Stability of solutions in different variables (Q843639)

The problem of stability in various variables has been known since the nineteenth centure. A. M. Lyapunov noted that the stability of a solution of a system in some variables does not imply the stability of the same solution in other variables in general. In this paper the following problem is considered. Let \(x_p(t)\) and \(y_p(t)=\Phi(x_p(t))\) be the solutions of the systems \[ dx/dt=f(x,t),\quad x\in\Omega\subset \mathbb{R}^n,\tag{1} \] \[ dy/dt=h(y,t),\quad y\in\Phi(\Omega)\subset \mathbb{R}^n,\quad y=\Phi(x),\tag{2} \] where \(\Omega\) is an open set and \(y=\Phi(x)\) is a smooth nondegenerate change of variables reducing system (1) to system (2). The author derives additional conditions which imply that the asymptotic (respectively exponential) stability of the solution \(y_p(t)\) implies the asymptotic (respectively exponential) stability of the solution \(x_p(t)\).