On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

Given a three-dimensional dynamical system on the interval $t_0<t<+\infty$, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighborhood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue $\lambda=\varepsilon+i\beta$, with $\beta\gg\varepsilon>0$ and a real eigenvalue $\delta<0$ with $|\delta|\gg\varepsilon$. On the arbitrary interval $[t_0,+\infty)$, an approximate solution is sought as a polynomial $P_N(\varepsilon)$ in powers of the small parameter $\varepsilon$ with coefficients from Hölder function spaces. It is proved that there exist $\varepsilon_N$ and $C_N$ depending on the initial data such that, for $0<\varepsilon<\varepsilon_N$, the difference between the exact and approximate solutions does not exceed $C_{N^{\varepsilon^{N+1}}}$.