The authors consider the general nonlinear stochastic differential equation \[ dx(t)= b(x(t),t, r(t))\,dt+ g(x(t),t, r(t))\,dw(t), \] where \(x\in\mathbb{R}^n\) is the state vector, \(w\in\mathbb{R}^m\) is the standard vector Wiener process, \(b\in\mathbb{R}^n\) and \(g\in\mathbb{R}^{n\times m}\) are nonlinear vector and matrix functions corresponding to stochastic Liénard-type equation, respectively; \(r(t)\), \(t\geq 0\) is a right-continuous Markov chain taking values in a finite state space \(S=\{1,\dots, N\}\) with generator \(\Gamma= [-\gamma_{ij}]\) given by \[ P\{r(t+ \delta)= j|r(t)= i\}= \begin{cases} \gamma_{ij}\delta+ o(\delta),\quad &\text{if } i\neq j,\\ 1+\gamma_{ii}\delta+ o(\delta),\quad &\text{if }i\neq j,\end{cases} \] where \(\delta> 0\), \(\gamma_{ij}\geq 0\) is the transition rate from \(i\) to \(j\) if \(i\neq j\), \(\gamma_{ii}=-\sum_{i\neq j}\gamma_{ij}\). It is assumed that the Markov chain is irreducible and independent of \(w(t)\).
Using Lyapunov technique explicit conditions for asymptotic stochastic stability are derived and compared with earlier results obtained by Mao and his coworkers. Five examples illustrate the obtained results.