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 < +∞, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighbor-hood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue λ = ɛ + i β, with β ≫ ɛ > 0 and a real eigenvalue with δ < 0 with |δ| ≫ ɛ. On the arbitrary interval [ t 0 , +∞), an approximate solution is sought as a polynomial P N (ɛ) in powers of the small parameter with coefficients from Hölder function spaces. It is proved that there exist ɛ N and C N depending on the initial data such that, for 0 < ɛ < ɛ N , the difference between the exact and approximate solutions does not exceed C N ɛ N +1 .