infty)\), \(\tau_i(t_0)= t_0\), such that \(|x(\tau_1(t), x_0)- x(\tau_2(t), y_0)|< \varepsilon\) holds for all \(t\geq t_0\). The authors present sufficient conditions for Zhukovskij stability and Zhukovskij instability in terms of the stability properties of the trivial solution of the system \(dz/dt= A(z(t, x_0))z\), where \[ A(x)= {\partial f\over \partial x} (x)- 2|f(x)|^{- 2} f(x)f^T(x) J(x),\;J(x)= {1\over 2} \Biggl[{\partial f\over \partial x} (x)+ \Biggl({\partial f\over \partial x} (x)\Biggr)^T\Biggr], \] and in terms of the eigenvalues of the matrix \(J(x)\).