Invariant surfaces of periodic systems with conservative cubic first approximation

Abstract

Two classes of time-periodic systems of ordinary differential equations with a small parameter e ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems $${{\dot {x}}_{i}}$$ = $$ - {{\gamma }_{i}}{{y}_{i}}{{\varepsilon }^{\nu }}$$ , $${{\dot {y}}_{i}}$$ = γi( $$x_{i}^{3}$$ – $${{\eta }_{i}}{{x}_{i}}$$ )eν (i = $$\overline {1,n} $$ , ν = 0, 1) have 1 to 3n singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.