Comparative time graininess and asymptotic stability of dynamical systems (Q1922432)

The following \(d\)-dimensional dynamical system on a measure chain \((T, \preceq, \mu)\) is considered: \[ x^\Delta(t)= f(x(t)),\quad t\in T, \tag{1} \] for a continuously differentiable mapping \(f: \mathbb{R}^d\to \mathbb{R}^d\), where \(x^\Delta(t)\) is the chain measure derivative of \(x: T\to \mathbb{R}^d\) at \(t\in T\) defined by \[ x^\Delta(t)= \lim_{\begin{smallmatrix} \tau\to t\\ \tau\in T, \sigma(\tau)\neq t\end{smallmatrix}} {x(\sigma(\tau))- x(t)\over \mu(\sigma(\tau), t)}. \] At the same time the authors consider the linearized Euler scheme with equal time step \(h^*> 0\): \[ z_{n+ 1}= z_n+ h^* \nabla f(0) z_n,\quad n= 0, 1,2,\dots\;.\tag{2} \] First of all, the authors show that if the null solution of the linear Euler scheme (2) is asymptotically stable, then the null solution of the linearization about \(x= 0\) of equation (1), namely \[ z^\Delta(t)= Az(t),\quad t\in T, \tag{3} \] is also asymptotically stable for any regular measure chain \((T, \preceq, \mu)\), where \(A= \nabla f(0)\). The main result is to show that if the null solution of the linear Euler scheme (2) is asymptotically stable, then the null solution of the nonlinear differential-difference equation (1) is locally asymptotically stable for any regular measure chain \((T, \preceq, \mu)\) with time graininess \(\mu^*\leq h^*\). The proof uses a Lyapunov function characterizing the asymptotic stability of the discrete-time dynamical system (2).