The swinging Atwood's machine is a conservative mechanical system with two degrees of freedom and its equations of motion may be easily written in the form \begin{equation} r \ddot{\varphi}=-\sin \varphi-2 \dot{r} \dot{\varphi},\ \ \ \ (2+\varepsilon) \ddot{r}=-\varepsilon -(1-\cos \varphi)+r \dot{\varphi}^2. \label{eq:1} \end{equation} Here the variables $r, \varphi$ describe geometrical configuration of the system, and parameter $\varepsilon=(m_2-m_1)/m_1$ determines a difference of two masses $m_1\leq m_2$ attached to opposite ends of a massless inextensible thread. Note that differential equations (\ref{eq:1}) are essentially nonlinear and their solution cannot be written in symbolic form, in general. Numerical analysis shows that the system may demonstrate different kinds of motion. In the absence of oscillations $(\varphi=0)$ when only one degree of freedom is excited the system demonstrates a uniformly accelerated motion of two masses. An equilibrium state $\varphi=0, r=const$ may exist only in the case of equal masses $(\varepsilon = 0)$ but it is unstable and even very small perturbations enforce the system to leave it. However, for sufficiently small difference of masses $(0<\varepsilon \ll 1)$ one can choose such initial conditions that motion of the system is periodic and both masses oscillate near some equilibrium positions. These oscillations are nonlinear and their amplitudes and frequencies are completely determined by the value of parameter $\varepsilon$. As for $\varepsilon > 0$ the system has no equilibrium state, the periodic motion may be considered as a state of its dynamic equilibrium when owing to oscillations the smaller mass $m_1$ counterweights the larger mass $m_2$. Such behaviour of the system is very interesting and is not possible in the absence of oscillations. In the present talk, we show that small perturbations of the initial conditions determining the periodic solution to equations (\ref{eq:1}) lead only to small oscillations of the system near equilibrium. It gives an example of mechanical system, where the state of dynamical equilibrium is stabilized by means of oscillations.