On solutions to two-dimensional systems realizing the transition from an unstable equilibrium to a stable cycle

For a two-dimensional dynamical system at $-\infty<t<+\infty$, a process describing the transition from an arbitrary neighborhood of an unstable equilibrium to a stable limit cycle is studied. The system is reduced to the Poincaré normal form. An approximate solution is constructed as a polynomial of degree $2N$ containing only even degrees of the small parameter $\varepsilon$. The functional classes to which the coefficients of this polynomial belong are described. The function space containing the exact solution differing from the approximate one by $O(\varepsilon^{2N+1})$ is determined.