On transition processes for the Van der Pol equation

For the classical Van der Pol equation at $-\infty<t<+\infty$, solutions describing the transition from the state of unstable equilibrium to the stable limit cycle are studied. Formal series in powers of a small parameter are constructed. It is shown that the coefficients of the series are periodic functions of a relatively fast independent variable, and an exact description of the coefficient dependence on the slow independent variable is given. It is proved that, for sufficiently small values of the parameter, the exact solution exists in the same functional class as the terms of the formal series, beginning with the second term, and that the formal series are asymptotic with respect to the small parameter for the exact solution.